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<title>Marton’s conjecture in abelian groups with bounded torsion</title>
<link>https://terrytao.wordpress.com/2024/04/04/martons-conjecture-in-abelian-groups-with-bounded-torsion/</link>
<comments>https://terrytao.wordpress.com/2024/04/04/martons-conjecture-in-abelian-groups-with-bounded-torsion/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Thu, 04 Apr 2024 22:05:52 +0000</pubDate>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[paper]]></category>
<category><![CDATA[additive combinatorics]]></category>
<category><![CDATA[Ben Green]]></category>
<category><![CDATA[Freddie Manners]]></category>
<category><![CDATA[Polynomial Freiman-Ruzsa conjecture]]></category>
<category><![CDATA[Shannon entropy]]></category>
<category><![CDATA[Timothy Gowers]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14442</guid>
<description><![CDATA[Tim Gowers, Ben Green, Freddie Manners, and I have just uploaded to the arXiv our paper “Marton’s conjecture in abelian groups with bounded torsion“. This paper fully resolves a conjecture of Katalin Marton (the bounded torsion case of the Polynomial Freiman–Ruzsa conjecture (first proposed by Katalin Marton): Theorem 1 (Marton’s conjecture) Let be an abelian […]]]></description>
<content:encoded><![CDATA[<a href="https://www.dpmms.cam.ac.uk/~wtg10/">Tim Gowers</a>, <a href="https://people.maths.ox.ac.uk/greenbj/">Ben Green</a>, <a href="https://mathweb.ucsd.edu/~fmanners/">Freddie Manners</a>, and I have just uploaded to the arXiv our paper “<a href="https://arxiv.org/abs/2404.02244">Marton’s conjecture in abelian groups with bounded torsion</a>“. This paper fully resolves a conjecture of Katalin Marton (the bounded torsion case of the <a href="https://terrytao.wordpress.com/2007/03/11/ben-green-the-polynomial-freiman-ruzsa-conjecture/">Polynomial Freiman–Ruzsa conjecture</a> (first proposed by Katalin Marton):
<blockquote>Theorem 1 (Marton’s conjecture) <a name="main"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7BG+%3D+%28G%2C%2B%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG+%3D+%28G%2C%2B%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG+%3D+%28G%2C%2B%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G = (G,+)}" class="latex" /> be an abelian <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" />-torsion group (thus, <img src="https://s0.wp.com/latex.php?latex=%7Bmx%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bmx%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bmx%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{mx=0}" class="latex" /> for all <img src="https://s0.wp.com/latex.php?latex=%7Bx+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x \in G}" class="latex" />), and let <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \subset G}" class="latex" /> be such that <img src="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A+A| \leq K|A|}" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> can be covered by at most <img src="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7BO%28m%5E3%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7BO%28m%5E3%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7BO%28m%5E3%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(2K)^{O(m^3)}}" class="latex" /> translates of a subgroup <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> of cardinality at most <img src="https://s0.wp.com/latex.php?latex=%7B%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A|}" class="latex" />. Moreover, <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> is contained in <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell A - \ell A}" class="latex" /> for some <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+m+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+m+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+m+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell \ll (2 + m \log K)^{O(m^3 \log m)}}" class="latex" />.</blockquote>
We had <a href="https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/">previously established</a> the <img src="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=2}" class="latex" /> case of this result, with the number of translates bounded by <img src="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(2K)^{12}}" class="latex" /> (which was subsequently improved to <img src="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B11%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B11%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B11%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(2K)^{11}}" class="latex" /> by Jyun-Jie Liao), but without the additional containment <img src="https://s0.wp.com/latex.php?latex=%7BH+%5Csubset+%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH+%5Csubset+%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH+%5Csubset+%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H \subset \ell A - \ell A}" class="latex" />. It remains a challenge to replace <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell}" class="latex" /> by a bounded constant (such as <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" />); this is essentially the “polynomial Bogolyubov conjecture”, which is still open. The <img src="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=2}" class="latex" /> result has been formalized in the proof assistant language Lean, as discussed in <a href="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/">this previous blog post</a>. As a consequence of this result, many of the applications of the previous theorem may now be extended from characteristic <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" /> to higher characteristic.
Our proof techniques are a modification of those in our previous paper, and in particular continue to be based on the theory of Shannon entropy. For inductive purposes, it turns out to be convenient to work with the following version of the conjecture (which, up to <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" />-dependent constants, is actually equivalent to the above theorem):
<blockquote>Theorem 2 (Marton’s conjecture, entropy form) <a name="main-entropy"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> be an abelian <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" />-torsion group, and let <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1,\dots,X_m}" class="latex" /> be independent finitely supported random variables on <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, such that
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+%5Cleq+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+%5Cleq+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+%5Cleq+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}[X_1+\dots+X_m] - \frac{1}{m} \sum_{i=1}^m {\bf H}[X_i] \leq \log K," class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+H%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+H%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+H%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf H}}" class="latex" /> denotes Shannon entropy. Then there is a uniform random variable <img src="https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U_H}" class="latex" /> on a subgroup <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> such that
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+d%5BX_i%3B+U_H%5D+%5Cll+m%5E3+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+d%5BX_i%3B+U_H%5D+%5Cll+m%5E3+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+d%5BX_i%3B+U_H%5D+%5Cll+m%5E3+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{1}{m} \sum_{i=1}^m d[X_i; U_H] \ll m^3 \log K," class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d}" class="latex" /> denotes the entropic Ruzsa distance (see <a href="https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/">previous blog post</a> for a definition); furthermore, if all the <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> take values in some symmetric set <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" />, then <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> lies in <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell+S%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell+S%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell+S%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell S}" class="latex" /> for some <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell \ll (2 + \log K)^{O(m^3 \log m)}}" class="latex" />.</blockquote>
As a first approximation, one should think of all the <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> as identically distributed, and having the uniform distribution on <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" />, as this is the case that is actually relevant for implying Theorem <a href="#main">1</a>; however, the recursive nature of the proof of Theorem <a href="#main-entropy">2</a> requires one to manipulate the <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> separately. It also is technically convenient to work with <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" /> independent variables, rather than just a pair of variables as we did in the <img src="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=2}" class="latex" /> case; this is perhaps the biggest additional technical complication needed to handle higher characteristics.
The strategy, as with the previous paper, is to attempt an entropy decrement argument: to try to locate modifications <img src="https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X'_1,\dots,X'_m}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1,\dots,X_m}" class="latex" /> that are reasonably close (in Ruzsa distance) to the original random variables, while decrementing the “multidistance”
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}[X_1+\dots+X_m] - \frac{1}{m} \sum_{i=1}^m {\bf H}[X_i] " class="latex" />
which turns out to be a convenient metric for progress (for instance, this quantity is non-negative, and vanishes if and only if the <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> are all translates of a uniform random variable <img src="https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U_H}" class="latex" /> on a subgroup <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" />). In the previous paper we modified the corresponding functional to minimize by some additional terms in order to improve the exponent <img src="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{12}" class="latex" />, but as we are not attempting to completely optimize the constants, we did not do so in the current paper (and as such, our arguments here give a slightly different way of establishing the <img src="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=2}" class="latex" /> case, albeit with somewhat worse exponents).
As before, we search for such improved random variables <img src="https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X'_1,\dots,X'_m}" class="latex" /> by introducing more independent random variables – we end up taking an array of <img src="https://s0.wp.com/latex.php?latex=%7Bm%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m^2}" class="latex" /> random variables <img src="https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y_{i,j}}" class="latex" /> for <img src="https://s0.wp.com/latex.php?latex=%7Bi%2Cj%3D1%2C%5Cdots%2Cm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%2Cj%3D1%2C%5Cdots%2Cm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%2Cj%3D1%2C%5Cdots%2Cm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i,j=1,\dots,m}" class="latex" />, with each <img src="https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y_{i,j}}" class="latex" /> a copy of <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" />, and forming various sums of these variables and conditioning them against other sums. Thanks to the magic of Shannon entropy inequalities, it turns out that it is guaranteed that at least one of these modifications will decrease the multidistance, except in an “endgame” situation in which certain random variables are nearly (conditionally) independent of each other, in the sense that certain conditional mutual informations are small. In particular, in the endgame scenario, the row sums <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_j+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_j+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_j+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_j Y_{i,j}}" class="latex" /> of our array will end up being close to independent of the column sums <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_i+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_i+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_i+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_i Y_{i,j}}" class="latex" />, subject to conditioning on the total sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{i,j} Y_{i,j}}" class="latex" />. Not coincidentally, this type of conditional independence phenomenon also shows up when considering row and column sums of iid independent gaussian random variables, as a specific feature of the gaussian distribution. It is related to the more familiar observation that if <img src="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X,Y}" class="latex" /> are two independent copies of a Gaussian random variable, then <img src="https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X+Y}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BX-Y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX-Y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX-Y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X-Y}" class="latex" /> are also independent of each other.
Up until now, the argument does not use the <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" />-torsion hypothesis, nor the fact that we work with an <img src="https://s0.wp.com/latex.php?latex=%7Bm+%5Ctimes+m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm+%5Ctimes+m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm+%5Ctimes+m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m \times m}" class="latex" /> array of random variables as opposed to some other shape of array. But now the torsion enters in a key role, via the obvious identity
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%2Cj%7D+i+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+j+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+%28-i-j%29+Y_%7Bi%2Cj%7D+%3D+0.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%2Cj%7D+i+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+j+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+%28-i-j%29+Y_%7Bi%2Cj%7D+%3D+0.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%2Cj%7D+i+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+j+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+%28-i-j%29+Y_%7Bi%2Cj%7D+%3D+0.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{i,j} i Y_{i,j} + \sum_{i,j} j Y_{i,j} + \sum_{i,j} (-i-j) Y_{i,j} = 0." class="latex" />
In the endgame, the any pair of these three random variables are close to independent (after conditioning on the total sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{i,j} Y_{i,j}}" class="latex" />). Applying some “entropic Ruzsa calculus” (and in particular an entropic version of the Balog–Szeméredi–Gowers inequality), one can then arrive at a new random variable <img src="https://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U}" class="latex" /> of small entropic doubling that is reasonably close to all of the <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> in Ruzsa distance, which provides the final way to reduce the multidistance.
Besides the polynomial Bogolyubov conjecture mentioned above (which we do not know how to address by entropy methods), the other natural question is to try to develop a characteristic zero version of this theory in order to establish the polynomial Freiman–Ruzsa conjecture over torsion-free groups, which in our language asserts (roughly speaking) that random variables of small entropic doubling are close (in Ruzsa distance) to a discrete Gaussian random variable, with good bounds. The above machinery is consistent with this conjecture, in that it produces lots of independent variables related to the original variable, various linear combinations of which obey the same sort of entropy estimates that gaussian random variables would exhibit, but what we are missing is a way to get back from these entropy estimates to an assertion that the random variables really are close to Gaussian in some sense. In continuous settings, Gaussians are known to extremize the entropy for a given variance, and of course we have the central limit theorem that shows that averages of random variables typically converge to a Gaussian, but it is not clear how to adapt these phenomena to the discrete Gaussian setting (without the circular reasoning of assuming the polynoimal Freiman–Ruzsa conjecture to begin with).
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<slash:comments>14</slash:comments>
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<media:title type="html">Terry</media:title>
</media:content>
</item>
<item>
<title>AI Mathematical Olympiad – Progress Prize Competition now open</title>
<link>https://terrytao.wordpress.com/2024/04/02/ai-mathematical-olympiad-progress-prize-competition-now-open/</link>
<comments>https://terrytao.wordpress.com/2024/04/02/ai-mathematical-olympiad-progress-prize-competition-now-open/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Tue, 02 Apr 2024 23:34:55 +0000</pubDate>
<category><![CDATA[advertising]]></category>
<category><![CDATA[math.GM]]></category>
<category><![CDATA[Artificial Intelligence]]></category>
<category><![CDATA[international mathematical olympiad]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14434</guid>
<description><![CDATA[The first progress prize competition for the AI Mathematical Olympiad has now launched. (Disclosure: I am on the advisory committee for the prize.) This is a competition in which contestants submit an AI model which, after the submissions deadline on June 27, will be tested (on a fixed computational resource, without internet access) on a […]]]></description>
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The first progress prize competition for the <a href="https://aimoprize.com/">AI Mathematical Olympiad</a> has <a href="https://www.kaggle.com/competitions/ai-mathematical-olympiad-prize/overview">now launched</a>. (Disclosure: I am on the <a href="https://aimoprize.com/updates/2024-02-07-advisory-committee-announced">advisory committee</a> for the prize.) This is a competition in which contestants submit an AI model which, after the submissions deadline on June 27, will be tested (on a fixed computational resource, without internet access) on a set of 50 “private” test math problems, each of which has an answer as an integer between 0 and 999. Prior to the close of submission, the models can be tested on 50 “public” test math problems (where the results of the model are public, but not the problems themselves), as well as 10 training problems that are available to all contestants. As of this time of writing, the <a href="https://www.kaggle.com/competitions/ai-mathematical-olympiad-prize/leaderboard">leaderboard </a>shows that the best-performing model has solved 4 out of 50 of the questions (a standard benchmark, Gemma 7B, had previously solved 3 out of 50). A total of $<img src="https://s0.wp.com/latex.php?latex=2%5E%7B20%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=2%5E%7B20%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=2%5E%7B20%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="2^{20}" class="latex" /> ($1.048 million) has been allocated for various prizes associated to this competition. More detailed rules can be found <a href="https://www.kaggle.com/competitions/ai-mathematical-olympiad-prize/rules">here</a>.


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<slash:comments>29</slash:comments>
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<media:title type="html">Terry</media:title>
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</item>
<item>
<title>Talks at the JMM</title>
<link>https://terrytao.wordpress.com/2024/03/17/talks-at-the-jmm/</link>
<comments>https://terrytao.wordpress.com/2024/03/17/talks-at-the-jmm/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Mon, 18 Mar 2024 05:07:28 +0000</pubDate>
<category><![CDATA[advertising]]></category>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[math.HO]]></category>
<category><![CDATA[math.NT]]></category>
<category><![CDATA[talk]]></category>
<category><![CDATA[Joint mathematics meetings]]></category>
<category><![CDATA[machine assisted proof]]></category>
<category><![CDATA[multiplicative functions]]></category>
<category><![CDATA[periodic tiling conjecture]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14419</guid>
<description><![CDATA[Earlier this year, I gave a series of lectures at the Joint Mathematics Meetings at San Francisco. I am uploading here the slides for these talks: I also have written a text version of the first talk, which has been submitted to the Notices of the American Mathematical Society.]]></description>
<content:encoded><![CDATA[
Earlier this year, I gave a series of lectures at the <a href="https://www.jointmathematicsmeetings.org/meetings/national/jmm2024/2300_program.html">Joint Mathematics Meetings at San Francisco</a>. I am uploading here the slides for these talks:
<ul>
<li>“<a href="https://terrytao.files.wordpress.com/2024/03/machine-jan-3.pdf">Machine assisted proof</a>” (<a href="https://www.youtube.com/watch?v=AayZuuDDKP0">Video here</a>)</li>
<li>“<a href="https://terrytao.files.wordpress.com/2024/03/periodic-tiling-jan-4.pdf">Translational tilings of Euclidean space</a>” (<a href="https://www.youtube.com/watch?v=QQBqJZtWIhA">Video here</a>)</li>
<li>“<a href="https://terrytao.files.wordpress.com/2024/03/correlations-jan-5.pdf">Correlations of multiplicative functions</a>” (<a href="https://www.youtube.com/watch?v=t_plilnbAtM">Video here</a>)</li>
</ul>
I also have written a <a href="https://terrytao.files.wordpress.com/2024/03/machine-assisted-proof-notices.pdf">text version of the first talk</a>, which has been submitted to the Notices of the American Mathematical Society.
]]></content:encoded>
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<slash:comments>11</slash:comments>
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<media:title type="html">Terry</media:title>
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<item>
<title>A generalized Cauchy-Schwarz inequality via the Gibbs variational formula</title>
<link>https://terrytao.wordpress.com/2023/12/10/a-generalized-cauchy-schwarz-inequality-via-the-gibbs-variational-formula/</link>
<comments>https://terrytao.wordpress.com/2023/12/10/a-generalized-cauchy-schwarz-inequality-via-the-gibbs-variational-formula/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Mon, 11 Dec 2023 03:23:52 +0000</pubDate>
<category><![CDATA[expository]]></category>
<category><![CDATA[math.CA]]></category>
<category><![CDATA[math.PR]]></category>
<category><![CDATA[Anthony Carbery]]></category>
<category><![CDATA[Cauchy-Schwarz]]></category>
<category><![CDATA[Gibbs variational formula]]></category>
<category><![CDATA[Shannon entropy]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14307</guid>
<description><![CDATA[Let be a non-empty finite set. If is a random variable taking values in , the Shannon entropy of is defined as There is a nice variational formula that lets one compute logs of sums of exponentials in terms of this entropy: Lemma 1 (Gibbs variational formula) Let be a function. Then Proof: Note that […]]]></description>
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Let <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" /> be a non-empty finite set. If <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> is a random variable taking values in <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" />, the Shannon entropy <img src="https://s0.wp.com/latex.php?latex=%7BH%5BX%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%5BX%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%5BX%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H[X]}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> is defined as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+H%5BX%5D+%3D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+H%5BX%5D+%3D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+H%5BX%5D+%3D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle H[X] = -\sum_{s \in S} {\bf P}[X = s] \log {\bf P}[X = s]." class="latex" />
There is a nice variational formula that lets one compute logs of sums of exponentials in terms of this entropy:

<blockquote>Lemma 1 (Gibbs variational formula) <a name="gibbs"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7Bf%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f: S \rightarrow {\bf R}}" class="latex" /> be a function. Then <a name="efs"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+%5Csup_X+%7B%5Cbf+E%7D+f%28X%29+%2B+%7B%5Cbf+H%7D%5BX%5D.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+%5Csup_X+%7B%5Cbf+E%7D+f%28X%29+%2B+%7B%5Cbf+H%7D%5BX%5D.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+%5Csup_X+%7B%5Cbf+E%7D+f%28X%29+%2B+%7B%5Cbf+H%7D%5BX%5D.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \log \sum_{s \in S} \exp(f(s)) = \sup_X {\bf E} f(X) + {\bf H}[X]. \ \ \ \ \ (1)" class="latex" />
</a> </blockquote>


Proof: Note that shifting <img src="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f}" class="latex" /> by a constant affects both sides of <a href="#efs">(1)</a> the same way, so we may normalize <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{s \in S} \exp(f(s)) = 1}" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=%7B%5Cexp%28f%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cexp%28f%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cexp%28f%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\exp(f(s))}" class="latex" /> is now the probability distribution of some random variable <img src="https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y}" class="latex" />, and the inequality can be rewritten as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++0+%3D+%5Csup_X+%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BY+%3D+s%5D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++0+%3D+%5Csup_X+%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BY+%3D+s%5D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++0+%3D+%5Csup_X+%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BY+%3D+s%5D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 0 = \sup_X \sum_{s \in S} {\bf P}[X = s] \log {\bf P}[Y = s] -\sum_{s \in S} {\bf P}[X = s] \log {\bf P}[X = s]." class="latex" />
But this is precisely the <a href="https://en.wikipedia.org/wiki/Gibbs%27_inequality">Gibbs inequality</a>. (The expression inside the supremum can also be written as <img src="https://s0.wp.com/latex.php?latex=%7B-D_%7BKL%7D%28X%7C%7CY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B-D_%7BKL%7D%28X%7C%7CY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B-D_%7BKL%7D%28X%7C%7CY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{-D_{KL}(X||Y)}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7BD_%7BKL%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BD_%7BKL%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BD_%7BKL%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{D_{KL}}" class="latex" /> denotes <a href="https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence">Kullback-Leibler divergence</a>. One can also interpret this inequality as a special case of <a href="https://en.wikipedia.org/wiki/Convex_conjugate#Fenchel's_inequality">the Fenchel–Young inequality</a> relating the conjugate convex functions <img src="https://s0.wp.com/latex.php?latex=%7Bx+%5Cmapsto+e%5Ex%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx+%5Cmapsto+e%5Ex%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx+%5Cmapsto+e%5Ex%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x \mapsto e^x}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7By+%5Cmapsto+y+%5Clog+y+-+y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By+%5Cmapsto+y+%5Clog+y+-+y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By+%5Cmapsto+y+%5Clog+y+-+y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y \mapsto y \log y - y}" class="latex" />.) <img src="https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\Box" class="latex" />

In this note I would like to use this variational formula (which is also known as the Donsker-Varadhan variational formula) to give another proof of the following <a href="https://zbmath.org/1043.05011">inequality of Carbery</a>.

<blockquote>Theorem 2 (Generalized Cauchy-Schwarz inequality) Let <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \geq 0}" class="latex" />, let <img src="https://s0.wp.com/latex.php?latex=%7BS%2C+T_1%2C%5Cdots%2CT_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%2C+T_1%2C%5Cdots%2CT_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%2C+T_1%2C%5Cdots%2CT_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S, T_1,\dots,T_n}" class="latex" /> be finite non-empty sets, and let <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%3A+S+%5Crightarrow+T_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%3A+S+%5Crightarrow+T_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%3A+S+%5Crightarrow+T_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_i: S \rightarrow T_i}" class="latex" /> be functions for each <img src="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i=1,\dots,n}" class="latex" />. Let <img src="https://s0.wp.com/latex.php?latex=%7BK%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K: S \rightarrow {\bf R}^+}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bf_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_i: T_i \rightarrow {\bf R}^+}" class="latex" /> be positive functions for each <img src="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i=1,\dots,n}" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+%5Cprod_%7Bi%3D1%7D%5En+f_i%28%5Cpi_i%28s%29%29+%5Cleq+Q+%5Cprod_%7Bi%3D1%7D%5En+%28%5Csum_%7Bt_i+%5Cin+T_i%7D+f_i%28t_i%29%5E%7Bn%2B1%7D%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+%5Cprod_%7Bi%3D1%7D%5En+f_i%28%5Cpi_i%28s%29%29+%5Cleq+Q+%5Cprod_%7Bi%3D1%7D%5En+%28%5Csum_%7Bt_i+%5Cin+T_i%7D+f_i%28t_i%29%5E%7Bn%2B1%7D%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+%5Cprod_%7Bi%3D1%7D%5En+f_i%28%5Cpi_i%28s%29%29+%5Cleq+Q+%5Cprod_%7Bi%3D1%7D%5En+%28%5Csum_%7Bt_i+%5Cin+T_i%7D+f_i%28t_i%29%5E%7Bn%2B1%7D%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{s \in S} K(s) \prod_{i=1}^n f_i(\pi_i(s)) \leq Q \prod_{i=1}^n (\sum_{t_i \in T_i} f_i(t_i)^{n+1})^{1/(n+1)}" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q}" class="latex" /> is the quantity <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++Q+%3A%3D+%28%5Csum_%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+%5COmega_n%7D+K%28s_0%29+%5Cdots+K%28s_n%29%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++Q+%3A%3D+%28%5Csum_%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+%5COmega_n%7D+K%28s_0%29+%5Cdots+K%28s_n%29%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++Q+%3A%3D+%28%5Csum_%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+%5COmega_n%7D+K%28s_0%29+%5Cdots+K%28s_n%29%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle Q := (\sum_{(s_0,\dots,s_n) \in \Omega_n} K(s_0) \dots K(s_n))^{1/(n+1)}" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Omega_n}" class="latex" /> is the set of all tuples <img src="https://s0.wp.com/latex.php?latex=%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+S%5E%7Bn%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+S%5E%7Bn%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+S%5E%7Bn%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(s_0,\dots,s_n) \in S^{n+1}}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%28s_%7Bi-1%7D%29+%3D+%5Cpi_i%28s_i%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%28s_%7Bi-1%7D%29+%3D+%5Cpi_i%28s_i%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%28s_%7Bi-1%7D%29+%3D+%5Cpi_i%28s_i%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_i(s_{i-1}) = \pi_i(s_i)}" class="latex" /> for <img src="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i=1,\dots,n}" class="latex" />. </blockquote>


Thus for instance, the identity is trivial for <img src="https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n=0}" class="latex" />. When <img src="https://s0.wp.com/latex.php?latex=%7Bn%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n=1}" class="latex" />, the inequality reads <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+%5Cleq+%28%5Csum_%7Bs_0%2Cs_1+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%7D+K%28s_0%29+K%28s_1%29%29%5E%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+%5Cleq+%28%5Csum_%7Bs_0%2Cs_1+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%7D+K%28s_0%29+K%28s_1%29%29%5E%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+%5Cleq+%28%5Csum_%7Bs_0%2Cs_1+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%7D+K%28s_0%29+K%28s_1%29%29%5E%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{s \in S} K(s) f_1(\pi_1(s)) \leq (\sum_{s_0,s_1 \in S: \pi_1(s_0)=\pi_1(s_1)} K(s_0) K(s_1))^{1/2}" class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28+%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E2%29%5E%7B1%2F2%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28+%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E2%29%5E%7B1%2F2%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28+%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E2%29%5E%7B1%2F2%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle ( \sum_{t_1 \in T_1} f_1(t_1)^2)^{1/2}," class="latex" />
which is easily proven by Cauchy-Schwarz, while for <img src="https://s0.wp.com/latex.php?latex=%7Bn%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n=2}" class="latex" /> the inequality reads <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+f_2%28%5Cpi_2%28s%29%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+f_2%28%5Cpi_2%28s%29%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+f_2%28%5Cpi_2%28s%29%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{s \in S} K(s) f_1(\pi_1(s)) f_2(\pi_2(s)) " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%28%5Csum_%7Bs_0%2Cs_1%2C+s_2+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%3B+%5Cpi_2%28s_1%29%3D%5Cpi_2%28s_2%29%7D+K%28s_0%29+K%28s_1%29+K%28s_2%29%29%5E%7B1%2F3%7D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%28%5Csum_%7Bs_0%2Cs_1%2C+s_2+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%3B+%5Cpi_2%28s_1%29%3D%5Cpi_2%28s_2%29%7D+K%28s_0%29+K%28s_1%29+K%28s_2%29%29%5E%7B1%2F3%7D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%28%5Csum_%7Bs_0%2Cs_1%2C+s_2+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%3B+%5Cpi_2%28s_1%29%3D%5Cpi_2%28s_2%29%7D+K%28s_0%29+K%28s_1%29+K%28s_2%29%29%5E%7B1%2F3%7D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \leq (\sum_{s_0,s_1, s_2 \in S: \pi_1(s_0)=\pi_1(s_1); \pi_2(s_1)=\pi_2(s_2)} K(s_0) K(s_1) K(s_2))^{1/3} " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%28%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E3%29%5E%7B1%2F3%7D+%28%5Csum_%7Bt_2+%5Cin+T_2%7D+f_2%28t_2%29%5E3%29%5E%7B1%2F3%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%28%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E3%29%5E%7B1%2F3%7D+%28%5Csum_%7Bt_2+%5Cin+T_2%7D+f_2%28t_2%29%5E3%29%5E%7B1%2F3%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%28%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E3%29%5E%7B1%2F3%7D+%28%5Csum_%7Bt_2+%5Cin+T_2%7D+f_2%28t_2%29%5E3%29%5E%7B1%2F3%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle (\sum_{t_1 \in T_1} f_1(t_1)^3)^{1/3} (\sum_{t_2 \in T_2} f_2(t_2)^3)^{1/3}" class="latex" />
which can also be proven by elementary means. However even for <img src="https://s0.wp.com/latex.php?latex=%7Bn%3D3%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%3D3%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%3D3%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n=3}" class="latex" />, the existing proofs require the “tensor power trick” in order to reduce to the case when the <img src="https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_i}" class="latex" /> are step functions (in which case the inequality can be proven elementarily, as discussed in the above paper of Carbery).

We now prove this inequality. We write <img src="https://s0.wp.com/latex.php?latex=%7BK%28s%29+%3D+%5Cexp%28k%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%28s%29+%3D+%5Cexp%28k%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%28s%29+%3D+%5Cexp%28k%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K(s) = \exp(k(s))}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bf_i%28t_i%29+%3D+%5Cexp%28g_i%28t_i%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_i%28t_i%29+%3D+%5Cexp%28g_i%28t_i%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_i%28t_i%29+%3D+%5Cexp%28g_i%28t_i%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_i(t_i) = \exp(g_i(t_i))}" class="latex" /> for some functions <img src="https://s0.wp.com/latex.php?latex=%7Bk%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k: S \rightarrow {\bf R}}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bg_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bg_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bg_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{g_i: T_i \rightarrow {\bf R}}" class="latex" />. If we take logarithms in the inequality to be proven and apply Lemma <a href="#gibbs">1</a>, the inequality becomes <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_X+%7B%5Cbf+E%7D+k%28X%29+%2B+%5Csum_%7Bi%3D1%7D%5En+g_i%28%5Cpi_i%28X%29%29+%2B+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_X+%7B%5Cbf+E%7D+k%28X%29+%2B+%5Csum_%7Bi%3D1%7D%5En+g_i%28%5Cpi_i%28X%29%29+%2B+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_X+%7B%5Cbf+E%7D+k%28X%29+%2B+%5Csum_%7Bi%3D1%7D%5En+g_i%28%5Cpi_i%28X%29%29+%2B+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sup_X {\bf E} k(X) + \sum_{i=1}^n g_i(\pi_i(X)) + {\bf H}[X] " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csup_%7B%28X_0%2C%5Cdots%2CX_n%29%7D+%7B%5Cbf+E%7D+k%28X_0%29%2B%5Cdots%2Bk%28X_n%29+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csup_%7B%28X_0%2C%5Cdots%2CX_n%29%7D+%7B%5Cbf+E%7D+k%28X_0%29%2B%5Cdots%2Bk%28X_n%29+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csup_%7B%28X_0%2C%5Cdots%2CX_n%29%7D+%7B%5Cbf+E%7D+k%28X_0%29%2B%5Cdots%2Bk%28X_n%29+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \leq \frac{1}{n+1} \sup_{(X_0,\dots,X_n)} {\bf E} k(X_0)+\dots+k(X_n) + {\bf H}[X_0,\dots,X_n] " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csum_%7Bi%3D1%7D%5En+%5Csup_%7BY_i%7D+%28n%2B1%29+%7B%5Cbf+E%7D+g_i%28Y_i%29+%2B+%7B%5Cbf+H%7D%5BY_i%5D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csum_%7Bi%3D1%7D%5En+%5Csup_%7BY_i%7D+%28n%2B1%29+%7B%5Cbf+E%7D+g_i%28Y_i%29+%2B+%7B%5Cbf+H%7D%5BY_i%5D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csum_%7Bi%3D1%7D%5En+%5Csup_%7BY_i%7D+%28n%2B1%29+%7B%5Cbf+E%7D+g_i%28Y_i%29+%2B+%7B%5Cbf+H%7D%5BY_i%5D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + \frac{1}{n+1} \sum_{i=1}^n \sup_{Y_i} (n+1) {\bf E} g_i(Y_i) + {\bf H}[Y_i]" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> ranges over random variables taking values in <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_0,\dots,X_n}" class="latex" /> range over tuples of random variables taking values in <img src="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Omega_n}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%7BY_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y_i}" class="latex" /> range over random variables taking values in <img src="https://s0.wp.com/latex.php?latex=%7BT_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T_i}" class="latex" />. Comparing the suprema, the claim now reduces to

<blockquote>Lemma 3 (Conditional expectation computation) Let <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> be an <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" />-valued random variable. Then there exists a <img src="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Omega_n}" class="latex" />-valued random variable <img src="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_0,\dots,X_n)}" class="latex" />, where each <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> has the same distribution as <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%28n%2B1%29+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%28n%2B1%29+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%28n%2B1%29+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}[X_0,\dots,X_n] = (n+1) {\bf H}[X] " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle - {\bf H}[\pi_1(X)] - \dots - {\bf H}[\pi_n(X)]." class="latex" />
</blockquote>


Proof: We induct on <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" />. When <img src="https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n=0}" class="latex" /> we just take <img src="https://s0.wp.com/latex.php?latex=%7BX_0+%3D+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_0+%3D+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_0+%3D+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_0 = X}" class="latex" />. Now suppose that <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \geq 1}" class="latex" />, and the claim has already been proven for <img src="https://s0.wp.com/latex.php?latex=%7Bn-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n-1}" class="latex" />, thus one has already obtained a tuple <img src="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29+%5Cin+%5COmega_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29+%5Cin+%5COmega_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29+%5Cin+%5COmega_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_0,\dots,X_{n-1}) \in \Omega_{n-1}}" class="latex" /> with each <img src="https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_0,\dots,X_{n-1}}" class="latex" /> having the same distribution as <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%3D+n+%7B%5Cbf+H%7D%5BX%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_%7Bn-1%7D%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%3D+n+%7B%5Cbf+H%7D%5BX%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_%7Bn-1%7D%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%3D+n+%7B%5Cbf+H%7D%5BX%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_%7Bn-1%7D%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}[X_0,\dots,X_{n-1}] = n {\bf H}[X] - {\bf H}[\pi_1(X)] - \dots - {\bf H}[\pi_{n-1}(X)]." class="latex" />
By hypothesis, <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X_{n-1})}" class="latex" /> has the same distribution as <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X)}" class="latex" />. For each value <img src="https://s0.wp.com/latex.php?latex=%7Bt_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t_n}" class="latex" /> attained by <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X)}" class="latex" />, we can take conditionally independent copies of <img src="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_0,\dots,X_{n-1})}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> conditioned to the events <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X_{n-1}) = t_n}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X) = t_n}" class="latex" /> respectively, and then concatenate them to form a tuple <img src="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_0,\dots,X_n)}" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Omega_n}" class="latex" />, with <img src="https://s0.wp.com/latex.php?latex=%7BX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_n}" class="latex" /> a further copy of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> that is conditionally independent of <img src="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_0,\dots,X_{n-1})}" class="latex" /> relative to <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X_{n-1}) = \pi_n(X)}" class="latex" />. One can the use the <a href="https://en.wikipedia.org/wiki/Conditional_entropy#Chain_rule">entropy chain rul</a>e to compute <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%7C+%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%7C+%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%7C+%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}[X_0,\dots,X_n] = {\bf H}[\pi_n(X_n)] + {\bf H}[X_0,\dots,X_n| \pi_n(X_n)]" class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle = {\bf H}[\pi_n(X_n)] + {\bf H}[X_0,\dots,X_{n-1}| \pi_n(X_n)] + {\bf H}[X_n| \pi_n(X_n)] " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_%7Bn-1%7D%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_%7Bn-1%7D%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_%7Bn-1%7D%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle = {\bf H}[\pi_n(X)] + {\bf H}[X_0,\dots,X_{n-1}| \pi_n(X_{n-1})] + {\bf H}[X_n| \pi_n(X_n)] " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%28%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_%7Bn-1%7D%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%28%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_%7Bn-1%7D%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%28%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_%7Bn-1%7D%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle = {\bf H}[\pi_n(X)] + ({\bf H}[X_0,\dots,X_{n-1}] - {\bf H}[\pi_n(X_{n-1})]) " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+%28%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+%28%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+%28%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + ({\bf H}[X_n] - {\bf H}[\pi_n(X_n)]) " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%2B+%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%2B+%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%2B+%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle ={\bf H}[X_0,\dots,X_{n-1}] + {\bf H}[X_n] - {\bf H}[\pi_n(X_n)]" class="latex" />
and the claim now follows from the induction hypothesis. <img src="https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\Box" class="latex" />

With a little more effort, one can replace <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" /> by a more general measure space (and use differential entropy in place of Shannon entropy), to recover Carbery’s inequality in full generality; we leave the details to the interested reader.

]]></content:encoded>
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<media:title type="html">Terry</media:title>
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</item>
<item>
<title>A slightly longer Lean 4 proof tour</title>
<link>https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/</link>
<comments>https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Wed, 06 Dec 2023 07:10:07 +0000</pubDate>
<category><![CDATA[expository]]></category>
<category><![CDATA[math.CA]]></category>
<category><![CDATA[Lean4]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14195</guid>
<description><![CDATA[In my previous post, I walked through the task of formally deducing one lemma from another in Lean 4. The deduction was deliberately chosen to be short and only showcased a small number of Lean tactics. Here I would like to walk through the process I used for a slightly longer proof I worked out […]]]></description>
<content:encoded><![CDATA[
In my <a href="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/">previous post</a>, I walked through the task of formally deducing one lemma from another in <a href="https://en.wikipedia.org/wiki/Lean_(proof_assistant)">Lean 4</a>. The deduction was deliberately chosen to be short and only showcased a small number of Lean tactics. Here I would like to walk through the process I used for a slightly longer proof I worked out recently, after seeing the following <a href="https://twitter.com/damekdavis/status/1730983634570510819">challenge from Damek Davis</a>: to formalize (in a civilized fashion) the proof of the following lemma:
<blockquote class="wp-block-quote is-layout-flow wp-block-quote-is-layout-flow">
Lemma. Let <img src="https://s0.wp.com/latex.php?latex=%5C%7Ba_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5C%7Ba_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5C%7Ba_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\{a_k\}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%5C%7BD_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5C%7BD_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5C%7BD_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\{D_k\}" class="latex" /> be sequences of real numbers indexed by natural numbers <img src="https://s0.wp.com/latex.php?latex=k%3D0%2C1%2C%5Cdots&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=k%3D0%2C1%2C%5Cdots&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=k%3D0%2C1%2C%5Cdots&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="k=0,1,\dots" class="latex" />, with <img src="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k" class="latex" /> non-increasing and <img src="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_k" class="latex" /> non-negative. Suppose also that <img src="https://s0.wp.com/latex.php?latex=a_k+%5Cleq+D_k+-+D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k+%5Cleq+D_k+-+D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k+%5Cleq+D_k+-+D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k \leq D_k - D_{k+1}" class="latex" /> for all <img src="https://s0.wp.com/latex.php?latex=k+%5Cgeq+0&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=k+%5Cgeq+0&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=k+%5Cgeq+0&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="k \geq 0" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=a_k+%5Cleq+%5Cfrac%7BD_0%7D%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k+%5Cleq+%5Cfrac%7BD_0%7D%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k+%5Cleq+%5Cfrac%7BD_0%7D%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k \leq \frac{D_0}{k+1}" class="latex" /> for all <img src="https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="k" class="latex" />.
</blockquote>
Here I tried to draw upon the lessons I had learned from the PFR formalization project, and to first set up a human readable proof of the lemma before starting the Lean formalization – a lower-case “blueprint” rather than the fancier <a href="https://github.com/PatrickMassot/leanblueprint">Blueprint</a> used in the PFR project. The main idea of the proof here is to use the telescoping series identity
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D+%3D+D_0+-+D_%7Bk%2B1%7D.&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D+%3D+D_0+-+D_%7Bk%2B1%7D.&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D+%3D+D_0+-+D_%7Bk%2B1%7D.&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{i=0}^k D_i - D_{i+1} = D_0 - D_{k+1}." class="latex" />
Since <img src="https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_{k+1}" class="latex" /> is non-negative, and <img src="https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_i \leq D_i - D_{i+1}" class="latex" /> by hypothesis, we have
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+a_i+%5Cleq+D_0&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+a_i+%5Cleq+D_0&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+a_i+%5Cleq+D_0&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{i=0}^k a_i \leq D_0" class="latex" />
but by the monotone hypothesis on <img src="https://s0.wp.com/latex.php?latex=a_i&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_i&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_i&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_i" class="latex" /> the left-hand side is at least <img src="https://s0.wp.com/latex.php?latex=%28k%2B1%29+a_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%28k%2B1%29+a_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%28k%2B1%29+a_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="(k+1) a_k" class="latex" />, giving the claim.
This is already a human-readable proof, but in order to formalize it more easily in Lean, I decided to rewrite it as a chain of inequalities, starting at <img src="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k" class="latex" /> and ending at <img src="https://s0.wp.com/latex.php?latex=D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_0 / (k+1)" class="latex" />. With a little bit of pen and paper effort, I obtained
<img src="https://s0.wp.com/latex.php?latex=a_k+%3D+%28k%2B1%29+a_k+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k+%3D+%28k%2B1%29+a_k+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k+%3D+%28k%2B1%29+a_k+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k = (k+1) a_k / (k+1)" class="latex" />
(by field identities)
<img src="https://s0.wp.com/latex.php?latex=%3D+%28%5Csum_%7Bi%3D0%7D%5Ek+a_k%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%3D+%28%5Csum_%7Bi%3D0%7D%5Ek+a_k%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%3D+%28%5Csum_%7Bi%3D0%7D%5Ek+a_k%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="= (\sum_{i=0}^k a_k) / (k+1)" class="latex" />
(by the formula for summing a constant)
<img src="https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+a_i%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+a_i%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+a_i%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\leq (\sum_{i=0}^k a_i) / (k+1)" class="latex" />
(by the monotone hypothesis)
<img src="https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\leq (\sum_{i=0}^k D_i - D_{i+1}) / (k+1)" class="latex" />
(by the hypothesis <img src="https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_i \leq D_i - D_{i+1}" class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%3D+%28D_0+-+D_%7Bk%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%3D+%28D_0+-+D_%7Bk%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%3D+%28D_0+-+D_%7Bk%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="= (D_0 - D_{k+1}) / (k+1)" class="latex" />
(by telescoping series)
<img src="https://s0.wp.com/latex.php?latex=%5Cleq+D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cleq+D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cleq+D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\leq D_0 / (k+1)" class="latex" />
(by the non-negativity of <img src="https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_{k+1}" class="latex" />).
I decided that this was a good enough blueprint for me to work with. The next step is to formalize the statement of the lemma in Lean. For this quick project, it was convenient to use the <a href="https://live.lean-lang.org/">online Lean playground</a>, rather than my local IDE, so the screenshots will look a little different from those in the previous post. (If you like, you can follow this tour in that playground, by clicking on the screenshots of the Lean code.) I start by importing Lean’s math library, and starting an example of a statement to state and prove:
<img src="image/png;base64,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" alt=""/>
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)"><img width="256" height="124" data-attachment-id="14279" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-26-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-26.png" data-orig-size="256,124" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-26" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-26.png?w=256" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-26.png?w=256" src="https://terrytao.files.wordpress.com/2023/12/image-26.png?w=256" alt="" class="wp-image-14279" srcset="https://terrytao.files.wordpress.com/2023/12/image-26.png 256w, https://terrytao.files.wordpress.com/2023/12/image-26.png?w=150 150w" sizes="(max-width: 256px) 100vw, 256px" /></a></figure>
Now we have to declare the hypotheses and variables. The main variables here are the sequences <img src="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_k" class="latex" />, which in Lean are best modeled by functions <code>a</code>, <code>D</code> from the natural numbers ℕ to the reals ℝ. (One can choose to “hardwire” the non-negativity hypothesis into the <img src="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_k" class="latex" /> by making <code>D</code> take values in the nonnegative reals <img src="https://s0.wp.com/latex.php?latex=%7B%5Cbf+R%7D%5E%2B&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cbf+R%7D%5E%2B&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cbf+R%7D%5E%2B&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="{\bf R}^+" class="latex" /> (denoted <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/NNReal.html#NNReal">NNReal</a></code> in Lean), but this turns out to be inconvenient, because the laws of algebra and summation that we will need are clunkier on the non-negative reals (which are not even a group) than on the reals (which are a field). So we add in the variables:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)"><img width="452" height="159" data-attachment-id="14205" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-26/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image.png" data-orig-size="452,159" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image.png?w=452" src="https://terrytao.files.wordpress.com/2023/12/image.png?w=452" alt="" class="wp-image-14205" srcset="https://terrytao.files.wordpress.com/2023/12/image.png 452w, https://terrytao.files.wordpress.com/2023/12/image.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image.png?w=300 300w" sizes="(max-width: 452px) 100vw, 452px" /></a></figure>
Now we add in the hypotheses, which in Lean convention are usually given names starting with <code>h</code>. This is fairly straightforward; the one thing is that the property of being monotone decreasing already has a name in Lean’s Mathlib, namely <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Basic.html#Antitone">Antitone</a></code>, and it is generally a good idea to use the Mathlib provided terminology (because that library contains a lot of useful lemmas about such terms).
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)"><img width="1024" height="121" data-attachment-id="14289" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-27/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-27.png" data-orig-size="1519,180" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-27" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-27.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-27.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-27.png?w=1024" alt="" class="wp-image-14289" srcset="https://terrytao.files.wordpress.com/2023/12/image-27.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-27.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-27.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-27.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-27.png 1519w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
One thing to note here is that Lean is quite good at filling in implied ranges of variables. Because <code>a</code> and <code>D</code> have the natural numbers ℕ as their domain, the dummy variable <code>k</code> in these hypotheses is automatically being quantified over ℕ. We could have made this quantification explicit if we so chose, for instance using <code>∀ k : ℕ, 0 ≤ D k</code> instead of <code>∀ k, 0 ≤ D k</code>, but it is not necessary to do so. Also note that Lean does not require parentheses when applying functions: we write <code>D k</code> here rather than <code>D(k)</code> (which in fact does not compile in Lean unless one puts a space between the <code>D</code> and the parentheses). This is slightly different from standard mathematical notation, but is not too difficult to get used to.
This looks like the end of the hypotheses, so we could now add a colon to move to the conclusion, and then add that conclusion:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)"><img loading="lazy" width="1024" height="128" data-attachment-id="14209" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-1-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-1.png" data-orig-size="1491,187" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-1" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-1.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-1.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-1.png?w=1024" alt="" class="wp-image-14209" srcset="https://terrytao.files.wordpress.com/2023/12/image-1.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-1.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-1.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-1.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-1.png 1491w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
This is a perfectly fine Lean statement. But it turns out that when proving a universally quantified statement such as <code>∀ k, a k ≤ D 0 / (k + 1)</code>, the first step is almost always to open up the quantifier to introduce the variable <code>k</code> (using the Lean command <code>intro k</code>). Because of this, it is slightly more efficient to hide the universal quantifier by placing the variable <code>k</code> in the hypotheses, rather than in the quantifier (in which case we have to now specify that it is a natural number, as Lean can no longer deduce this from context):
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)"><img loading="lazy" width="1024" height="117" data-attachment-id="14212" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-2-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-2.png" data-orig-size="1628,187" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-2" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-2.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-2.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-2.png?w=1024" alt="" class="wp-image-14212" srcset="https://terrytao.files.wordpress.com/2023/12/image-2.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-2.png?w=1019 1019w, https://terrytao.files.wordpress.com/2023/12/image-2.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-2.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-2.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-2.png 1628w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
At this point Lean is complaining of an unexpected end of input: the example has been stated, but not proved. We will temporarily mollify Lean by adding a <code>sorry</code> as the purported proof:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20sorry"><img loading="lazy" width="1024" height="114" data-attachment-id="14214" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-3-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-3.png" data-orig-size="1622,182" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-3" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-3.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-3.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-3.png?w=1024" alt="" class="wp-image-14214" srcset="https://terrytao.files.wordpress.com/2023/12/image-3.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-3.png?w=1016 1016w, https://terrytao.files.wordpress.com/2023/12/image-3.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-3.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-3.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-3.png 1622w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Now Lean is content, other than giving a warning (as indicated by the yellow squiggle under the <code>example</code>) that the proof contains a sorry.
It is now time to follow the blueprint. The Lean tactic for proving an inequality via chains of other inequalities is known as <code><a href="https://leanprover-community.github.io/extras/calc.html">calc</a></code>. We use the blueprint to fill in the <code>calc</code> that we want, leaving the justifications of each step as “<code>sorry</code>”s for now:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%20%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="302" data-attachment-id="14222" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-6-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-6.png" data-orig-size="1628,481" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-6" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-6.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-6.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-6.png?w=1024" alt="" class="wp-image-14222" srcset="https://terrytao.files.wordpress.com/2023/12/image-6.png?w=1022 1022w, https://terrytao.files.wordpress.com/2023/12/image-6.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-6.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-6.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-6.png 1628w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Here, we “<code>open</code>“ed the <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Basic.html">Finset</a></code> namespace in order to easily access <code>Finset</code>‘s <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Basic.html#Finset.range">range</a></code> function, with <code>range k</code> basically being the finite set of natural numbers <img src="https://s0.wp.com/latex.php?latex=%5C%7B0%2C%5Cdots%2Ck-1%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5C%7B0%2C%5Cdots%2Ck-1%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5C%7B0%2C%5Cdots%2Ck-1%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\{0,\dots,k-1\}" class="latex" />, and also “<code>open</code>“ed the <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Basic.html">BigOperators</a></code> namespace to access the familiar ∑ notation for (finite) summation, in order to make the steps in the Lean code resemble the blueprint as much as possible. One could avoid opening these namespaces, but then expressions such as <code>∑ i in range (k+1), a i</code> would instead have to be written as something like <code>Finset.sum (Finset.range (k+1)) (fun i ↦ a i)</code>, which looks a lot less like like standard mathematical writing. The proof structure here may remind some readers of the “two column proofs” that are somewhat popular in American high school geometry classes.
Now we have six sorries to fill. Navigating to the first sorry, Lean tells us the ambient hypotheses, and the goal that we need to prove to fill that sorry:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-5.png"><img loading="lazy" width="672" height="235" data-attachment-id="14219" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-5-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-5.png" data-orig-size="672,235" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-5" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-5.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-5.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-5.png?w=672" alt="" class="wp-image-14219" srcset="https://terrytao.files.wordpress.com/2023/12/image-5.png 672w, https://terrytao.files.wordpress.com/2023/12/image-5.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-5.png?w=300 300w" sizes="(max-width: 672px) 100vw, 672px" /></a></figure>
The ⊢ symbol here is Lean’s marker for the goal. The uparrows ↑ are coercion symbols, indicating that the natural number <code>k</code> has to be converted to a real number in order to interact via arithmetic operations with other real numbers such as <code>a k</code>, but we can ignore these coercions for this tour (for this proof, it turns out Lean will basically manage them automatically without need for any explicit intervention by a human). 
The goal here is a self-evident algebraic identity; it involves division, so one has to check that the denominator is non-zero, but this is self-evident. In Lean, a convenient way to establish algebraic identities is to use the tactic <code><a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#field_simp">field_simp</a></code> to clear denominators, and then <code><a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#ring">ring</a></code> to verify any identity that is valid for commutative rings. This works, and clears the first <code>sorry</code>:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%20%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="299" data-attachment-id="14223" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-7-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-7.png" data-orig-size="1630,476" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-7" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-7.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-7.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-7.png?w=1024" alt="" class="wp-image-14223" srcset="https://terrytao.files.wordpress.com/2023/12/image-7.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-7.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-7.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-7.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-7.png 1630w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
<code>field_simp</code>, by the way, is smart enough to deduce on its own that the denominator <code>k+1</code> here is manifestly non-zero (and in fact positive); no human intervention is required to point this out. Similarly for other “clearing denominator” steps that we will encounter in the other parts of the proof.
Now we navigate to the next `sorry`. Lean tells us the hypotheses and goals:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-8.png"><img loading="lazy" width="772" height="277" data-attachment-id="14226" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-8-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-8.png" data-orig-size="772,277" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-8" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-8.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-8.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-8.png?w=772" alt="" class="wp-image-14226" srcset="https://terrytao.files.wordpress.com/2023/12/image-8.png 772w, https://terrytao.files.wordpress.com/2023/12/image-8.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-8.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-8.png?w=768 768w" sizes="(max-width: 772px) 100vw, 772px" /></a></figure>
We can reduce the goal by canceling out the common denominator <code>↑k+1</code>. Here we can use the handy Lean tactic <code><a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#congr">congr</a></code>, which tries to match two sides of an equality goal as much as possible, and leave any remaining discrepancies between the two sides as further goals to be proven. Applying <code>congr</code>, the goal reduces to
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-9.png"><img loading="lazy" width="768" height="226" data-attachment-id="14229" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-9-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-9.png" data-orig-size="768,226" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-9" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-9.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-9.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-9.png?w=768" alt="" class="wp-image-14229" srcset="https://terrytao.files.wordpress.com/2023/12/image-9.png 768w, https://terrytao.files.wordpress.com/2023/12/image-9.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-9.png?w=300 300w" sizes="(max-width: 768px) 100vw, 768px" /></a></figure>
Here one might imagine that this is something that one can prove by induction. But this particular sort of identity – summing a constant over a finite set – is already covered by Mathlib. Indeed, searching for <code>Finset</code>, <code>sum</code>, and <code>const</code> soon leads us to the <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Basic.html#Finset.sum_const">Finset.sum_const</a></code> lemma here. But there is an even more convenient path to take here, which is to apply the powerful tactic <code>simp</code>, which tries to simplify the goal as much as possible using all the “<code>simp</code> lemmas” Mathlib has to offer (of which <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Basic.html#Finset.sum_const">Finset.sum_const</a></code> is an example, but there are thousands of others). As it turns out, <code>simp</code> completely kills off this identity, without any further human intervention:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%20%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="318" data-attachment-id="14233" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-10-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-10.png" data-orig-size="1645,511" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-10" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-10.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-10.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-10.png?w=1024" alt="" class="wp-image-14233" srcset="https://terrytao.files.wordpress.com/2023/12/image-10.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-10.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-10.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-10.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-10.png 1645w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Now we move on to the next sorry, and look at our goal:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-11.png"><img loading="lazy" width="785" height="267" data-attachment-id="14235" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-11-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-11.png" data-orig-size="785,267" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-11" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-11.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-11.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-11.png?w=785" alt="" class="wp-image-14235" srcset="https://terrytao.files.wordpress.com/2023/12/image-11.png 785w, https://terrytao.files.wordpress.com/2023/12/image-11.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-11.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-11.png?w=768 768w" sizes="(max-width: 785px) 100vw, 785px" /></a></figure>
<code>congr</code> doesn’t work here because we have an inequality instead of an equality, but there is a powerful relative <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GCongr/Core.html">gcongr</a></code> of <code>congr</code> that is perfectly suited for inequalities. It can also open up sums, products, and integrals, reducing global inequalities between such quantities into pointwise inequalities. If we invoke <code>gcongr with i hi</code> (where we tell <code>gcongr</code> to use <code>i</code> for the variable opened up, and <code>hi</code> for the constraint this variable will satisfy), we arrive at a greatly simplified goal (and a new ambient variable and hypothesis):
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-12.png"><img loading="lazy" width="762" height="244" data-attachment-id="14239" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-12-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-12.png" data-orig-size="762,244" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-12" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-12.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-12.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-12.png?w=762" alt="" class="wp-image-14239" srcset="https://terrytao.files.wordpress.com/2023/12/image-12.png 762w, https://terrytao.files.wordpress.com/2023/12/image-12.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-12.png?w=300 300w" sizes="(max-width: 762px) 100vw, 762px" /></a></figure>
Now we need to use the monotonicity hypothesis on <code>a</code>, which we have named <code>ha</code> here. Looking at the <a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Basic.html">documentation for Antitone</a>, one finds a lemma that looks applicable here:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-13.png"><img loading="lazy" width="1024" height="166" data-attachment-id="14241" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-13-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-13.png" data-orig-size="1560,253" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-13" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-13.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-13.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-13.png?w=1024" alt="" class="wp-image-14241" srcset="https://terrytao.files.wordpress.com/2023/12/image-13.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-13.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-13.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-13.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-13.png 1560w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
One can apply this lemma in this case by writing <code>apply Antitone.imp ha</code>, but because <code>ha</code> is already of type <code>Antitone</code>, we can abbreviate this to <code>apply ha.imp</code>. (Actually, as indicated in the documentation, due to the way <code>Antitone</code> is defined, we can even just use <code>apply ha</code> here.) This reduces the goal nicely:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-14.png"><img loading="lazy" width="675" height="255" data-attachment-id="14244" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-14-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-14.png" data-orig-size="675,255" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-14" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-14.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-14.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-14.png?w=675" alt="" class="wp-image-14244" srcset="https://terrytao.files.wordpress.com/2023/12/image-14.png 675w, https://terrytao.files.wordpress.com/2023/12/image-14.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-14.png?w=300 300w" sizes="(max-width: 675px) 100vw, 675px" /></a></figure>
The goal is now very close to the hypothesis <code>hi</code>. One could now look up the documentation for <code>Finset.range</code> to see how to unpack <code>hi</code>, but as before <code>simp</code> can do this for us. Invoking <code>simp at hi</code>, we obtain
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-15.png"><img loading="lazy" width="671" height="263" data-attachment-id="14247" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-15-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-15.png" data-orig-size="671,263" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-15" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-15.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-15.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-15.png?w=671" alt="" class="wp-image-14247" srcset="https://terrytao.files.wordpress.com/2023/12/image-15.png 671w, https://terrytao.files.wordpress.com/2023/12/image-15.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-15.png?w=300 300w" sizes="(max-width: 671px) 100vw, 671px" /></a></figure>
Now the goal and hypothesis are very close indeed. Here we can just close the goal using the <code><a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#linarith">linarith</a></code> tactic used in the <a href="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/">previous tour</a>:
<figure class="wp-block-image size-large is-resized"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="304" data-attachment-id="14250" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-16-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-16.png" data-orig-size="1744,519" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-16" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-16.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-16.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-16.png?w=1024" alt="" class="wp-image-14250" style="width:840px;height:auto" srcset="https://terrytao.files.wordpress.com/2023/12/image-16.png?w=1022 1022w, https://terrytao.files.wordpress.com/2023/12/image-16.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-16.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-16.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-16.png 1744w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
The next sorry can be resolved by similar methods, using the hypothesis <code>hD</code> applied at the variable <code>i</code>:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20exact%20hD%20i%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="290" data-attachment-id="14252" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-17-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-17.png" data-orig-size="1739,493" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-17" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-17.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-17.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-17.png?w=1024" alt="" class="wp-image-14252" srcset="https://terrytao.files.wordpress.com/2023/12/image-17.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-17.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-17.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-17.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-17.png 1739w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Now for the penultimate sorry. As in a previous step, we can use <code>congr</code> to remove the denominator, leaving us in this state:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-18.png"><img loading="lazy" width="764" height="249" data-attachment-id="14254" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-18-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-18.png" data-orig-size="764,249" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-18" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-18.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-18.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-18.png?w=764" alt="" class="wp-image-14254" srcset="https://terrytao.files.wordpress.com/2023/12/image-18.png 764w, https://terrytao.files.wordpress.com/2023/12/image-18.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-18.png?w=300 300w" sizes="(max-width: 764px) 100vw, 764px" /></a></figure>
This is a telescoping series identity. One could try to prove it by induction, or one could try to see if this identity is already in Mathlib. Searching for <code>Finset</code>, <code>sum</code>, and <code>sub</code> will <a href="https://leanprover-community.github.io/mathlib4_docs/search.html?sitesearch=https%3A%2F%2Fleanprover-community.github.io%2Fmathlib4_docs&q=finset+sum+sub">locate the right tool</a> (as the fifth hit), but a simpler way to proceed here is to use the <code>exact?</code> tactic we saw in the <a href="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/">previous tour</a>:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-19.png"><img loading="lazy" width="755" height="156" data-attachment-id="14256" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-19-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-19.png" data-orig-size="755,156" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-19" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-19.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-19.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-19.png?w=755" alt="" class="wp-image-14256" srcset="https://terrytao.files.wordpress.com/2023/12/image-19.png 755w, https://terrytao.files.wordpress.com/2023/12/image-19.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-19.png?w=300 300w" sizes="(max-width: 755px) 100vw, 755px" /></a></figure>
A brief check of the documentation for <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Basic.html#Finset.sum_range_sub'">sum_range_sub'</a></code> confirms that this is what we want. Actually we can just use <code>apply sum_range_sub'</code> here, as the <code><a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#apply">apply</a></code> tactic is smart enough to fill in the missing arguments:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20exact%20hD%20i%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20apply%20sum_range_sub'%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="279" data-attachment-id="14259" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-20-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-20.png" data-orig-size="1744,476" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-20" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-20.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-20.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-20.png?w=1024" alt="" class="wp-image-14259" srcset="https://terrytao.files.wordpress.com/2023/12/image-20.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-20.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-20.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-20.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-20.png 1744w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
One last <code>sorry</code> to go. As before, we use <code>gcongr</code> to cancel denominators, leaving us with
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-21.png"><img loading="lazy" width="662" height="217" data-attachment-id="14261" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-21-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-21.png" data-orig-size="662,217" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-21" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-21.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-21.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-21.png?w=662" alt="" class="wp-image-14261" srcset="https://terrytao.files.wordpress.com/2023/12/image-21.png 662w, https://terrytao.files.wordpress.com/2023/12/image-21.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-21.png?w=300 300w" sizes="(max-width: 662px) 100vw, 662px" /></a></figure>
This looks easy, because the hypothesis <code>hpos</code> will tell us that <code>D (k+1)</code> is nonnegative; specifically, the instance <code>hpos (k+1)</code> of that hypothesis will state exactly this. The <code>linarith</code> tactic will then resolve this goal once it is told about this particular instance:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20exact%20hD%20i%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20apply%20sum_range_sub'%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%3B%20linarith%20%5Bhpos%20(k%2B1)%5D"><img loading="lazy" width="1024" height="294" data-attachment-id="14264" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-22-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-22.png" data-orig-size="1782,513" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-22" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-22.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-22.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-22.png?w=1024" alt="" class="wp-image-14264" srcset="https://terrytao.files.wordpress.com/2023/12/image-22.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-22.png?w=1021 1021w, https://terrytao.files.wordpress.com/2023/12/image-22.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-22.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-22.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-22.png 1782w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
We now have a complete proof – no more yellow squiggly line in the example. There are two warnings though – there are two variables <code>i</code> and <code>hi</code> introduced in the proof that Lean’s “linter” has noticed are not actually used in the proof. So we can rename them with underscores to tell Lean that we are okay with them not being used:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20_i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20gcongr%20with%20i%20_%3B%20exact%20hD%20i%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20apply%20sum_range_sub'%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%3B%20linarith%20%5Bhpos%20(k%2B1)%5D"><img loading="lazy" width="1024" height="278" data-attachment-id="14266" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-23-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-23.png" data-orig-size="1763,479" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-23" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-23.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-23.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-23.png?w=1024" alt="" class="wp-image-14266" srcset="https://terrytao.files.wordpress.com/2023/12/image-23.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-23.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-23.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-23.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-23.png 1763w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
This is a perfectly fine proof, but upon noticing that many of the steps are similar to each other, one can do a bit of “code golf” as in the <a href="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/">previous tour</a> to compactify the proof a bit:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20a%20k%20%3D%20(%E2%88%91%20_i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%3A%3D%20by%20simp%3B%20field_simp%3B%20ring%0D%0A%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20le_trans%20_%20(hD%20i)%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%3A%3D%20by%20gcongr%3B%20rw%20%5Bsum_range_sub'%5D%3B%20linarith%20%5Bhpos%20(k%2B1)%5D"><img loading="lazy" width="1024" height="250" data-attachment-id="14269" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-25-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-25.png" data-orig-size="1642,401" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-25" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-25.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-25.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-25.png?w=1024" alt="" class="wp-image-14269" srcset="https://terrytao.files.wordpress.com/2023/12/image-25.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-25.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-25.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-25.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-25.png 1642w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
With enough familiarity with the Lean language, this proof actually tracks quite closely with (an optimized version of) the human blueprint.
This concludes the tour of a lengthier Lean proving exercise. I am finding the pre-planning step of the proof (using an informal “blueprint” to break the proof down into extremely granular pieces) to make the formalization process significantly easier than in the past (when I often adopted a sequential process of writing one line of code at a time without first sketching out a skeleton of the argument). (The proof here took only about 15 minutes to create initially, although for this blog post I had to recreate it with screenshots and supporting links, which took significantly more time.) I believe that a realistic near-term goal for AI is to be able to fill in automatically a significant fraction of the sorts of atomic “<code>sorry</code>“s of the size one saw in this proof, allowing one to convert a blueprint to a formal Lean proof even more rapidly.
One final remark: in this tour I filled in the “<code>sorry</code>“s in the order in which they appeared, but there is actually no requirement that one does this, and once one has used a blueprint to atomize a proof into self-contained smaller pieces, one can fill them in in any order. Importantly for a group project, these micro-tasks can be parallelized, with different contributors claiming whichever “<code>sorry</code>” they feel they are qualified to solve, and working independently of each other. (And, because Lean can automatically verify if their proof is correct, there is no need to have a pre-existing bond of trust with these contributors in order to accept their contributions.) Furthermore, because the specification of a “<code>sorry</code>” someone can make a meaningful contribution to the proof by working on an extremely localized component of it without needing the mathematical expertise to understand the global argument. This is not particularly important in this simple case, where the entire lemma is not too hard to understand to a trained mathematician, but can become quite relevant for complex formalization projects.
]]></content:encoded>
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<item>
<title>Formalizing the proof of PFR in Lean4 using Blueprint: a short tour</title>
<link>https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/</link>
<comments>https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Sat, 18 Nov 2023 23:45:56 +0000</pubDate>
<category><![CDATA[expository]]></category>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[Blueprint]]></category>
<category><![CDATA[Lean4]]></category>
<category><![CDATA[Polynomial Freiman-Ruzsa conjecture]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14054</guid>
<description><![CDATA[Since the release of my preprint with Tim, Ben, and Freddie proving the Polynomial Freiman-Ruzsa (PFR) conjecture over , I (together with Yael Dillies and Bhavik Mehta) have started a collaborative project to formalize this argument in the proof assistant language Lean4. It has been less than a week since the project was launched, but […]]]></description>
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Since the release of <a href="https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/">my preprint with Tim, Ben, and Freddie</a> proving the Polynomial Freiman-Ruzsa (PFR) conjecture over <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmathbb+F%7D_2&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmathbb+F%7D_2&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmathbb+F%7D_2&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="{\mathbb F}_2" class="latex" />, I (together with <a href="https://github.com/YaelDillies">Yael Dillies</a> and <a href="https://github.com/b-mehta">Bhavik Mehta</a>) have started a <a href="https://teorth.github.io/pfr/">collaborative project</a> to formalize this argument in the proof assistant language <a href="https://en.wikipedia.org/wiki/Lean_(proof_assistant)">Lean4</a>. It has been less than a week since the project was launched, but it is proceeding quite well, with a significant fraction of the paper already either fully or partially formalized. The project has been greatly assisted by the <a href="https://github.com/PatrickMassot/leanblueprint">Blueprint tool</a> of <a href="https://www.imo.universite-paris-saclay.fr/~patrick.massot/en/">Patrick Massot</a>, which allows one to write a human-readable “blueprint” of the proof that is linked to the Lean formalization; similar blueprints have been <a href="https://leanprover-community.github.io/liquid/">used for other projects</a>, such as Scholze’s <a href="https://github.com/leanprover-community/lean-liquid">liquid tensor experiment</a>. For the PFR project, the blueprint can be found <a href="https://teorth.github.io/pfr/blueprint/">here</a>. One feature of the blueprint that I find particularly appealing is the dependency graph that is automatically generated from the blueprint, and can provide a rough snapshot of how far along the formalization has advanced. For PFR, the latest state of the dependency graph can be found <a href="https://teorth.github.io/pfr/blueprint/dep_graph_document.html">here</a>. At the current time of writing, the graph looks like this:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image.png"><img loading="lazy" width="1024" height="367" data-attachment-id="14060" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image.png" data-orig-size="2692,965" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image.png?w=1024" alt="" class="wp-image-14060" srcset="https://terrytao.files.wordpress.com/2023/11/image.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image.png?w=2048 2048w, https://terrytao.files.wordpress.com/2023/11/image.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image.png?w=768 768w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
The color coding of the various bubbles (for lemmas) and rectangles (for definitions) is explained in the legend to <a href="https://teorth.github.io/pfr/blueprint/dep_graph_document.html">the dependency graph</a>, but roughly speaking the green bubbles/rectangles represent lemmas or definitions that have been fully formalized, and the blue ones represent lemmas or definitions which are ready to be formalized (their statements, but not proofs, have already been formalized, as well as those of all prerequisite lemmas and proofs). The goal is to get all the bubbles leading up to and including the “<a href="https://teorth.github.io/pfr/blueprint/sect0007.html#pfr">pfr</a>” bubble at the bottom colored in green.
In this post I would like to give a quick “tour” of the project, to give a sense of how it operates. If one clicks on the “<a href="https://teorth.github.io/pfr/blueprint/sect0007.html#pfr">pfr</a>” bubble at the bottom of the dependency graph, we get the following:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-1.png"><img loading="lazy" width="1024" height="265" data-attachment-id="14064" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-1/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-1.png" data-orig-size="2390,620" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-1" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-1.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-1.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-1.png?w=1024" alt="" class="wp-image-14064" srcset="https://terrytao.files.wordpress.com/2023/11/image-1.png?w=1022 1022w, https://terrytao.files.wordpress.com/2023/11/image-1.png?w=2043 2043w, https://terrytao.files.wordpress.com/2023/11/image-1.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-1.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-1.png?w=768 768w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Here, Blueprint is displaying a human-readable form of the PFR statement. This is coming from the <a href="https://teorth.github.io/pfr/blueprint/sect0007.html#pfr">corresponding portion of the blueprint</a>, which also comes with a human-readable proof of this statement that relies on other statements in the project:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-2.png"><img loading="lazy" width="1024" height="449" data-attachment-id="14066" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-2.png" data-orig-size="1744,765" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-2" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-2.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-2.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-2.png?w=1024" alt="" class="wp-image-14066" srcset="https://terrytao.files.wordpress.com/2023/11/image-2.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-2.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-2.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-2.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-2.png 1744w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
(I have cropped out the second half of the proof here, as it is not relevant to the discussion.)
Observe that the “<a href="https://teorth.github.io/pfr/blueprint/sect0007.html#pfr">pfr</a>” bubble is white, but has a green border. This means that the statement of PFR has been formalized in Lean, but not the proof; and the proof itself is not ready to be formalized, because some of the prerequisites (in particular, “<a href="https://teorth.github.io/pfr/blueprint/sect0006.html#entropy-pfr">entropy-pfr</a>” (Theorem 6.16)) do not even have their statements formalized yet. If we click on the “Lean” link below the description of PFR in the dependency graph, we are lead to the (auto-generated) <a href="https://teorth.github.io/pfr/docs/PFR/main.html#PFR_conjecture">Lean documentation for this assertion</a>:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-3.png"><img loading="lazy" width="1024" height="281" data-attachment-id="14068" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-3/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-3.png" data-orig-size="1533,421" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-3" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-3.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-3.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-3.png?w=1024" alt="" class="wp-image-14068" srcset="https://terrytao.files.wordpress.com/2023/11/image-3.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-3.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-3.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-3.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-3.png 1533w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
This is what a typical theorem in Lean looks like (after a procedure known as “pretty printing”). There are a number of hypotheses stated before the colon, for instance that <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" /> is a finite elementary abelian group of order <img src="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="2" class="latex" /> (this is how we have chosen to formalize the finite field vector spaces <img src="https://s0.wp.com/latex.php?latex=%7B%5Cbf+F%7D_2%5En&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cbf+F%7D_2%5En&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cbf+F%7D_2%5En&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="{\bf F}_2^n" class="latex" />), that <img src="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="A" class="latex" /> is a non-empty subset of <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" /> (the hypothesis that <img src="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="A" class="latex" /> is non-empty was not stated in the LaTeX version of the conjecture, but we realized it was necessary in the formalization, and will update the LaTeX blueprint shortly to reflect this) with the cardinality of <img src="https://s0.wp.com/latex.php?latex=A%2BA&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A%2BA&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A%2BA&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="A+A" class="latex" /> less than <img src="https://s0.wp.com/latex.php?latex=K&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=K&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=K&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="K" class="latex" /> times the cardinality of <img src="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="A" class="latex" />, and the statement after the colon is the conclusion: that <img src="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="A" class="latex" /> can be contained in the sum <img src="https://s0.wp.com/latex.php?latex=c%2BH&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=c%2BH&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=c%2BH&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="c+H" class="latex" /> of a subgroup <img src="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="H" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" /> and a set <img src="https://s0.wp.com/latex.php?latex=c&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=c&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=c&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="c" class="latex" /> of cardinality at most <img src="https://s0.wp.com/latex.php?latex=2K%5E%7B12%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=2K%5E%7B12%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=2K%5E%7B12%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="2K^{12}" class="latex" />.
The astute reader may notice that the above theorem seems to be missing one or two details, for instance it does not explicitly assert that <img src="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="H" class="latex" /> is a subgroup. This is because the “pretty printing” suppresses some of the information in the actual statement of the theorem, which can be seen by clicking on the “<a href="https://github.com/teorth/pfr/blob/6b2c357dc922ff00dec5dd6873b841925b0e22ef//PFR/main.lean#L17-L21">Source</a>” link:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-4.png"><img loading="lazy" width="1024" height="174" data-attachment-id="14072" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-4/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-4.png" data-orig-size="1737,296" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-4" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-4.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-4.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-4.png?w=1024" alt="" class="wp-image-14072" srcset="https://terrytao.files.wordpress.com/2023/11/image-4.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-4.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-4.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-4.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-4.png 1737w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Here we see that <img src="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="H" class="latex" /> is required to have the “type” of an additive subgroup of <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" />. (Lean’s language revolves very strongly around <a href="https://en.wikipedia.org/wiki/Type_theory">types</a>, but for this tour we will not go into detail into what a type is exactly.) The prominent “sorry” at the bottom of this theorem asserts that a proof is not yet provided for this theorem, but the intention of course is to replace this “sorry” with an actual proof eventually.
Filling in this “sorry” is too hard to do right now, so let’s look for a simpler task to accomplish instead. Here is a simple intermediate lemma “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-nonneg">ruzsa-nonneg</a>” that shows up in the proof:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-5.png"><img loading="lazy" width="1024" height="371" data-attachment-id="14075" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-5/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-5.png" data-orig-size="1647,597" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-5" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-5.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-5.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-5.png?w=1024" alt="" class="wp-image-14075" srcset="https://terrytao.files.wordpress.com/2023/11/image-5.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-5.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-5.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-5.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-5.png 1647w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
The expression <img src="https://s0.wp.com/latex.php?latex=d%5BX%3B+Y%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=d%5BX%3B+Y%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=d%5BX%3B+Y%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="d[X; Y]" class="latex" /> refers to something called the entropic Ruzsa distance between <img src="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=Y&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Y&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Y&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="Y" class="latex" />, which is something that is defined <a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruz-dist-def">elsewhere in the project</a>, but for the current discussion it is not important to know its precise definition, other than that it is a real number. The bubble is blue with a green border, which means that the statement has been formalized, and the proof is ready to be formalized also. The blueprint dependency graph indicates that this lemma can be deduced from just one preceding lemma, called “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-diff">ruzsa-diff</a>“:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-6.png"><img loading="lazy" width="1024" height="333" data-attachment-id="14078" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-6/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-6.png" data-orig-size="1716,559" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-6" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-6.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-6.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-6.png?w=1024" alt="" class="wp-image-14078" srcset="https://terrytao.files.wordpress.com/2023/11/image-6.png?w=1022 1022w, https://terrytao.files.wordpress.com/2023/11/image-6.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-6.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-6.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-6.png 1716w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
“<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-diff">ruzsa-diff</a>” is also blue and bordered in green, so it has the same current status as “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-nonneg">ruzsa-nonneg</a>“: the statement is formalized, and the proof is ready to be formalized also, but the proof has not been written in Lean yet. The quantity <img src="https://s0.wp.com/latex.php?latex=H%5BX%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H%5BX%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H%5BX%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="H[X]" class="latex" />, by the way, refers to the <a href="https://en.wikipedia.org/wiki/Entropy_(information_theory)">Shannon entropy</a> of <img src="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X" class="latex" />, defined <a href="https://teorth.github.io/pfr/blueprint/sect0002.html#entropy-def">elsewhere in the project</a>, but for this discussion we do not need to know its definition, other than to know that it is a real number.
Looking at Lemma 3.11 and Lemma 3.13 it is clear how the former will imply the latter: the quantity <img src="https://s0.wp.com/latex.php?latex=%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="|H[X] - H[Y]|" class="latex" /> is clearly non-negative! (There is a factor of <img src="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="2" class="latex" /> present in Lemma 3.11, but it can be easily canceled out.) So it should be an easy task to fill in the proof of Lemma 3.13 assuming Lemma 3.11, even if we still don’t know how to prove Lemma 3.11 yet. Let’s first look at the Lean code for each lemma. Lemma 3.11 is formalized <a href="https://github.com/teorth/pfr/blob/6b2c357dc922ff00dec5dd6873b841925b0e22ef//PFR/ruzsa_distance.lean#L118-L118">as follows</a>:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-7.png"><img loading="lazy" width="1024" height="80" data-attachment-id="14082" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-7/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-7.png" data-orig-size="1185,93" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-7" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-7.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-7.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-7.png?w=1024" alt="" class="wp-image-14082" srcset="https://terrytao.files.wordpress.com/2023/11/image-7.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-7.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-7.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-7.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-7.png 1185w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Again we have a “sorry” to indicate that this lemma does not currently have a proof. The Lean notation (as well as the name of the lemma) differs a little from the LaTeX version for technical reasons that we will not go into here. (Also, the variables <img src="https://s0.wp.com/latex.php?latex=X%2C+%5Cmu%2C+Y%2C+%5Cmu%27&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X%2C+%5Cmu%2C+Y%2C+%5Cmu%27&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X%2C+%5Cmu%2C+Y%2C+%5Cmu%27&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X, \mu, Y, \mu'" class="latex" /> are introduced at an earlier stage in the Lean file; again, we will ignore this point for the ensuing discussion.) Meanwhile, Lemma 3.13 is <a href="https://github.com/teorth/pfr/blob/6b2c357dc922ff00dec5dd6873b841925b0e22ef//PFR/ruzsa_distance.lean#L127-L128">currently formalized as</a>
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-8.png"><img loading="lazy" width="822" height="108" data-attachment-id="14085" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-8/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-8.png" data-orig-size="822,108" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-8" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-8.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-8.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-8.png?w=822" alt="" class="wp-image-14085" srcset="https://terrytao.files.wordpress.com/2023/11/image-8.png 822w, https://terrytao.files.wordpress.com/2023/11/image-8.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-8.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-8.png?w=768 768w" sizes="(max-width: 822px) 100vw, 822px" /></a></figure>
OK, I’m now going to try to fill in the latter “sorry”. In my local copy of the <a href="https://github.com/teorth/pfr">PFR github repository</a>, I open up the <a href="https://github.com/teorth/pfr/blob/master/PFR/ruzsa_distance.lean">relevant Lean file</a> in my editor (<a href="https://code.visualstudio.com/">Visual Studio Code</a>, with the <a href="https://github.com/leanprover/vscode-lean4">lean4 extension</a>) and navigate to the “sorry” of “rdist_nonneg”. The accompanying “Lean infoview” then shows the current state of the Lean proof:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-9.png"><img loading="lazy" width="605" height="1024" data-attachment-id="14087" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-9/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-9.png" data-orig-size="614,1040" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-9" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-9.png?w=177" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-9.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-9.png?w=605" alt="" class="wp-image-14087" srcset="https://terrytao.files.wordpress.com/2023/11/image-9.png?w=605 605w, https://terrytao.files.wordpress.com/2023/11/image-9.png?w=89 89w, https://terrytao.files.wordpress.com/2023/11/image-9.png?w=177 177w, https://terrytao.files.wordpress.com/2023/11/image-9.png 614w" sizes="(max-width: 605px) 100vw, 605px" /></a></figure>
Here we see a number of ambient hypotheses (e.g., that <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" /> is an additive commutative group, that <img src="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X" class="latex" /> is a map from <img src="https://s0.wp.com/latex.php?latex=%5COmega&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5COmega&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5COmega&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\Omega" class="latex" /> to <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" />, and so forth; many of these hypotheses are not actually relevant for this particular lemma), and at the bottom we see the goal we wish to prove.
OK, so now I’ll try to prove the claim. This is accomplished by applying a series of “tactics” to transform the goal and/or hypotheses. The first step I’ll do is to put in the factor of <img src="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="2" class="latex" /> that is needed to apply Lemma 3.11. This I will do with the “suffices” tactic, writing in the proof
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-10.png"><img loading="lazy" width="812" height="204" data-attachment-id="14090" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-10/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-10.png" data-orig-size="812,204" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-10" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-10.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-10.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-10.png?w=812" alt="" class="wp-image-14090" srcset="https://terrytao.files.wordpress.com/2023/11/image-10.png 812w, https://terrytao.files.wordpress.com/2023/11/image-10.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-10.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-10.png?w=768 768w" sizes="(max-width: 812px) 100vw, 812px" /></a></figure>
I now have two goals (and two “sorries”): one to show that <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq 2 d[X;Y]" class="latex" /> implies <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+d%5BX%2CY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+d%5BX%2CY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+d%5BX%2CY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq d[X,Y]" class="latex" />, and the other to show that <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq 2 d[X;Y]" class="latex" />. (The yellow squiggly underline indicates that this lemma has not been fully proven yet due to the presence of “sorry”s. The dot “.” is a syntactic marker that is useful to separate the two goals from each other, but you can ignore it for this tour.) The Lean tactic “suffices” corresponds, roughly speaking, to the phrase “It suffices to show that …” (or more precisely, “It suffices to show that … . To see this, … . It remains to verify the claim …”) in Mathematical English. For my own education, I wrote a “Lean phrasebook” of further correspondences between lines of Lean code and sentences or phrases in Mathematical English, which can be found <a href="https://docs.google.com/spreadsheets/d/1Gsn5al4hlpNc_xKoXdU6XGmMyLiX4q-LFesFVsMlANo/edit#gid=0">here</a>.
Let’s fill in the first “sorry”. The tactic state now looks like this (cropping out some irrelevant hypotheses):
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-11.png"><img loading="lazy" width="532" height="206" data-attachment-id="14093" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-11/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-11.png" data-orig-size="532,206" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-11" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-11.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-11.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-11.png?w=532" alt="" class="wp-image-14093" srcset="https://terrytao.files.wordpress.com/2023/11/image-11.png 532w, https://terrytao.files.wordpress.com/2023/11/image-11.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-11.png?w=300 300w" sizes="(max-width: 532px) 100vw, 532px" /></a></figure>
Here I can use a handy tactic “<a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#linarith">linarith</a>“, which solves any goal that can be derived by linear arithmetic from existing hypotheses:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-12.png"><img loading="lazy" width="829" height="218" data-attachment-id="14095" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-12/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-12.png" data-orig-size="829,218" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-12" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-12.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-12.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-12.png?w=829" alt="" class="wp-image-14095" srcset="https://terrytao.files.wordpress.com/2023/11/image-12.png 829w, https://terrytao.files.wordpress.com/2023/11/image-12.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-12.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-12.png?w=768 768w" sizes="(max-width: 829px) 100vw, 829px" /></a></figure>
This works, and now the tactic state reports no goals left to prove on this branch, so we move on to the remaining sorry, in which the goal is now to prove <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq 2 d[X;Y]" class="latex" />:
<figure class="wp-block-image size-large is-resized"><a href="https://terrytao.files.wordpress.com/2023/11/image-13.png"><img loading="lazy" width="507" height="177" data-attachment-id="14097" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-13/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-13.png" data-orig-size="507,177" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-13" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-13.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-13.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-13.png?w=507" alt="" class="wp-image-14097" style="width:840px;height:auto" srcset="https://terrytao.files.wordpress.com/2023/11/image-13.png 507w, https://terrytao.files.wordpress.com/2023/11/image-13.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-13.png?w=300 300w" sizes="(max-width: 507px) 100vw, 507px" /></a></figure>
Here we will try to invoke Lemma 3.11. I add the following lines of code:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-14.png"><img loading="lazy" width="1024" height="301" data-attachment-id="14099" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-14/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-14.png" data-orig-size="1042,307" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-14" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-14.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-14.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-14.png?w=1024" alt="" class="wp-image-14099" srcset="https://terrytao.files.wordpress.com/2023/11/image-14.png?w=1022 1022w, https://terrytao.files.wordpress.com/2023/11/image-14.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-14.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-14.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-14.png 1042w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
The Lean tactic “<a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#have">have</a>” roughly corresponds to the Mathematical English phrase “We have the statement…” or “We claim the statement…”; like “suffices”, it splits a goal into two subgoals, though in the reversed order to “suffices”.
I again have two subgoals, one to prove the bound <img src="https://s0.wp.com/latex.php?latex=%7CH%5BX%5D-H%5BY%5D%7C+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7CH%5BX%5D-H%5BY%5D%7C+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7CH%5BX%5D-H%5BY%5D%7C+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="|H[X]-H[Y]| \leq 2 d[X;Y]" class="latex" /> (which I will call “h”), and then to deduce the previous goal <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq 2 d[X;Y]" class="latex" /> from <img src="https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="h" class="latex" />. For the first, I know I should invoke the lemma “diff_ent_le_rdist” that is encoding Lemma 3.11. One way to do this is to try the tactic “exact?”, which will automatically search to see if the goal can already be deduced immediately from an existing lemma. It reports:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-15.png"><img loading="lazy" width="553" height="127" data-attachment-id="14101" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-15/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-15.png" data-orig-size="553,127" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-15" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-15.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-15.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-15.png?w=553" alt="" class="wp-image-14101" srcset="https://terrytao.files.wordpress.com/2023/11/image-15.png 553w, https://terrytao.files.wordpress.com/2023/11/image-15.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-15.png?w=300 300w" sizes="(max-width: 553px) 100vw, 553px" /></a></figure>
So I try this (by clicking on the suggested code, which automatically pastes it into the right location), and it works, leaving me with the final “sorry”:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-16.png"><img loading="lazy" width="1020" height="279" data-attachment-id="14103" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-16/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-16.png" data-orig-size="1020,279" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-16" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-16.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-16.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-16.png?w=1020" alt="" class="wp-image-14103" srcset="https://terrytao.files.wordpress.com/2023/11/image-16.png 1020w, https://terrytao.files.wordpress.com/2023/11/image-16.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-16.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-16.png?w=768 768w" sizes="(max-width: 1020px) 100vw, 1020px" /></a></figure>
The lean tactic “<a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#exact">exact</a>” corresponds, roughly speaking, to the Mathematical English phrase “But this is exactly …”.
At this point I should mention that I also have the <a href="https://github.com/features/copilot">Github Copilot</a> extension to Visual Studio Code installed. This is an AI which acts as an advanced autocomplete that can suggest possible lines of code as one types. In this case, it offered a suggestion which was almost correct (the second line is what we need, whereas the first is not necessary, and in fact does not even compile in Lean):
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-25.png"><img loading="lazy" width="1024" height="276" data-attachment-id="14135" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-25/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-25.png" data-orig-size="1193,322" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-25" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-25.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-25.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-25.png?w=1024" alt="" class="wp-image-14135" srcset="https://terrytao.files.wordpress.com/2023/11/image-25.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-25.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-25.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-25.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-25.png 1193w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
In any event, “exact?” worked in this case, so I can ignore the suggestion of Copilot this time (it has been very useful in other cases though). I apply the “exact?” tactic a second time and follow its suggestion to establish the matching bound <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq |H[X] - H[Y]|" class="latex" />:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-17.png"><img loading="lazy" width="1013" height="354" data-attachment-id="14105" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-17/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-17.png" data-orig-size="1013,354" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-17" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-17.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-17.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-17.png?w=1013" alt="" class="wp-image-14105" srcset="https://terrytao.files.wordpress.com/2023/11/image-17.png 1013w, https://terrytao.files.wordpress.com/2023/11/image-17.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-17.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-17.png?w=768 768w" sizes="(max-width: 1013px) 100vw, 1013px" /></a></figure>
(One can find documention for the “abs_nonneg” method <a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Abs.html#abs_nonneg">here</a>. Copilot, by the way, was also able to resolve this step, albeit with a slightly different syntax; there are also several other search engines available to locate this method as well, such as <a href="https://moogle-morphlabs.vercel.app/search/raw?q=absolute%20value%20is%20nonnegative">Moogle</a>. One of the main purposes of the <a href="https://leanprover-community.github.io/contribute/naming.html">Lean naming conventions for lemmas</a>, by the way, is to facilitate the location of methods such as “abs_nonneg”, which is easier figure out how to search for than a method named (say) “Lemma 1.2.1”.) To fill in the final “sorry”, I try “exact?” one last time, to figure out how to combine <img src="https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="h" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=h%27&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=h%27&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=h%27&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="h'" class="latex" /> to give the desired goal, and it works!
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-18.png"><img loading="lazy" width="1024" height="362" data-attachment-id="14107" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-18/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-18.png" data-orig-size="1027,364" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-18" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-18.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-18.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-18.png?w=1024" alt="" class="wp-image-14107" srcset="https://terrytao.files.wordpress.com/2023/11/image-18.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-18.png?w=1021 1021w, https://terrytao.files.wordpress.com/2023/11/image-18.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-18.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-18.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-18.png 1027w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Note that all the squiggly underlines have disappeared, indicating that Lean has accepted this as a valid proof. The documentation for “ge_trans” may be found <a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Order/Defs.html#ge_trans">here</a>. The reader may observe that this method uses the <img src="https://s0.wp.com/latex.php?latex=%5Cgeq&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cgeq&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cgeq&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\geq" class="latex" /> relation rather than the <img src="https://s0.wp.com/latex.php?latex=%5Cleq&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cleq&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cleq&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\leq" class="latex" /> relation, but in Lean the assertions <img src="https://s0.wp.com/latex.php?latex=X+%5Cgeq+Y&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X+%5Cgeq+Y&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X+%5Cgeq+Y&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X \geq Y" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=Y+%5Cleq+X&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Y+%5Cleq+X&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Y+%5Cleq+X&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="Y \leq X" class="latex" /> are “<a href="https://www.ma.imperial.ac.uk/~buzzard/xena/formalising-mathematics-2022/Part_B/equality.html#:~:text=Definitional%20equality,-Definitional%20equality%20is&text=In%20Lean%2C%20%C2%ACP%20is,equal%2C%20they%20are%20definitionally%20equal.">definitionally equal</a>“, allowing tactics such as “exact” to use them interchangeably. “exact <a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Order/Defs.html#le_trans">le_trans</a> h’ h” would also have worked in this instance.
It is possible to compactify this proof quite a bit by cutting out several intermediate steps (a procedure sometimes known as “<a href="https://en.wikipedia.org/wiki/Code_golf">code golf</a>“):
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-19.png"><img loading="lazy" width="1024" height="130" data-attachment-id="14109" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-19/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-19.png" data-orig-size="1190,152" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-19" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-19.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-19.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-19.png?w=1024" alt="" class="wp-image-14109" srcset="https://terrytao.files.wordpress.com/2023/11/image-19.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-19.png?w=1018 1018w, https://terrytao.files.wordpress.com/2023/11/image-19.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-19.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-19.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-19.png 1190w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
And now the proof is done! In the end, it was literally a “one-line proof”, which makes sense given how close Lemma 3.11 and Lemma 3.13 were to each other.
The current version of Blueprint does not automatically verify the proof (even though it does compile in Lean), so we have to manually update the blueprint as well. The LaTeX for Lemma 3.13 <a href="https://github.com/teorth/pfr/blob/master/blueprint/src/chapter/distance.tex">currently looks like this:</a>
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-20.png"><img loading="lazy" width="1024" height="309" data-attachment-id="14111" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-20/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-20.png" data-orig-size="1130,341" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-20" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-20.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-20.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-20.png?w=1024" alt="" class="wp-image-14111" srcset="https://terrytao.files.wordpress.com/2023/11/image-20.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-20.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-20.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-20.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-20.png 1130w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
I add the “\leanok” macro to the proof, to flag that the proof has now been formalized:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-22.png"><img loading="lazy" width="1024" height="284" data-attachment-id="14114" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-22/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-22.png" data-orig-size="1217,338" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-22" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-22.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-22.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-22.png?w=1024" alt="" class="wp-image-14114" srcset="https://terrytao.files.wordpress.com/2023/11/image-22.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-22.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-22.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-22.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-22.png 1217w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
I then push everything back up to the master Github repository. The blueprint will take quite some time (about half an hour) to rebuild, but eventually it does, and the dependency graph (which Blueprint has for some reason decided to rearrange a bit) now shows “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-nonneg">ruzsa-nonneg</a>” in green:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-23.png"><img loading="lazy" width="1024" height="443" data-attachment-id="14121" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-23/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-23.png" data-orig-size="1643,711" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-23" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-23.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-23.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-23.png?w=1024" alt="" class="wp-image-14121" srcset="https://terrytao.files.wordpress.com/2023/11/image-23.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-23.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-23.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-23.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-23.png 1643w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
And so the formalization of PFR moves a little bit closer to completion. (Of course, this was a particularly easy lemma to formalize, that I chose to illustrate the process; one can imagine that most other lemmas will take a bit more work.) Note that while “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-nonneg">ruzsa-nonneg</a>” is now colored in green, we don’t yet have a full proof of this result, because the lemma “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-diff">ruzsa-diff</a>” that it relies on is not green. Nevertheless, the proof is locally complete at this point; hopefully at some point in the future, the predecessor results will also be locally proven, at which point this result will be completely proven. Note how this blueprint structure allows one to work on different parts of the proof asynchronously; it is not necessary to wait for earlier stages of the argument to be fully formalized to start working on later stages, although I anticipate a small amount of interaction between different components as we iron out any bugs or slight inaccuracies in the blueprint. (For instance, I am suspecting that we may need to add some measurability hypotheses on the random variables <img src="https://s0.wp.com/latex.php?latex=X%2C+Y&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X%2C+Y&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X%2C+Y&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X, Y" class="latex" /> in the above two lemmas to make them completely true, but this is something that should emerge organically as the formalization process continues.)
That concludes the brief tour! If you are interested in learning more about the project, you can follow the <a href="https://leanprover.zulipchat.com/#narrow/stream/412902-Polynomial-Freiman-Ruzsa-conjecture">Zulip chat stream</a>; you can also <a href="https://leanprover-community.github.io/get_started.html">download Lean</a> and <a href="https://leanprover-community.github.io/install/project.html">work on the PFR project yourself</a>, using a local copy of the Github repository and sending pull requests to the master copy if you have managed to fill in one or more of the “sorry”s in the current version (but if you plan to work on anything more large scale than filling in a small lemma, it is good to announce your intention on the Zulip chat to avoid duplication of effort) . (One key advantage of working with a project based around a proof assistant language such as Lean is that it makes large-scale mathematical collaboration possible without necessarily having a pre-established level of trust amongst the collaborators; my fellow repository maintainers and I have already approved several pull requests from contributors that had not previously met, as the code was verified to be correct and we could see that it advanced the project. Conversely, as the above example should hopefully demonstrate, it is possible for a contributor to work on one small corner of the project without necessarily needing to understand all the mathematics that goes into the project as a whole.)
If one just wants to experiment with Lean without going to the effort of downloading it, you can playing try the “<a href="https://adam.math.hhu.de/">Natural Number Game</a>” for a gentle introduction to the language, or the <a href="https://live.lean-lang.org/">Lean4 playground</a> for an online Lean server. Further resources to learn Lean4 may be found <a href="https://leanprover-community.github.io/learn.html">here</a>.
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<title>On a conjecture of Marton</title>
<link>https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/</link>
<comments>https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Mon, 13 Nov 2023 17:41:05 +0000</pubDate>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[paper]]></category>
<category><![CDATA[additive combinatorics]]></category>
<category><![CDATA[Ben Green]]></category>
<category><![CDATA[Freddie Manners]]></category>
<category><![CDATA[Freiman's theorem]]></category>
<category><![CDATA[Polynomial Freiman-Ruzsa conjecture]]></category>
<category><![CDATA[Shannon entropy]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14046</guid>
<description><![CDATA[Tim Gowers, Ben Green, Freddie Manners, and I have just uploaded to the arXiv our paper “On a conjecture of Marton“. This paper establishes a version of the notorious Polynomial Freiman–Ruzsa conjecture (first proposed by Katalin Marton): Theorem 1 (Polynomial Freiman–Ruzsa conjecture) Let be such that . Then can be covered by at most translates […]]]></description>
<content:encoded><![CDATA[

<a href="https://www.dpmms.cam.ac.uk/~wtg10/">Tim Gowers</a>, <a href="https://people.maths.ox.ac.uk/greenbj/">Ben Green</a>, <a href="https://mathweb.ucsd.edu/~fmanners/">Freddie Manners</a>, and I have just uploaded to the arXiv our paper “<a href="https://arxiv.org/abs/2311.05762">On a conjecture of Marton</a>“. This paper establishes a version of the notorious <a href="https://terrytao.wordpress.com/2007/03/11/ben-green-the-polynomial-freiman-ruzsa-conjecture/">Polynomial Freiman–Ruzsa conjecture</a> (first proposed by Katalin Marton):

<blockquote>Theorem 1 (Polynomial Freiman–Ruzsa conjecture) Let <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \subset {\bf F}_2^n}" class="latex" /> be such that <img src="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A+A| \leq K|A|}" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> can be covered by at most <img src="https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2K^{12}}" class="latex" /> translates of a subspace <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_2^n}" class="latex" /> of cardinality at most <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" />. </blockquote>


The previous best known result towards this conjecture was by Konyagin (as communicated in <a HREF="https://zbmath.org/1337.11014">this paper of Sanders</a>), who obtained a similar result but with <img src="https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2K^{12}}" class="latex" /> replaced by <img src="https://s0.wp.com/latex.php?latex=%7B%5Cexp%28O_%5Cvarepsilon%28%5Clog%5E%7B3%2B%5Cvarepsilon%7D+K%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cexp%28O_%5Cvarepsilon%28%5Clog%5E%7B3%2B%5Cvarepsilon%7D+K%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cexp%28O_%5Cvarepsilon%28%5Clog%5E%7B3%2B%5Cvarepsilon%7D+K%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\exp(O_\varepsilon(\log^{3+\varepsilon} K))}" class="latex" /> for any <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon>0}" class="latex" /> (assuming that say <img src="https://s0.wp.com/latex.php?latex=%7BK+%5Cgeq+3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK+%5Cgeq+3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK+%5Cgeq+3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K \geq 3/2}" class="latex" /> to avoid some degeneracies as <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> approaches <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" />, which is not the difficult case of the conjecture). The conjecture (with <img src="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{12}" class="latex" /> replaced by an unspecified constant <img src="https://s0.wp.com/latex.php?latex=%7BC%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BC%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BC%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{C}" class="latex" />) has a number of equivalent forms; see this <a href="https://zbmath.org/1155.11306">survey of Green</a>, and these papers <a href="https://zbmath.org/1274.11158">of Lovett</a> and <a href="https://zbmath.org/1229.11132">of Green and myself</a> for some examples; in particular, as discussed in the latter two references, the constants in the inverse <img src="https://s0.wp.com/latex.php?latex=%7BU%5E3%28%7B%5Cbf+F%7D_2%5En%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU%5E3%28%7B%5Cbf+F%7D_2%5En%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU%5E3%28%7B%5Cbf+F%7D_2%5En%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U^3({\bf F}_2^n)}" class="latex" /> theorem are now polynomial in nature (although we did not try to optimize the constant).

The exponent <img src="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{12}" class="latex" /> here was the product of a large number of optimizations to the argument (our original exponent here was closer to <img src="https://s0.wp.com/latex.php?latex=%7B1000%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1000%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1000%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1000}" class="latex" />), but can be improved even further with additional effort (our current argument, for instance, allows one to replace it with <img src="https://s0.wp.com/latex.php?latex=%7B7%2B%5Csqrt%7B17%7D+%3D+11.123%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B7%2B%5Csqrt%7B17%7D+%3D+11.123%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B7%2B%5Csqrt%7B17%7D+%3D+11.123%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{7+\sqrt{17} = 11.123\dots}" class="latex" />, but we decided to state our result using integer exponents instead).

In this paper we will focus exclusively on the characteristic <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" /> case (so we will be cavalier in identifying addition and subtraction), but in a followup paper we will establish similar results in other finite characteristics.

Much of the prior progress on this sort of result has proceeded via Fourier analysis. Perhaps surprisingly, our approach uses no Fourier analysis whatsoever, being conducted instead entirely in “physical space”. Broadly speaking, it follows a natural strategy, which is to induct on the doubling constant <img src="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C%2F%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C%2F%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C%2F%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A+A|/|A|}" class="latex" />. Indeed, suppose for instance that one could show that every set <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> of doubling constant <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> was “commensurate” in some sense to a set <img src="https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A'}" class="latex" /> of doubling constant at most <img src="https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K^{0.99}}" class="latex" />. One measure of commensurability, for instance, might be the Ruzsa distance <img src="https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%27%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CA%27%7C%5E%7B1%2F2%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%27%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CA%27%7C%5E%7B1%2F2%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%27%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CA%27%7C%5E%7B1%2F2%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\log \frac{|A+A'|}{|A|^{1/2} |A'|^{1/2}}}" class="latex" />, which one might hope to control by <img src="https://s0.wp.com/latex.php?latex=%7BO%28%5Clog+K%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28%5Clog+K%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28%5Clog+K%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(\log K)}" class="latex" />. Then one could iterate this procedure until doubling constant dropped below say <img src="https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{3/2}" class="latex" />, at which point the conjecture is known to hold (there is an elementary argument that if <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> has doubling constant less than <img src="https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{3/2}" class="latex" />, then <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" /> is in fact a subspace of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_2^n}" class="latex" />). One can then use several applications of the Ruzsa triangle inequality <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Cfrac%7B%7CA%2BC%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D+%5Cleq+%5Clog+%5Cfrac%7B%7CA%2BB%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CB%7C%5E%7B1%2F2%7D%7D+%2B+%5Clog+%5Cfrac%7B%7CB%2BC%7C%7D%7B%7CB%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Cfrac%7B%7CA%2BC%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D+%5Cleq+%5Clog+%5Cfrac%7B%7CA%2BB%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CB%7C%5E%7B1%2F2%7D%7D+%2B+%5Clog+%5Cfrac%7B%7CB%2BC%7C%7D%7B%7CB%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Cfrac%7B%7CA%2BC%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D+%5Cleq+%5Clog+%5Cfrac%7B%7CA%2BB%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CB%7C%5E%7B1%2F2%7D%7D+%2B+%5Clog+%5Cfrac%7B%7CB%2BC%7C%7D%7B%7CB%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \log \frac{|A+C|}{|A|^{1/2} |C|^{1/2}} \leq \log \frac{|A+B|}{|A|^{1/2} |B|^{1/2}} + \log \frac{|B+C|}{|B|^{1/2} |C|^{1/2}}" class="latex" />
to conclude (the fact that we reduce <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> to <img src="https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K^{0.99}}" class="latex" /> means that the various Ruzsa distances that need to be summed are controlled by a convergent geometric series).

There are a number of possible ways to try to “improve” a set <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> of not too large doubling by replacing it with a commensurate set of better doubling. We note two particular potential improvements:

<ul> <li>(i) Replacing <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" />. For instance, if <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> was a random subset (of density <img src="https://s0.wp.com/latex.php?latex=%7B1%2FK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2FK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2FK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1/K}" class="latex" />) of a large subspace <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_2^n}" class="latex" />, then replacing <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" /> usually drops the doubling constant from <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> down to nearly <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> (under reasonable choices of parameters). </li><li>(ii) Replacing <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" /> for a “typical” <img src="https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h \in A+A}" class="latex" />. For instance, if <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> was the union of <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> random cosets of a subspace <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of large codimension, then replacing <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" /> again usually drops the doubling constant from <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> down to nearly <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" />.
</li></ul>

Unfortunately, there are sets <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> where neither of the above two operations (i), (ii) significantly improves the doubling constant. For instance, if <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> is a random density <img src="https://s0.wp.com/latex.php?latex=%7B1%2F%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2F%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2F%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1/\sqrt{K}}" class="latex" /> subset of <img src="https://s0.wp.com/latex.php?latex=%7B%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sqrt{K}}" class="latex" /> random translates of a medium-sized subspace <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" />, one can check that the doubling constant stays close to <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> if one applies either operation (i) or operation (ii). But in this case these operations don’t actually worsen the doubling constant much either, and by applying some combination of (i) and (ii) (either intersecting <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" /> with a translate, or taking a sumset of <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" /> with itself) one can start lowering the doubling constant again.

This begins to suggest a potential strategy: show that at least one of the operations (i) or (ii) will improve the doubling constant, or at least not worsen it too much; and in the latter case, perform some more complicated operation to locate the desired doubling constant improvement.

A sign that this strategy might have a chance of working is provided by the following heuristic argument. If <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> has doubling constant <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" />, then the Cartesian product <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \times A}" class="latex" /> has doubling constant <img src="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K^2}" class="latex" />. On the other hand, by using the projection map <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+%7B%5Cbf+F%7D_2%5En+%5Ctimes+%7B%5Cbf+F%7D_2%5En+%5Crightarrow+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+%7B%5Cbf+F%7D_2%5En+%5Ctimes+%7B%5Cbf+F%7D_2%5En+%5Crightarrow+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+%7B%5Cbf+F%7D_2%5En+%5Ctimes+%7B%5Cbf+F%7D_2%5En+%5Crightarrow+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi: {\bf F}_2^n \times {\bf F}_2^n \rightarrow {\bf F}_2^n}" class="latex" /> defined by <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28x%2Cy%29+%3A%3D+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28x%2Cy%29+%3A%3D+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%28x%2Cy%29+%3A%3D+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi(x,y) := x+y}" class="latex" />, we see that <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \times A}" class="latex" /> projects to <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28A+%5Ctimes+A%29+%3D+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28A+%5Ctimes+A%29+%3D+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%28A+%5Ctimes+A%29+%3D+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi(A \times A) = A+A}" class="latex" />, with fibres <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%5E%7B-1%7D%28%5C%7Bh%5C%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%5E%7B-1%7D%28%5C%7Bh%5C%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%5E%7B-1%7D%28%5C%7Bh%5C%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi^{-1}(\{h\})}" class="latex" /> being essentially a copy of <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" />. So, morally, <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \times A}" class="latex" /> also behaves like a “skew product” of <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" /> and the fibres <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" />, which suggests (non-rigorously) that the doubling constant <img src="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K^2}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \times A}" class="latex" /> is also something like the doubling constant of <img src="https://s0.wp.com/latex.php?latex=%7BA+%2B+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%2B+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%2B+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A + A}" class="latex" />, times the doubling constant of a typical fibre <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" />. This would imply that at least one of <img src="https://s0.wp.com/latex.php?latex=%7BA+%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A +A}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" /> would have doubling constant at most <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" />, and thus that at least one of operations (i), (ii) would not worsen the doubling constant.

Unfortunately, this argument does not seem to be easily made rigorous using the traditional doubling constant; even the significantly weaker statement that <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" /> has doubling constant at most <img src="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K^2}" class="latex" /> is false (see comments for more discussion). However, it turns out (as discussed in <a href="https://terrytao.wordpress.com/2023/06/27/sumsets-and-entropy-revisited/">this recent paper of myself with Green and Manners</a>) that things are much better. Here, the analogue of a subset <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_2^n}" class="latex" /> is a random variable <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> taking values in <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_2^n}" class="latex" />, and the analogue of the (logarithmic) doubling constant <img src="https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%7C%7D%7B%7CA%7C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%7C%7D%7B%7CA%7C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%7C%7D%7B%7CA%7C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\log \frac{|A+A|}{|A|}}" class="latex" /> is the entropic doubling constant <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3A%3D+%7B%5Cbf+H%7D%28X_1%2BX_2%29-%7B%5Cbf+H%7D%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3A%3D+%7B%5Cbf+H%7D%28X_1%2BX_2%29-%7B%5Cbf+H%7D%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3A%3D+%7B%5Cbf+H%7D%28X_1%2BX_2%29-%7B%5Cbf+H%7D%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X;X) := {\bf H}(X_1+X_2)-{\bf H}(X)}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1,X_2}" class="latex" /> are independent copies of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" />. If <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> is a random variable in some additive group <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+G+%5Crightarrow+H%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+G+%5Crightarrow+H%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+G+%5Crightarrow+H%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi: G \rightarrow H}" class="latex" /> is a homomorphism, one then has what we call the fibring inequality <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%3BX%29+%5Cgeq+d%28%5Cpi%28X%29%3B%5Cpi%28X%29%29+%2B+d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%3BX%29+%5Cgeq+d%28%5Cpi%28X%29%3B%5Cpi%28X%29%29+%2B+d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%3BX%29+%5Cgeq+d%28%5Cpi%28X%29%3B%5Cpi%28X%29%29+%2B+d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle d(X;X) \geq d(\pi(X);\pi(X)) + d(X|\pi(X); X|\pi(X))," class="latex" />
where the conditional doubling constant <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X|\pi(X); X|\pi(X))}" class="latex" /> is defined as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29+%3D+%7B%5Cbf+H%7D%28X_1+%2B+X_2+%7C+%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29+-+%7B%5Cbf+H%7D%28+X+%7C+%5Cpi%28X%29+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29+%3D+%7B%5Cbf+H%7D%28X_1+%2B+X_2+%7C+%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29+-+%7B%5Cbf+H%7D%28+X+%7C+%5Cpi%28X%29+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29+%3D+%7B%5Cbf+H%7D%28X_1+%2B+X_2+%7C+%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29+-+%7B%5Cbf+H%7D%28+X+%7C+%5Cpi%28X%29+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle d(X|\pi(X); X|\pi(X)) = {\bf H}(X_1 + X_2 | \pi(X_1), \pi(X_2)) - {\bf H}( X | \pi(X) )." class="latex" />
Indeed, from the chain rule for Shannon entropy one has <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X%29%29+%2B+%7B%5Cbf+H%7D%28X%7C%5Cpi%28X%29%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X%29%29+%2B+%7B%5Cbf+H%7D%28X%7C%5Cpi%28X%29%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X%29%29+%2B+%7B%5Cbf+H%7D%28X%7C%5Cpi%28X%29%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}(X) = {\bf H}(\pi(X)) + {\bf H}(X|\pi(X))" class="latex" />
and <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1%2BX_2%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%2B+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1%2BX_2%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%2B+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1%2BX_2%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%2B+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}(X_1+X_2) = {\bf H}(\pi(X_1) + \pi(X_2)) + {\bf H}(X_1 + X_2|\pi(X_1) + \pi(X_2))" class="latex" />
while from the non-negativity of conditional mutual information one has <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%5Cgeq+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%5Cgeq+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%5Cgeq+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}(X_1 + X_2|\pi(X_1) + \pi(X_2)) \geq {\bf H}(X_1 + X_2|\pi(X_1), \pi(X_2))" class="latex" />
and it is an easy matter to combine all these identities and inequalities to obtain the claim.

Applying this inequality with <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> replaced by two independent copies <img src="https://s0.wp.com/latex.php?latex=%7B%28X_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_1,X_2)}" class="latex" /> of itself, and using the addition map <img src="https://s0.wp.com/latex.php?latex=%7B%28x%2Cy%29+%5Cmapsto+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28x%2Cy%29+%5Cmapsto+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28x%2Cy%29+%5Cmapsto+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(x,y) \mapsto x+y}" class="latex" /> for <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi}" class="latex" />, we obtain in particular that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%2CX_2%7CX_1%2BX_2%3B+X_1%2CX_2%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%2CX_2%7CX_1%2BX_2%3B+X_1%2CX_2%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%2CX_2%7CX_1%2BX_2%3B+X_1%2CX_2%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 2 d(X;X) \geq d(X_1+X_2; X_1+X_2) + d(X_1,X_2|X_1+X_2; X_1,X_2|X_1+X_2)" class="latex" />
or (since <img src="https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_2}" class="latex" /> is determined by <img src="https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1}" class="latex" /> once one fixes <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2}" class="latex" />) <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 2 d(X;X) \geq d(X_1+X_2; X_1+X_2) + d(X_1|X_1+X_2; X_1|X_1+X_2)." class="latex" />
So if <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3D+%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3D+%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3D+%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X;X) = \log K}" class="latex" />, then at least one of <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%2BX_2%3B+X_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%2BX_2%3B+X_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X_1%2BX_2%3B+X_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X_1+X_2; X_1+X_2)}" class="latex" /> or <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X_1|X_1+X_2; X_1|X_1+X_2)}" class="latex" /> will be less than or equal to <img src="https://s0.wp.com/latex.php?latex=%7B%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\log K}" class="latex" />. This is the entropy analogue of at least one of (i) or (ii) improving, or at least not degrading the doubling constant, although there are some minor technicalities involving how one deals with the conditioning to <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2}" class="latex" /> in the second term <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X_1|X_1+X_2; X_1|X_1+X_2)}" class="latex" /> that we will gloss over here (one can pigeonhole the instances of <img src="https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1}" class="latex" /> to different events <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2=x}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2=x'}" class="latex" />, and “depolarise” the induction hypothesis to deal with distances <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X%3BY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X;Y)}" class="latex" /> between pairs of random variables <img src="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X,Y}" class="latex" /> that do not necessarily have the same distribution). Furthermore, we can even calculate the defect in the above inequality: a careful inspection of the above argument eventually reveals that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%3D+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%3D+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%3D+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 2 d(X;X) = d(X_1+X_2; X_1+X_2) + d(X_1|X_1+X_2; X_1|X_1+X_2)" class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%7B%5Cbf+I%7D%28+X_1+%2B+X_2+%3A+X_1+%2B+X_3+%7C+X_1+%2B+X_2+%2B+X_3+%2B+X_4%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%7B%5Cbf+I%7D%28+X_1+%2B+X_2+%3A+X_1+%2B+X_3+%7C+X_1+%2B+X_2+%2B+X_3+%2B+X_4%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%7B%5Cbf+I%7D%28+X_1+%2B+X_2+%3A+X_1+%2B+X_3+%7C+X_1+%2B+X_2+%2B+X_3+%2B+X_4%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + {\bf I}( X_1 + X_2 : X_1 + X_3 | X_1 + X_2 + X_3 + X_4)" class="latex" />
where we now take four independent copies <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%2CX_3%2CX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%2CX_3%2CX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%2CX_3%2CX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1,X_2,X_3,X_4}" class="latex" />. This leads (modulo some technicalities) to the following interesting conclusion: if neither (i) nor (ii) leads to an improvement in the entropic doubling constant, then <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BX_2%2BX_3%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_2%2BX_3%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_2%2BX_3%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_2+X_3}" class="latex" /> are conditionally independent relative to <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2+X_3+X_4}" class="latex" />. This situation (or an approximation to this situation) is what we refer to in the paper as the “endgame”.

A version of this endgame conclusion is in fact valid in any characteristic. But in characteristic <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" />, we can take advantage of the identity <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28X_1%2BX_2%29+%2B+%28X_2%2BX_3%29+%3D+X_1+%2B+X_3.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28X_1%2BX_2%29+%2B+%28X_2%2BX_3%29+%3D+X_1+%2B+X_3.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28X_1%2BX_2%29+%2B+%28X_2%2BX_3%29+%3D+X_1+%2B+X_3.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle (X_1+X_2) + (X_2+X_3) = X_1 + X_3." class="latex" />
Conditioning on <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2+X_3+X_4}" class="latex" />, and using symmetry we now conclude that if we are in the endgame exactly (so that the mutual information is zero), then the independent sum of two copies of <img src="https://s0.wp.com/latex.php?latex=%7B%28X_1%2BX_2%7CX_1%2BX_2%2BX_3%2BX_4%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_1%2BX_2%7CX_1%2BX_2%2BX_3%2BX_4%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_1%2BX_2%7CX_1%2BX_2%2BX_3%2BX_4%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_1+X_2|X_1+X_2+X_3+X_4)}" class="latex" /> has exactly the same distribution; in particular, the entropic doubling constant here is zero, which is certainly a reduction in the doubling constant.

To deal with the situation where the conditional mutual information is small but not completely zero, we have to use an entropic version of the Balog-Szemeredi-Gowers lemma, but fortunately this was already worked out in an <a href="https://zbmath.org/1239.11015">old paper of mine</a> (although in order to optimise the final constant, we ended up using a slight variant of that lemma).

I am planning to formalize this paper in the <a href="https://leanprover-community.github.io/index.html">Lean4 language</a>. Further discussion of this project will take place on <a href="https://leanprover.zulipchat.com/#narrow/stream/412902-Polynomial-Freiman-Ruzsa-conjecture">this Zulip stream</a>, and the project itself will be held at <a href="https://github.com/teorth/pfr">this Github repository.</a>

]]></content:encoded>
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<media:title type="html">Terry</media:title>
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<item>
<title>A Maclaurin type inequality</title>
<link>https://terrytao.wordpress.com/2023/10/10/a-maclaurin-type-inequality/</link>
<comments>https://terrytao.wordpress.com/2023/10/10/a-maclaurin-type-inequality/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Tue, 10 Oct 2023 16:47:52 +0000</pubDate>
<category><![CDATA[math.CA]]></category>
<category><![CDATA[paper]]></category>
<category><![CDATA[Maclaurin inequality]]></category>
<category><![CDATA[symmetric polynomials]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14016</guid>
<description><![CDATA[I have just uploaded to the arXiv my paper “A Maclaurin type inequality“. This paper concerns a variant of the Maclaurin inequality for the elementary symmetric means of real numbers . This inequality asserts that whenever and are non-negative. It can be proven as a consequence of the Newton inequality valid for all and arbitrary […]]]></description>
<content:encoded><![CDATA[

I have just uploaded to the arXiv my paper “<a href="http://arxiv.org/abs/2310.05328">A Maclaurin type inequality</a>“. This paper concerns a variant of the <a href="https://en.wikipedia.org/wiki/Maclaurin%27s_inequality">Maclaurin inequality</a> for the elementary symmetric means <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++s_k%28y%29+%3A%3D+%5Cfrac%7B1%7D%7B%5Cbinom%7Bn%7D%7Bk%7D%7D+%5Csum_%7B1+%5Cleq+i_1+%3C+%5Cdots+%3C+i_k+%5Cleq+n%7D+y_%7Bi_1%7D+%5Cdots+y_%7Bi_k%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++s_k%28y%29+%3A%3D+%5Cfrac%7B1%7D%7B%5Cbinom%7Bn%7D%7Bk%7D%7D+%5Csum_%7B1+%5Cleq+i_1+%3C+%5Cdots+%3C+i_k+%5Cleq+n%7D+y_%7Bi_1%7D+%5Cdots+y_%7Bi_k%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++s_k%28y%29+%3A%3D+%5Cfrac%7B1%7D%7B%5Cbinom%7Bn%7D%7Bk%7D%7D+%5Csum_%7B1+%5Cleq+i_1+%3C+%5Cdots+%3C+i_k+%5Cleq+n%7D+y_%7Bi_1%7D+%5Cdots+y_%7Bi_k%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle s_k(y) := \frac{1}{\binom{n}{k}} \sum_{1 \leq i_1 < \dots < i_k \leq n} y_{i_1} \dots y_{i_k}" class="latex" />
of <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" /> real numbers <img src="https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y_1,\dots,y_n}" class="latex" />. This inequality asserts that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++s_%5Cell%28y%29%5E%7B1%2F%5Cell%7D+%5Cleq+s_k%28y%29%5E%7B1%2Fk%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++s_%5Cell%28y%29%5E%7B1%2F%5Cell%7D+%5Cleq+s_k%28y%29%5E%7B1%2Fk%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++s_%5Cell%28y%29%5E%7B1%2F%5Cell%7D+%5Cleq+s_k%28y%29%5E%7B1%2Fk%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle s_\ell(y)^{1/\ell} \leq s_k(y)^{1/k}" class="latex" />
whenever <img src="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+k+%5Cleq+%5Cell+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+k+%5Cleq+%5Cell+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+k+%5Cleq+%5Cell+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1 \leq k \leq \ell \leq n}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y_1,\dots,y_n}" class="latex" /> are non-negative. It can be proven as a consequence of the <a href="https://en.wikipedia.org/wiki/Newton%27s_inequalities">Newton inequality</a> <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++s_%7Bk-1%7D%28y%29+s_%7Bk%2B1%7D%28y%29+%5Cleq+s_k%28y%29%5E2+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++s_%7Bk-1%7D%28y%29+s_%7Bk%2B1%7D%28y%29+%5Cleq+s_k%28y%29%5E2+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++s_%7Bk-1%7D%28y%29+s_%7Bk%2B1%7D%28y%29+%5Cleq+s_k%28y%29%5E2+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle s_{k-1}(y) s_{k+1}(y) \leq s_k(y)^2 " class="latex" />
valid for all <img src="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+k+%3C+n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+k+%3C+n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+k+%3C+n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1 \leq k < n}" class="latex" /> and arbitrary real <img src="https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y_1,\dots,y_n}" class="latex" /> (in particular, here the <img src="https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y_i}" class="latex" /> are allowed to be negative). Note that the <img src="https://s0.wp.com/latex.php?latex=%7Bk%3D1%2C+n%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%3D1%2C+n%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%3D1%2C+n%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k=1, n=2}" class="latex" /> case of this inequality is just the arithmetic mean-geometric mean inequality <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++y_1+y_2+%5Cleq+%28%5Cfrac%7By_1%2By_2%7D%7B2%7D%29%5E2%3B&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++y_1+y_2+%5Cleq+%28%5Cfrac%7By_1%2By_2%7D%7B2%7D%29%5E2%3B&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++y_1+y_2+%5Cleq+%28%5Cfrac%7By_1%2By_2%7D%7B2%7D%29%5E2%3B&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle y_1 y_2 \leq (\frac{y_1+y_2}{2})^2;" class="latex" />
the general case of this inequality can be deduced from this special case by a number of standard manipulations (the most non-obvious of which is the operation of differentiating the real-rooted polynomial <img src="https://s0.wp.com/latex.php?latex=%7B%5Cprod_%7Bi%3D1%7D%5En+%28z-y_i%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cprod_%7Bi%3D1%7D%5En+%28z-y_i%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cprod_%7Bi%3D1%7D%5En+%28z-y_i%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\prod_{i=1}^n (z-y_i)}" class="latex" /> to obtain another real-rooted polynomial, thanks to <a href="https://en.wikipedia.org/wiki/Rolle%27s_theorem">Rolle’s theorem</a>; the key point is that this operation preserves all the elementary symmetric means up to <img src="https://s0.wp.com/latex.php?latex=%7Bs_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s_{n-1}}" class="latex" />). One can think of Maclaurin’s inequality as providing a refined version of the arithmetic mean-geometric mean inequality on <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" /> variables (which corresponds to the case <img src="https://s0.wp.com/latex.php?latex=%7Bk%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k=1}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell%3Dn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell%3Dn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell%3Dn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell=n}" class="latex" />).

Whereas Newton’s inequality works for arbitrary real <img src="https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y_i}" class="latex" />, the Maclaurin inequality breaks down once one or more of the <img src="https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y_i}" class="latex" /> are permitted to be negative. A key example occurs when <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" /> is even, half of the <img src="https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y_i}" class="latex" /> are equal to <img src="https://s0.wp.com/latex.php?latex=%7B%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{+1}" class="latex" />, and half are equal to <img src="https://s0.wp.com/latex.php?latex=%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{-1}" class="latex" />. Here, one can verify that the elementary symmetric means <img src="https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s_k(y)}" class="latex" /> vanish for odd <img src="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k}" class="latex" /> and are equal to <img src="https://s0.wp.com/latex.php?latex=%7B+%28-1%29%5E%7Bk%2F2%7D+%5Cfrac%7B%5Cbinom%7Bn%2F2%7D%7Bk%2F2%7D%7D%7B%5Cbinom%7Bn%7D%7Bk%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B+%28-1%29%5E%7Bk%2F2%7D+%5Cfrac%7B%5Cbinom%7Bn%2F2%7D%7Bk%2F2%7D%7D%7B%5Cbinom%7Bn%7D%7Bk%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B+%28-1%29%5E%7Bk%2F2%7D+%5Cfrac%7B%5Cbinom%7Bn%2F2%7D%7Bk%2F2%7D%7D%7B%5Cbinom%7Bn%7D%7Bk%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{ (-1)^{k/2} \frac{\binom{n/2}{k/2}}{\binom{n}{k}}}" class="latex" /> for even <img src="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k}" class="latex" />. In particular, some routine estimation then gives the order of magnitude bound <a name="ex"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Cs_k%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%7D%7D+%5Casymp+%5Cfrac%7Bk%5E%7B1%2F2%7D%7D%7Bn%5E%7B1%2F2%7D%7D+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Cs_k%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%7D%7D+%5Casymp+%5Cfrac%7Bk%5E%7B1%2F2%7D%7D%7Bn%5E%7B1%2F2%7D%7D+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Cs_k%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%7D%7D+%5Casymp+%5Cfrac%7Bk%5E%7B1%2F2%7D%7D%7Bn%5E%7B1%2F2%7D%7D+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |s_k(y)|^{\frac{1}{k}} \asymp \frac{k^{1/2}}{n^{1/2}} \ \ \ \ \ (1)" class="latex" />
</a> for <img src="https://s0.wp.com/latex.php?latex=%7B0+%3C+k+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%3C+k+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%3C+k+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 < k \leq n}" class="latex" /> even, thus giving a significant violation of the Maclaurin inequality even after putting absolute values around the <img src="https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s_k(y)}" class="latex" />. In particular, vanishing of one <img src="https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s_k(y)}" class="latex" /> does not imply vanishing of all subsequent <img src="https://s0.wp.com/latex.php?latex=%7Bs_%5Cell%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs_%5Cell%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs_%5Cell%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s_\ell(y)}" class="latex" />.

On the other hand, it was observed <a href="https://zbmath.org/1483.65014">by Gopalan and Yehudayoff</a> that if two consecutive values <img src="https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%2C+s_%7Bk%2B1%7D%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%2C+s_%7Bk%2B1%7D%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs_k%28y%29%2C+s_%7Bk%2B1%7D%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s_k(y), s_{k+1}(y)}" class="latex" /> are small, then this makes all subsequent values <img src="https://s0.wp.com/latex.php?latex=%7Bs_%5Cell%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs_%5Cell%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs_%5Cell%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s_\ell(y)}" class="latex" /> small as well. More precise versions of this statement were subsequently observed by <a href="https://zbmath.org/1433.68604">Meka-Reingold-Tal</a> and <a href="https://zbmath.org/?q=rf%3A7330813">Doron-Hatami-Hoza</a>, who obtained estimates of the shape <a name="l-bound"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Cs_%5Cell%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7B%5Cell%7D%7D+%5Cll+%5Cell%5E%7B1%2F2%7D+%5Cmax+%28%7Cs_k%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%7D%7D%2C+%7Cs_%7Bk%2B1%7D%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%2B1%7D%7D%29+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Cs_%5Cell%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7B%5Cell%7D%7D+%5Cll+%5Cell%5E%7B1%2F2%7D+%5Cmax+%28%7Cs_k%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%7D%7D%2C+%7Cs_%7Bk%2B1%7D%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%2B1%7D%7D%29+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Cs_%5Cell%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7B%5Cell%7D%7D+%5Cll+%5Cell%5E%7B1%2F2%7D+%5Cmax+%28%7Cs_k%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%7D%7D%2C+%7Cs_%7Bk%2B1%7D%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%2B1%7D%7D%29+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |s_\ell(y)|^{\frac{1}{\ell}} \ll \ell^{1/2} \max (|s_k(y)|^{\frac{1}{k}}, |s_{k+1}(y)|^{\frac{1}{k+1}}) \ \ \ \ \ (2)" class="latex" />
</a> whenever <img src="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+k+%5Cleq+%5Cell+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+k+%5Cleq+%5Cell+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+k+%5Cleq+%5Cell+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1 \leq k \leq \ell \leq n}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By_1%2C%5Cdots%2Cy_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y_1,\dots,y_n}" class="latex" /> are real (but possibly negative). For instance, setting <img src="https://s0.wp.com/latex.php?latex=%7Bk%3D1%2C+%5Cell%3Dn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%3D1%2C+%5Cell%3Dn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%3D1%2C+%5Cell%3Dn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k=1, \ell=n}" class="latex" /> we obtain the inequality <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28y_1+%5Cdots+y_n%29%5E%7B1%2Fn%7D+%5Cll+n%5E%7B1%2F2%7D+%5Cmax%28+%7Cs_1%28y%29%7C%2C+%7Cs_2%28y%29%7C%5E2%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28y_1+%5Cdots+y_n%29%5E%7B1%2Fn%7D+%5Cll+n%5E%7B1%2F2%7D+%5Cmax%28+%7Cs_1%28y%29%7C%2C+%7Cs_2%28y%29%7C%5E2%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28y_1+%5Cdots+y_n%29%5E%7B1%2Fn%7D+%5Cll+n%5E%7B1%2F2%7D+%5Cmax%28+%7Cs_1%28y%29%7C%2C+%7Cs_2%28y%29%7C%5E2%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle (y_1 \dots y_n)^{1/n} \ll n^{1/2} \max( |s_1(y)|, |s_2(y)|^2) " class="latex" />
which can be established by combining the arithmetic mean-geometric mean inequality <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28y_1+%5Cdots+y_n%29%5E%7B2%2Fn%7D+%5Cleq+%5Cfrac%7By_1%5E2+%2B+%5Cdots+%2B+y_n%5E2%7D%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28y_1+%5Cdots+y_n%29%5E%7B2%2Fn%7D+%5Cleq+%5Cfrac%7By_1%5E2+%2B+%5Cdots+%2B+y_n%5E2%7D%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28y_1+%5Cdots+y_n%29%5E%7B2%2Fn%7D+%5Cleq+%5Cfrac%7By_1%5E2+%2B+%5Cdots+%2B+y_n%5E2%7D%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle (y_1 \dots y_n)^{2/n} \leq \frac{y_1^2 + \dots + y_n^2}{n}" class="latex" />
with the <a href="https://en.wikipedia.org/wiki/Newton%27s_identities">Newton identity</a> <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++y_1%5E2+%2B+%5Cdots+%2B+y_n%5E2+%3D+n%5E2+s_1%28y%29%5E2+-+n%28n-1%29+s_2%28y%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++y_1%5E2+%2B+%5Cdots+%2B+y_n%5E2+%3D+n%5E2+s_1%28y%29%5E2+-+n%28n-1%29+s_2%28y%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++y_1%5E2+%2B+%5Cdots+%2B+y_n%5E2+%3D+n%5E2+s_1%28y%29%5E2+-+n%28n-1%29+s_2%28y%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle y_1^2 + \dots + y_n^2 = n^2 s_1(y)^2 - n(n-1) s_2(y)." class="latex" />
As with the proof of the Newton inequalities, the general case of <a href="#l-bound">(2)</a> can be obtained from this special case after some standard manipulations (including the differentiation operation mentioned previously).

However, if one inspects the bound <a href="#l-bound">(2)</a> against the bounds <a href="#ex">(1)</a> given by the key example, we see a mismatch – the right-hand side of <a href="#l-bound">(2)</a> is larger than the left-hand side by a factor of about <img src="https://s0.wp.com/latex.php?latex=%7Bk%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k^{1/2}}" class="latex" />. The main result of the paper rectifies this by establishing the optimal (up to constants) improvement <a name="l-bound-2"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Cs_%5Cell%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7B%5Cell%7D%7D+%5Cll+%5Cfrac%7B%5Cell%5E%7B1%2F2%7D%7D%7Bk%5E%7B1%2F2%7D%7D+%5Cmax+%28%7Cs_k%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%7D%7D%2C+%7Cs_%7Bk%2B1%7D%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%2B1%7D%7D%29+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Cs_%5Cell%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7B%5Cell%7D%7D+%5Cll+%5Cfrac%7B%5Cell%5E%7B1%2F2%7D%7D%7Bk%5E%7B1%2F2%7D%7D+%5Cmax+%28%7Cs_k%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%7D%7D%2C+%7Cs_%7Bk%2B1%7D%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%2B1%7D%7D%29+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7Cs_%5Cell%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7B%5Cell%7D%7D+%5Cll+%5Cfrac%7B%5Cell%5E%7B1%2F2%7D%7D%7Bk%5E%7B1%2F2%7D%7D+%5Cmax+%28%7Cs_k%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%7D%7D%2C+%7Cs_%7Bk%2B1%7D%28y%29%7C%5E%7B%5Cfrac%7B1%7D%7Bk%2B1%7D%7D%29+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |s_\ell(y)|^{\frac{1}{\ell}} \ll \frac{\ell^{1/2}}{k^{1/2}} \max (|s_k(y)|^{\frac{1}{k}}, |s_{k+1}(y)|^{\frac{1}{k+1}}) \ \ \ \ \ (3)" class="latex" />
</a> of <a href="#l-bound">(2)</a>. This answers a question <a href="https://mathoverflow.net/questions/446254/maclaurins-inequality-on-elementary-symmetric-polynomials-of-arbitrary-real-num">posed on MathOverflow</a>.

Unlike the previous arguments, we do not rely primarily on the arithmetic mean-geometric mean inequality. Instead, our primary tool is a new inequality <a name="smr"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bm%3D0%7D%5E%5Cell+%5Cbinom%7B%5Cell%7D%7Bm%7D+%7Cs_m%28y%29%7C+r%5Em+%5Cgeq+%281%2B+%7Cs_%5Cell%28y%29%7C%5E%7B2%2F%5Cell%7D+r%5E2%29%5E%7B%5Cell%2F2%7D%2C+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bm%3D0%7D%5E%5Cell+%5Cbinom%7B%5Cell%7D%7Bm%7D+%7Cs_m%28y%29%7C+r%5Em+%5Cgeq+%281%2B+%7Cs_%5Cell%28y%29%7C%5E%7B2%2F%5Cell%7D+r%5E2%29%5E%7B%5Cell%2F2%7D%2C+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bm%3D0%7D%5E%5Cell+%5Cbinom%7B%5Cell%7D%7Bm%7D+%7Cs_m%28y%29%7C+r%5Em+%5Cgeq+%281%2B+%7Cs_%5Cell%28y%29%7C%5E%7B2%2F%5Cell%7D+r%5E2%29%5E%7B%5Cell%2F2%7D%2C+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{m=0}^\ell \binom{\ell}{m} |s_m(y)| r^m \geq (1+ |s_\ell(y)|^{2/\ell} r^2)^{\ell/2}, \ \ \ \ \ (4)" class="latex" />
</a> valid for all <img src="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+%5Cell+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+%5Cell+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+%5Cell+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1 \leq \ell \leq n}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Br%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Br%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Br%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{r>0}" class="latex" />. Roughly speaking, the bound <a href="#l-bound-2">(3)</a> would follow from <a href="#smr">(4)</a> by setting <img src="https://s0.wp.com/latex.php?latex=%7Br+%5Casymp+%28k%2F%5Cell%29%5E%7B1%2F2%7D+%7Cs_%5Cell%28y%29%7C%5E%7B-1%2F%5Cell%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Br+%5Casymp+%28k%2F%5Cell%29%5E%7B1%2F2%7D+%7Cs_%5Cell%28y%29%7C%5E%7B-1%2F%5Cell%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Br+%5Casymp+%28k%2F%5Cell%29%5E%7B1%2F2%7D+%7Cs_%5Cell%28y%29%7C%5E%7B-1%2F%5Cell%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{r \asymp (k/\ell)^{1/2} |s_\ell(y)|^{-1/\ell}}" class="latex" />, provided that we can show that the <img src="https://s0.wp.com/latex.php?latex=%7Bm%3Dk%2Ck%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3Dk%2Ck%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3Dk%2Ck%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=k,k+1}" class="latex" /> terms of the left-hand side dominate the sum in this regime. This can be done, after a technical step of passing to tuples <img src="https://s0.wp.com/latex.php?latex=%7By%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y}" class="latex" /> which nearly optimize the required inequality <a href="#l-bound-2">(3)</a>.

We sketch the proof of the inequality <a href="#smr">(4)</a> as follows. One can use some standard manipulations reduce to the case where <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell%3Dn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell%3Dn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell%3Dn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell=n}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B%7Cs_n%28y%29%7C%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cs_n%28y%29%7C%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cs_n%28y%29%7C%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|s_n(y)|=1}" class="latex" />, and after replacing <img src="https://s0.wp.com/latex.php?latex=%7Br%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Br%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Br%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{r}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7B1%2Fr%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2Fr%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2Fr%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1/r}" class="latex" /> one is now left with establishing the inequality <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bm%3D0%7D%5En+%5Cbinom%7Bn%7D%7Bm%7D+%7Cs_m%28y%29%7C+r%5E%7Bn-m%7D+%5Cgeq+%281%2Br%5E2%29%5E%7Bn%2F2%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bm%3D0%7D%5En+%5Cbinom%7Bn%7D%7Bm%7D+%7Cs_m%28y%29%7C+r%5E%7Bn-m%7D+%5Cgeq+%281%2Br%5E2%29%5E%7Bn%2F2%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bm%3D0%7D%5En+%5Cbinom%7Bn%7D%7Bm%7D+%7Cs_m%28y%29%7C+r%5E%7Bn-m%7D+%5Cgeq+%281%2Br%5E2%29%5E%7Bn%2F2%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{m=0}^n \binom{n}{m} |s_m(y)| r^{n-m} \geq (1+r^2)^{n/2}." class="latex" />
Note that equality is attained in the previously discussed example with half of the <img src="https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y_i}" class="latex" /> equal to <img src="https://s0.wp.com/latex.php?latex=%7B%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%2B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{+1}" class="latex" /> and the other half equal to <img src="https://s0.wp.com/latex.php?latex=%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{-1}" class="latex" />, thanks to the binomial theorem.

To prove this identity, we consider the polynomial <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cprod_%7Bj%3D1%7D%5En+%28z+-+y_j%29+%3D+%5Csum_%7Bm%3D0%7D%5En+%28-1%29%5Em+%5Cbinom%7Bn%7D%7Bm%7D+s_k%28m%29+z%5E%7Bn-m%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cprod_%7Bj%3D1%7D%5En+%28z+-+y_j%29+%3D+%5Csum_%7Bm%3D0%7D%5En+%28-1%29%5Em+%5Cbinom%7Bn%7D%7Bm%7D+s_k%28m%29+z%5E%7Bn-m%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cprod_%7Bj%3D1%7D%5En+%28z+-+y_j%29+%3D+%5Csum_%7Bm%3D0%7D%5En+%28-1%29%5Em+%5Cbinom%7Bn%7D%7Bm%7D+s_k%28m%29+z%5E%7Bn-m%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \prod_{j=1}^n (z - y_j) = \sum_{m=0}^n (-1)^m \binom{n}{m} s_k(m) z^{n-m}." class="latex" />
Evaluating this polynomial at <img src="https://s0.wp.com/latex.php?latex=%7Bir%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bir%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bir%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{ir}" class="latex" />, taking absolute values, using the triangle inequality, and then taking logarithms, we conclude that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B1%7D%7B2%7D+%5Csum_%7Bj%3D1%7D%5En+%5Clog%28y_j%5E2+%2B+r%5E2%29+%5Cleq+%5Clog%28%5Csum_%7Bm%3D0%7D%5En+%5Cbinom%7Bn%7D%7Bm%7D+%7Cs_m%28y%29%7C+r%5E%7Bn-m%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B1%7D%7B2%7D+%5Csum_%7Bj%3D1%7D%5En+%5Clog%28y_j%5E2+%2B+r%5E2%29+%5Cleq+%5Clog%28%5Csum_%7Bm%3D0%7D%5En+%5Cbinom%7Bn%7D%7Bm%7D+%7Cs_m%28y%29%7C+r%5E%7Bn-m%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B1%7D%7B2%7D+%5Csum_%7Bj%3D1%7D%5En+%5Clog%28y_j%5E2+%2B+r%5E2%29+%5Cleq+%5Clog%28%5Csum_%7Bm%3D0%7D%5En+%5Cbinom%7Bn%7D%7Bm%7D+%7Cs_m%28y%29%7C+r%5E%7Bn-m%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{1}{2} \sum_{j=1}^n \log(y_j^2 + r^2) \leq \log(\sum_{m=0}^n \binom{n}{m} |s_m(y)| r^{n-m})." class="latex" />
A convexity argument gives the lower bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog%28y_j%5E2+%2B+r%5E2%29+%5Cgeq+%5Clog%281%2Br%5E2%29+%2B+%5Cfrac%7B2%7D%7B1%2Br%5E2%7D+%5Clog+%7Cy_j%7C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog%28y_j%5E2+%2B+r%5E2%29+%5Cgeq+%5Clog%281%2Br%5E2%29+%2B+%5Cfrac%7B2%7D%7B1%2Br%5E2%7D+%5Clog+%7Cy_j%7C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog%28y_j%5E2+%2B+r%5E2%29+%5Cgeq+%5Clog%281%2Br%5E2%29+%2B+%5Cfrac%7B2%7D%7B1%2Br%5E2%7D+%5Clog+%7Cy_j%7C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \log(y_j^2 + r^2) \geq \log(1+r^2) + \frac{2}{1+r^2} \log |y_j|" class="latex" />
while the normalization <img src="https://s0.wp.com/latex.php?latex=%7B%7Cs_n%28y%29%7C%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cs_n%28y%29%7C%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cs_n%28y%29%7C%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|s_n(y)|=1}" class="latex" /> gives <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bj%3D1%7D%5En+%5Clog+%7Cy_j%7C+%3D+0&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bj%3D1%7D%5En+%5Clog+%7Cy_j%7C+%3D+0&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bj%3D1%7D%5En+%5Clog+%7Cy_j%7C+%3D+0&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{j=1}^n \log |y_j| = 0" class="latex" />
and the claim follows.

]]></content:encoded>
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<slash:comments>13</slash:comments>
<media:content url="https://1.gravatar.com/avatar/d7f0e4a42bbbf58ffa656c92d4a32b2b6752e802dfa4cb919e5774dcfda006c0?s=96&d=identicon&r=PG" medium="image">
<media:title type="html">Terry</media:title>
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</item>
<item>
<title>Bounding sums or integrals of non-negative quantities</title>
<link>https://terrytao.wordpress.com/2023/09/30/bounding-sums-or-integrals-of-non-negative-quantities/</link>
<comments>https://terrytao.wordpress.com/2023/09/30/bounding-sums-or-integrals-of-non-negative-quantities/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Sat, 30 Sep 2023 23:21:38 +0000</pubDate>
<category><![CDATA[math.CA]]></category>
<category><![CDATA[tricks]]></category>
<category><![CDATA[asymptotic notation]]></category>
<category><![CDATA[asymptotics]]></category>
<category><![CDATA[estimation]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=13984</guid>
<description><![CDATA[A common task in analysis is to obtain bounds on sums or integrals where is some simple region (such as an interval) in one or more dimensions, and is an explicit (and elementary) non-negative expression involving one or more variables (such as or , and possibly also some additional parameters. Often, one would be content […]]]></description>
<content:encoded><![CDATA[

A common task in analysis is to obtain bounds on sums <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n \in A} f(n)" class="latex" />
or integrals <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_A f(x)\ dx " class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> is some simple region (such as an interval) in one or more dimensions, and <img src="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f}" class="latex" /> is an explicit (and <a href="https://en.wikipedia.org/wiki/Elementary_function">elementary</a>) non-negative expression involving one or more variables (such as <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" /> or <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" />, and possibly also some additional parameters. Often, one would be content with an order of magnitude upper bound such as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%5Cll+X&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%5Cll+X&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%5Cll+X&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n \in A} f(n) \ll X" class="latex" />
or <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+%5Cll+X&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+%5Cll+X&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+%5Cll+X&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_A f(x)\ dx \ll X" class="latex" />
where we use <img src="https://s0.wp.com/latex.php?latex=%7BX+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X \ll Y}" class="latex" /> (or <img src="https://s0.wp.com/latex.php?latex=%7BY+%5Cgg+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY+%5Cgg+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY+%5Cgg+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y \gg X}" class="latex" /> or <img src="https://s0.wp.com/latex.php?latex=%7BX+%3D+O%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX+%3D+O%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX+%3D+O%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X = O(Y)}" class="latex" />) to denote the bound <img src="https://s0.wp.com/latex.php?latex=%7B%7CX%7C+%5Cleq+CY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CX%7C+%5Cleq+CY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CX%7C+%5Cleq+CY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|X| \leq CY}" class="latex" /> for some constant <img src="https://s0.wp.com/latex.php?latex=%7BC%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BC%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BC%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{C}" class="latex" />; sometimes one wishes to also obtain the matching lower bound, thus obtaining <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%5Casymp+X&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%5Casymp+X&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%5Casymp+X&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n \in A} f(n) \asymp X" class="latex" />
or <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+%5Casymp+X&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+%5Casymp+X&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+%5Casymp+X&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_A f(x)\ dx \asymp X" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7BX+%5Casymp+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX+%5Casymp+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX+%5Casymp+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X \asymp Y}" class="latex" /> is synonymous with <img src="https://s0.wp.com/latex.php?latex=%7BX+%5Cll+Y+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX+%5Cll+Y+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX+%5Cll+Y+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X \ll Y \ll X}" class="latex" />. Finally, one may wish to obtain a more precise bound, such as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%3D+%281%2Bo%281%29%29+X&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%3D+%281%2Bo%281%29%29+X&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%3D+%281%2Bo%281%29%29+X&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n \in A} f(n) = (1+o(1)) X" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{o(1)}" class="latex" /> is a quantity that goes to zero as the parameters of the problem go to infinity (or some other limit). (For a deeper dive into asymptotic notation in general, see <a href="https://terrytao.wordpress.com/2022/05/10/partially-specified-mathematical-objects-ambient-parameters-and-asymptotic-notation/">this previous blog post</a>.)

Here are some typical examples of such estimation problems, drawn from recent questions on MathOverflow:
<ul> <li>(i) (From <a href="https://mathoverflow.net/questions/455537/does-this-dyadic-sum-converge">this question</a>) If <img src="https://s0.wp.com/latex.php?latex=%7Bd%2Cp+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%2Cp+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%2Cp+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d,p \geq 1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Ba%3Ed%2Fp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%3Ed%2Fp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%3Ed%2Fp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a>d/p}" class="latex" />, is the expression <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bj+%5Cin+%7B%5Cbf+Z%7D%7D+2%5E%7B%28%5Cfrac%7Bd%7D%7Bp%7D%2B1-a%29j%7D+%5Cint_0%5E%5Cinfty+e%5E%7B-2%5Ej+s%7D+%5Cfrac%7Bs%5Ea%7D%7B1%2Bs%5E%7B2a%7D%7D%5C+ds&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bj+%5Cin+%7B%5Cbf+Z%7D%7D+2%5E%7B%28%5Cfrac%7Bd%7D%7Bp%7D%2B1-a%29j%7D+%5Cint_0%5E%5Cinfty+e%5E%7B-2%5Ej+s%7D+%5Cfrac%7Bs%5Ea%7D%7B1%2Bs%5E%7B2a%7D%7D%5C+ds&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bj+%5Cin+%7B%5Cbf+Z%7D%7D+2%5E%7B%28%5Cfrac%7Bd%7D%7Bp%7D%2B1-a%29j%7D+%5Cint_0%5E%5Cinfty+e%5E%7B-2%5Ej+s%7D+%5Cfrac%7Bs%5Ea%7D%7B1%2Bs%5E%7B2a%7D%7D%5C+ds&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{j \in {\bf Z}} 2^{(\frac{d}{p}+1-a)j} \int_0^\infty e^{-2^j s} \frac{s^a}{1+s^{2a}}\ ds" class="latex" />
finite? </li><li>(ii) (From <a href="https://mathoverflow.net/questions/455545/how-to-show-that-the-trace-of-a-regularized-laplacian-defined-on-two-sphere-with">this question</a>) If <img src="https://s0.wp.com/latex.php?latex=%7Bh%2Cm+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh%2Cm+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh%2Cm+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h,m \geq 1}" class="latex" />, how can one show that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bd%3D0%7D%5E%5Cinfty+%5Cfrac%7B2d%2B1%7D%7B2h%5E2+%281+%2B+%5Cfrac%7Bd%28d%2B1%29%7D%7Bh%5E2%7D%29+%281+%2B+%5Cfrac%7Bd%28d%2B1%29%7D%7Bh%5E2m%5E2%7D%29%5E2%7D+%5Cll+1+%2B+%5Clog%28m%5E2%29%3F&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bd%3D0%7D%5E%5Cinfty+%5Cfrac%7B2d%2B1%7D%7B2h%5E2+%281+%2B+%5Cfrac%7Bd%28d%2B1%29%7D%7Bh%5E2%7D%29+%281+%2B+%5Cfrac%7Bd%28d%2B1%29%7D%7Bh%5E2m%5E2%7D%29%5E2%7D+%5Cll+1+%2B+%5Clog%28m%5E2%29%3F&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bd%3D0%7D%5E%5Cinfty+%5Cfrac%7B2d%2B1%7D%7B2h%5E2+%281+%2B+%5Cfrac%7Bd%28d%2B1%29%7D%7Bh%5E2%7D%29+%281+%2B+%5Cfrac%7Bd%28d%2B1%29%7D%7Bh%5E2m%5E2%7D%29%5E2%7D+%5Cll+1+%2B+%5Clog%28m%5E2%29%3F&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{d=0}^\infty \frac{2d+1}{2h^2 (1 + \frac{d(d+1)}{h^2}) (1 + \frac{d(d+1)}{h^2m^2})^2} \ll 1 + \log(m^2)?" class="latex" />
</li><li>(iii) (From <a href="https://mathoverflow.net/questions/454544/what-is-the-exact-asymptotic-bound-on-the-following-sum-of-polynomials#comment1176606_454544">this question</a>) Can one show that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk%3D1%7D%5E%7Bn-1%7D+%5Cfrac%7Bk%5E%7B2n-4k-3%7D%28n%5E2-2nk%2B2k%5E2%29%7D%7B%28n-k%29%5E%7B2n-4k-1%7D%7D+%3D+%28c%2Bo%281%29%29+%5Csqrt%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk%3D1%7D%5E%7Bn-1%7D+%5Cfrac%7Bk%5E%7B2n-4k-3%7D%28n%5E2-2nk%2B2k%5E2%29%7D%7B%28n-k%29%5E%7B2n-4k-1%7D%7D+%3D+%28c%2Bo%281%29%29+%5Csqrt%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk%3D1%7D%5E%7Bn-1%7D+%5Cfrac%7Bk%5E%7B2n-4k-3%7D%28n%5E2-2nk%2B2k%5E2%29%7D%7B%28n-k%29%5E%7B2n-4k-1%7D%7D+%3D+%28c%2Bo%281%29%29+%5Csqrt%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{k=1}^{n-1} \frac{k^{2n-4k-3}(n^2-2nk+2k^2)}{(n-k)^{2n-4k-1}} = (c+o(1)) \sqrt{n}" class="latex" />
as <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \rightarrow \infty}" class="latex" /> for an explicit constant <img src="https://s0.wp.com/latex.php?latex=%7Bc%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c}" class="latex" />, and what is this constant?
</li></ul>

Compared to other estimation tasks, such as that of controlling <a href="https://en.wikipedia.org/wiki/Oscillatory_integral">oscillatory integrals</a>, <a href="https://en.wikipedia.org/wiki/Exponential_sum">exponential sums</a>, <a href="https://en.wikipedia.org/wiki/Singular_integral">singular integrals</a>, or expressions involving one or more unknown functions (that are only known to lie in some function spaces, such as an <img src="https://s0.wp.com/latex.php?latex=%7BL%5Ep%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BL%5Ep%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BL%5Ep%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{L^p}" class="latex" /> space), high-dimensional geometry (or alternatively, large numbers of random variables), or number-theoretic structures (such as the primes), estimation of sums or integrals of non-negative elementary expressions is a relatively straightforward task, and can be accomplished by a variety of methods. The art of obtaining such estimates is typically not explicitly taught in textbooks, other than through some examples and exercises; it is typically picked up by analysts (or those working in adjacent areas, such as PDE, combinatorics, or theoretical computer science) as graduate students, while they work through their thesis or their first few papers in the subject.

Somewhat in the spirit of <a href="https://terrytao.wordpress.com/2010/10/21/245a-problem-solving-strategies/">this previous post on analysis problem solving strategies</a>, I am going to try here to collect some general principles and techniques that I have found useful for these sorts of problems. As with the previous post, I hope this will be something of a living document, and encourage others to add their own tips or suggestions in the comments.



 — 1. Asymptotic arithmetic — 

Asymptotic notation is designed so that many of the usual rules of algebra and inequality manipulation continue to hold, with the caveat that one has to be careful if subtraction or division is involved. For instance, if one knows that <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \ll X}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \ll Y}" class="latex" />, then one can immediately conclude that <img src="https://s0.wp.com/latex.php?latex=%7BA+%2B+B+%5Cll+X%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%2B+B+%5Cll+X%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%2B+B+%5Cll+X%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A + B \ll X+Y}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BAB+%5Cll+XY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BAB+%5Cll+XY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BAB+%5Cll+XY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{AB \ll XY}" class="latex" />, even if <img src="https://s0.wp.com/latex.php?latex=%7BA%2CB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2CB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2CB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A,B}" class="latex" /> are negative (note that the notation <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \ll X}" class="latex" /> or <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \ll Y}" class="latex" /> automatically forces <img src="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X,Y}" class="latex" /> to be non-negative). Equivalently, we have the rules <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++O%28X%29+%2B+O%28Y%29+%3D+O%28X%2BY%29%3B+%5Cquad+O%28X%29+%5Ccdot+O%28Y%29+%3D+O%28XY%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++O%28X%29+%2B+O%28Y%29+%3D+O%28X%2BY%29%3B+%5Cquad+O%28X%29+%5Ccdot+O%28Y%29+%3D+O%28XY%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++O%28X%29+%2B+O%28Y%29+%3D+O%28X%2BY%29%3B+%5Cquad+O%28X%29+%5Ccdot+O%28Y%29+%3D+O%28XY%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle O(X) + O(Y) = O(X+Y); \quad O(X) \cdot O(Y) = O(XY)" class="latex" />
and more generally we have the triangle inequality <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%5Calpha+O%28X_%5Calpha%29+%3D+O%28+%5Csum_%5Calpha+X_%5Calpha+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%5Calpha+O%28X_%5Calpha%29+%3D+O%28+%5Csum_%5Calpha+X_%5Calpha+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%5Calpha+O%28X_%5Calpha%29+%3D+O%28+%5Csum_%5Calpha+X_%5Calpha+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_\alpha O(X_\alpha) = O( \sum_\alpha X_\alpha )." class="latex" />
Again, we stress that this sort of rule implicitly requires the <img src="https://s0.wp.com/latex.php?latex=%7BX_%5Calpha%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_%5Calpha%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_%5Calpha%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_\alpha}" class="latex" /> to be non-negative, and that claims such as <img src="https://s0.wp.com/latex.php?latex=%7BO%28X%29+-+O%28Y%29+%3D+O%28X-Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28X%29+-+O%28Y%29+%3D+O%28X-Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28X%29+-+O%28Y%29+%3D+O%28X-Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(X) - O(Y) = O(X-Y)}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BO%28X%29%2FO%28Y%29+%3D+O%28X%2FY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28X%29%2FO%28Y%29+%3D+O%28X%2FY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28X%29%2FO%28Y%29+%3D+O%28X%2FY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(X)/O(Y) = O(X/Y)}" class="latex" /> are simply false. As a rule of thumb, if your calculations have arrived at a situation where a signed or oscillating sum or integral appears inside the big-O notation, or on the right-hand side of an estimate, without being “protected” by absolute value signs, then you have probably made a serious error in your calculations.

Another rule of inequalities that is inherited by asymptotic notation is that if one has two bounds <a name="nomin"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+X%3B+%5Cquad+A+%5Cll+Y+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+X%3B+%5Cquad+A+%5Cll+Y+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+X%3B+%5Cquad+A+%5Cll+Y+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A \ll X; \quad A \ll Y \ \ \ \ \ (1)" class="latex" />
</a> for the same quantity <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" />, then one can combine them into the unified asymptotic bound <a name="min"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Cmin%28X%2C+Y%29.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Cmin%28X%2C+Y%29.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Cmin%28X%2C+Y%29.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A \ll \min(X, Y). \ \ \ \ \ (2)" class="latex" />
</a> This is an example of a “free move”: a replacement of bounds that does not lose any of the strength of the original bounds, since of course <a href="#min">(2)</a> implies <a href="#nomin">(1)</a>. In contrast, other ways to combine the two bounds <a href="#nomin">(1)</a>, such as taking the geometric mean <a name="Avg"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+X%5E%7B1%2F2%7D+Y%5E%7B1%2F2%7D%2C+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+X%5E%7B1%2F2%7D+Y%5E%7B1%2F2%7D%2C+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+X%5E%7B1%2F2%7D+Y%5E%7B1%2F2%7D%2C+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A \ll X^{1/2} Y^{1/2}, \ \ \ \ \ (3)" class="latex" />
</a> while often convenient, are not “free”: the bounds <a href="#nomin">(1)</a> imply the averaged bound <a href="#Avg">(3)</a>, but the bound <a href="#Avg">(3)</a> does not imply <a href="#nomin">(1)</a>. On the other hand, the inequality <a href="#min">(2)</a>, while it does not concede any logical strength, can require more calculation to work with, often because one ends up splitting up cases such as <img src="https://s0.wp.com/latex.php?latex=%7BX+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX+%5Cll+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X \ll Y}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BX+%5Cgg+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX+%5Cgg+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX+%5Cgg+Y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X \gg Y}" class="latex" /> in order to simplify the minimum. So in practice, when trying to establish an estimate, one often starts with using conservative bounds such as <a href="#min">(2)</a> in order to maximize one’s chances of getting any proof (no matter how messy) of the desired estimate, and only after such a proof is found, one tries to look for more elegant approaches using less efficient bounds such as <a href="#Avg">(3)</a>.

For instance, suppose one wanted to show that the sum <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=-\infty}^\infty \frac{2^n}{(1+n^2) (1+2^{2n})}" class="latex" />
was convergent. Lower bounding the denominator term <img src="https://s0.wp.com/latex.php?latex=%7B1%2B2%5E%7B2n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2B2%5E%7B2n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2B2%5E%7B2n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1+2^{2n}}" class="latex" /> by <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> or by <img src="https://s0.wp.com/latex.php?latex=%7B2%5E%7B2n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5E%7B2n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5E%7B2n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^{2n}}" class="latex" />, one obtains the bounds <a name="m1"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B2%5En%7D%7B1%2Bn%5E2%7D+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B2%5En%7D%7B1%2Bn%5E2%7D+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B2%5En%7D%7B1%2Bn%5E2%7D+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{2^n}{(1+n^2) (1+2^{2n})} \ll \frac{2^n}{1+n^2} \ \ \ \ \ (4)" class="latex" />
</a> and also <a name="m2"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+2%5E%7B2n%7D%7D+%3D+%5Cfrac%7B2%5E%7B-n%7D%7D%7B1%2Bn%5E2%7D+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+2%5E%7B2n%7D%7D+%3D+%5Cfrac%7B2%5E%7B-n%7D%7D%7B1%2Bn%5E2%7D+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+2%5E%7B2n%7D%7D+%3D+%5Cfrac%7B2%5E%7B-n%7D%7D%7B1%2Bn%5E2%7D+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{2^n}{(1+n^2) (1+2^{2n})} \ll \frac{2^n}{(1+n^2) 2^{2n}} = \frac{2^{-n}}{1+n^2} \ \ \ \ \ (5)" class="latex" />
</a> so by applying <a href="#min">(2)</a> we obtain the unified bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+2%5E%7B2n%7D%7D+%3D+%5Cfrac%7B%5Cmin%282%5En%2C2%5E%7B-n%7D%29%7D%7B1%2Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+2%5E%7B2n%7D%7D+%3D+%5Cfrac%7B%5Cmin%282%5En%2C2%5E%7B-n%7D%29%7D%7B1%2Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+2%5E%7B2n%7D%7D+%3D+%5Cfrac%7B%5Cmin%282%5En%2C2%5E%7B-n%7D%29%7D%7B1%2Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{2^n}{(1+n^2) (1+2^{2n})} \ll \frac{2^n}{(1+n^2) 2^{2n}} = \frac{\min(2^n,2^{-n})}{1+n^2}." class="latex" />
To deal with this bound, we can split into the two contributions <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \geq 0}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7B2%5E%7B-n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5E%7B-n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5E%7B-n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^{-n}}" class="latex" /> dominates, and <img src="https://s0.wp.com/latex.php?latex=%7Bn+%3C+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%3C+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%3C+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n < 0}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7B2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^n}" class="latex" /> dominates. In the former case we see (from the ratio test, for instance) that the sum <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D0%7D%5E%5Cinfty+%5Cfrac%7B2%5E%7B-n%7D%7D%7B1%2Bn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D0%7D%5E%5Cinfty+%5Cfrac%7B2%5E%7B-n%7D%7D%7B1%2Bn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D0%7D%5E%5Cinfty+%5Cfrac%7B2%5E%7B-n%7D%7D%7B1%2Bn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=0}^\infty \frac{2^{-n}}{1+n^2}" class="latex" />
is absolutely convergent, and in the latter case we see that the sum <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D-%5Cinfty%7D%5E%7B-1%7D+%5Cfrac%7B2%5E%7Bn%7D%7D%7B1%2Bn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D-%5Cinfty%7D%5E%7B-1%7D+%5Cfrac%7B2%5E%7Bn%7D%7D%7B1%2Bn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D-%5Cinfty%7D%5E%7B-1%7D+%5Cfrac%7B2%5E%7Bn%7D%7D%7B1%2Bn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=-\infty}^{-1} \frac{2^{n}}{1+n^2}" class="latex" />
is also absolutely convergent, so the entire sum is absolutely convergent. But once one has this argument, one can try to streamline it, for instance by taking the geometric mean of <a href="#m1">(4)</a>, <a href="#m2">(5)</a> rather than the minimum to obtain the weaker bound <a name="crude"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B1%7D%7B1%2Bn%5E2%7D+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B1%7D%7B1%2Bn%5E2%7D+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B2%5En%7D%7B%281%2Bn%5E2%29+%281%2B2%5E%7B2n%7D%29%7D+%5Cll+%5Cfrac%7B1%7D%7B1%2Bn%5E2%7D+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{2^n}{(1+n^2) (1+2^{2n})} \ll \frac{1}{1+n^2} \ \ \ \ \ (6)" class="latex" />
</a> and now one can conclude without decomposition just by observing the absolute convergence of the doubly infinite sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7B1%2Bn%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7B1%2Bn%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3D-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7B1%2Bn%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n=-\infty}^\infty \frac{1}{1+n^2}}" class="latex" />. This is a less “efficient” estimate, because one has conceded a lot of the decay in the summand by using <a href="#crude">(6)</a> (the summand used to be exponentially decaying in <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" />, but is now only polynomially decaying), but it is still sufficient for the purpose of establishing absolute convergence.

One of the key advantages of dealing with order of magnitude estimates, as opposed to sharp inequalities, is that the arithmetic becomes <a href="https://en.wikipedia.org/wiki/Tropical_semiring">tropical</a>. More explicitly, we have the important rule <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++X+%2B+Y+%5Casymp+%5Cmax%28X%2CY%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++X+%2B+Y+%5Casymp+%5Cmax%28X%2CY%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++X+%2B+Y+%5Casymp+%5Cmax%28X%2CY%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle X + Y \asymp \max(X,Y)" class="latex" />
whenever <img src="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X,Y}" class="latex" /> are non-negative, since we clearly have <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmax%28X%2CY%29+%5Cleq+X%2BY+%5Cleq+2+%5Cmax%28X%2CY%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmax%28X%2CY%29+%5Cleq+X%2BY+%5Cleq+2+%5Cmax%28X%2CY%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmax%28X%2CY%29+%5Cleq+X%2BY+%5Cleq+2+%5Cmax%28X%2CY%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \max(X,Y) \leq X+Y \leq 2 \max(X,Y)." class="latex" />
In particular, if <img src="https://s0.wp.com/latex.php?latex=%7BY+%5Cleq+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY+%5Cleq+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY+%5Cleq+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y \leq X}" class="latex" />, then <img src="https://s0.wp.com/latex.php?latex=%7BO%28X%29+%2B+O%28Y%29+%3D+O%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28X%29+%2B+O%28Y%29+%3D+O%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28X%29+%2B+O%28Y%29+%3D+O%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(X) + O(Y) = O(X)}" class="latex" />. That is to say, given two orders of magnitudes, any term <img src="https://s0.wp.com/latex.php?latex=%7BO%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(Y)}" class="latex" /> of equal or lower order to a “main term” <img src="https://s0.wp.com/latex.php?latex=%7BO%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(X)}" class="latex" /> can be discarded. This is a very useful rule to keep in mind when trying to estimate sums or integrals, as it allows one to discard many terms that are not contributing to the final answer. It also interacts well with monotone operations, such as raising to a power <img src="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p}" class="latex" />; for instance, we have <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28X%2BY%29%5Ep+%5Casymp+%5Cmax%28X%2CY%29%5Ep+%3D+%5Cmax%28X%5Ep%2CY%5Ep%29+%5Casymp+X%5Ep+%2B+Y%5Ep&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28X%2BY%29%5Ep+%5Casymp+%5Cmax%28X%2CY%29%5Ep+%3D+%5Cmax%28X%5Ep%2CY%5Ep%29+%5Casymp+X%5Ep+%2B+Y%5Ep&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28X%2BY%29%5Ep+%5Casymp+%5Cmax%28X%2CY%29%5Ep+%3D+%5Cmax%28X%5Ep%2CY%5Ep%29+%5Casymp+X%5Ep+%2B+Y%5Ep&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle (X+Y)^p \asymp \max(X,Y)^p = \max(X^p,Y^p) \asymp X^p + Y^p" class="latex" />
if <img src="https://s0.wp.com/latex.php?latex=%7BX%2CY+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2CY+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2CY+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X,Y \geq 0}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p}" class="latex" /> is a fixed positive constant, whilst <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B1%7D%7BX%2BY%7D+%5Casymp+%5Cfrac%7B1%7D%7B%5Cmax%28X%2CY%29%7D+%3D+%5Cmin%28%5Cfrac%7B1%7D%7BX%7D%2C+%5Cfrac%7B1%7D%7BY%7D%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B1%7D%7BX%2BY%7D+%5Casymp+%5Cfrac%7B1%7D%7B%5Cmax%28X%2CY%29%7D+%3D+%5Cmin%28%5Cfrac%7B1%7D%7BX%7D%2C+%5Cfrac%7B1%7D%7BY%7D%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B1%7D%7BX%2BY%7D+%5Casymp+%5Cfrac%7B1%7D%7B%5Cmax%28X%2CY%29%7D+%3D+%5Cmin%28%5Cfrac%7B1%7D%7BX%7D%2C+%5Cfrac%7B1%7D%7BY%7D%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{1}{X+Y} \asymp \frac{1}{\max(X,Y)} = \min(\frac{1}{X}, \frac{1}{Y})" class="latex" />
if <img src="https://s0.wp.com/latex.php?latex=%7BX%2CY%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2CY%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2CY%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X,Y>0}" class="latex" />. Finally, this relation also sets up the fundamental divide and conquer strategy for estimation: if one wants to prove a bound such as <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \ll X}" class="latex" />, it will suffice to obtain a decomposition <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%3D+A_1+%2B+%5Cdots+%2B+A_k&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%3D+A_1+%2B+%5Cdots+%2B+A_k&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%3D+A_1+%2B+%5Cdots+%2B+A_k&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A = A_1 + \dots + A_k" class="latex" />
or at least an upper bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+A_1+%2B+%5Cdots+%2B+A_k&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+A_1+%2B+%5Cdots+%2B+A_k&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+A_1+%2B+%5Cdots+%2B+A_k&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A \ll A_1 + \dots + A_k" class="latex" />
of <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> by some bounded number of components <img src="https://s0.wp.com/latex.php?latex=%7BA_1%2C%5Cdots%2CA_k%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_1%2C%5Cdots%2CA_k%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_1%2C%5Cdots%2CA_k%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_1,\dots,A_k}" class="latex" />, and establish the bounds <img src="https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%2C+%5Cdots%2C+A_k+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%2C+%5Cdots%2C+A_k+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%2C+%5Cdots%2C+A_k+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_1 \ll X, \dots, A_k \ll X}" class="latex" /> separately. Typically the <img src="https://s0.wp.com/latex.php?latex=%7BA_1%2C%5Cdots%2CA_k%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_1%2C%5Cdots%2CA_k%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_1%2C%5Cdots%2CA_k%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_1,\dots,A_k}" class="latex" /> will be (morally at least) smaller than the original quantity <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> – for instance, if <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> is a sum of non-negative quantities, each of the <img src="https://s0.wp.com/latex.php?latex=%7BA_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_i}" class="latex" /> might be a subsum of those same quantities – which means that such a decomposition is a “free move”, in the sense that it does not risk making the problem harder. (This is because, if the original bound <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \ll X}" class="latex" /> is to be true, each of the new objectives <img src="https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%2C+%5Cdots%2C+A_k+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%2C+%5Cdots%2C+A_k+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%2C+%5Cdots%2C+A_k+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_1 \ll X, \dots, A_k \ll X}" class="latex" /> must also be true, and so the decomposition can only make the problem logically easier, not harder.) The only costs to such decomposition are that your proofs might be <img src="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k}" class="latex" /> times longer, as you may be repeating the same arguments <img src="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k}" class="latex" /> times, and that the implied constants in the <img src="https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%2C+%5Cdots%2C+A_k+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%2C+%5Cdots%2C+A_k+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%2C+%5Cdots%2C+A_k+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_1 \ll X, \dots, A_k \ll X}" class="latex" /> bounds may be worse than the implied constant in the original <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \ll X}" class="latex" /> bound. However, in many cases these costs are well worth the benefits of being able to simplify the problem into smaller pieces. As mentioned above, once one successfully executes a divide and conquer strategy, one can go back and try to reduce the number of decompositions, for instance by unifying components that are treated by similar methods, or by replacing strong but unwieldy estimates with weaker, but more convenient estimates.

The above divide and conquer strategy does not directly apply when one is decomposing into an unbounded number of pieces <img src="https://s0.wp.com/latex.php?latex=%7BA_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_j}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Bj%3D1%2C2%2C%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%3D1%2C2%2C%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%3D1%2C2%2C%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j=1,2,\dots}" class="latex" />. In such cases, one needs an additional gain in the index <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" /> that is summable in <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" /> in order to conclude. For instance, if one wants to establish a bound of the form <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \ll X}" class="latex" />, and one has located a decomposition or upper bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Csum_%7Bj%3D1%7D%5E%5Cinfty+A_j&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Csum_%7Bj%3D1%7D%5E%5Cinfty+A_j&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Csum_%7Bj%3D1%7D%5E%5Cinfty+A_j&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A \ll \sum_{j=1}^\infty A_j" class="latex" />
that looks promising for the problem, then it would suffice to obtain exponentially decaying bounds such as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+2%5E%7B-cj%7D+X&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+2%5E%7B-cj%7D+X&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+2%5E%7B-cj%7D+X&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A_j \ll 2^{-cj} X" class="latex" />
for all <img src="https://s0.wp.com/latex.php?latex=%7Bj+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j \geq 1}" class="latex" /> and some constant <img src="https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c>0}" class="latex" />, since this would imply <a name="2cj"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Csum_%7Bj%3D1%7D%5E%5Cinfty+2%5E%7B-cj%7D+X+%5Cll+X+%5C+%5C+%5C+%5C+%5C+%287%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Csum_%7Bj%3D1%7D%5E%5Cinfty+2%5E%7B-cj%7D+X+%5Cll+X+%5C+%5C+%5C+%5C+%5C+%287%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Csum_%7Bj%3D1%7D%5E%5Cinfty+2%5E%7B-cj%7D+X+%5Cll+X+%5C+%5C+%5C+%5C+%5C+%287%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A \ll \sum_{j=1}^\infty 2^{-cj} X \ll X \ \ \ \ \ (7)" class="latex" />
</a> thanks to the geometric series formula. (Here it is important that the implied constants in the asymptotic notation are uniform on <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" />; a <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" />-dependent bound such as <img src="https://s0.wp.com/latex.php?latex=%7BA_j+%5Cll_j+2%5E%7B-cj%7D+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_j+%5Cll_j+2%5E%7B-cj%7D+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_j+%5Cll_j+2%5E%7B-cj%7D+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_j \ll_j 2^{-cj} X}" class="latex" /> would be useless for this application, as then the growth of the implied constant in <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" /> could overwhelm the exponential decay in the <img src="https://s0.wp.com/latex.php?latex=%7B2%5E%7B-cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5E%7B-cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5E%7B-cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^{-cj}}" class="latex" /> factor). Exponential decay is in fact overkill; polynomial decay such as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+%5Cfrac%7BX%7D%7Bj%5E%7B1%2Bc%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+%5Cfrac%7BX%7D%7Bj%5E%7B1%2Bc%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+%5Cfrac%7BX%7D%7Bj%5E%7B1%2Bc%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A_j \ll \frac{X}{j^{1+c}}" class="latex" />
would already be sufficient, although harmonic decay such <a name="ajx"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+%5Cfrac%7BX%7D%7Bj%7D+%5C+%5C+%5C+%5C+%5C+%288%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+%5Cfrac%7BX%7D%7Bj%7D+%5C+%5C+%5C+%5C+%5C+%288%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+%5Cfrac%7BX%7D%7Bj%7D+%5C+%5C+%5C+%5C+%5C+%288%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A_j \ll \frac{X}{j} \ \ \ \ \ (8)" class="latex" />
</a> is not quite enough (the sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{j=1}^\infty \frac{1}{j}}" class="latex" /> diverges logarithmically), although in many such situations one could try to still salvage the bound by working a lot harder to squeeze some additional logarithmic factors out of one’s estimates. For instance, if one can improve <a href="#ajx">(8)</a> to <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+%5Cfrac%7BX%7D%7Bj+%5Clog%5E%7B1%2Bc%7D+j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+%5Cfrac%7BX%7D%7Bj+%5Clog%5E%7B1%2Bc%7D+j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A_j+%5Cll+%5Cfrac%7BX%7D%7Bj+%5Clog%5E%7B1%2Bc%7D+j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A_j \ll \frac{X}{j \log^{1+c} j}" class="latex" />
for all <img src="https://s0.wp.com/latex.php?latex=%7Bj+%5Cgeq+2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj+%5Cgeq+2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj+%5Cgeq+2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j \geq 2}" class="latex" /> and some constant <img src="https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c>0}" class="latex" />, since (by the integral test) the sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D2%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bj%5Clog%5E%7B1%2Bc%7D+j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D2%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bj%5Clog%5E%7B1%2Bc%7D+j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D2%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bj%5Clog%5E%7B1%2Bc%7D+j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{j=2}^\infty \frac{1}{j\log^{1+c} j}}" class="latex" /> converges (and one can treat the <img src="https://s0.wp.com/latex.php?latex=%7Bj%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j=1}" class="latex" /> term separately if one already has <a href="#ajx">(8)</a>).

Sometimes, when trying to prove an estimate such as <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \ll X}" class="latex" />, one has identified a promising decomposition with an unbounded number of terms <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Csum_%7Bj%3D1%7D%5EJ+A_j&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Csum_%7Bj%3D1%7D%5EJ+A_j&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%5Cll+%5Csum_%7Bj%3D1%7D%5EJ+A_j&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A \ll \sum_{j=1}^J A_j" class="latex" />
(where <img src="https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{J}" class="latex" /> is finite but unbounded) but is unsure of how to proceed next. Often the next thing to do is to study the extreme terms <img src="https://s0.wp.com/latex.php?latex=%7BA_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BA_J%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_J%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_J%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_J}" class="latex" /> of this decomposition, and first try to establish (the presumably simpler) tasks of showing that <img src="https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_1+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_1 \ll X}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BA_J+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_J+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_J+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_J \ll X}" class="latex" />. Often once one does so, it becomes clear how to combine the treatments of the two extreme cases to also treat the intermediate cases, obtaining a bound <img src="https://s0.wp.com/latex.php?latex=%7BA_j+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_j+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_j+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_j \ll X}" class="latex" /> for each individual term, leading to the inferior bound <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+JX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Cll+JX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Cll+JX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \ll JX}" class="latex" />; this can then be used as a starting point to hunt for additional gains, such as the exponential or polynomial gains mentioned previously, that could be used to remove this loss of <img src="https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{J}" class="latex" />. (There are more advanced techniques, such as those based on controlling moments such as the square function <img src="https://s0.wp.com/latex.php?latex=%7B%28%5Csum_%7Bj%3D1%7D%5EJ+%7CA_j%7C%5E2%29%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28%5Csum_%7Bj%3D1%7D%5EJ+%7CA_j%7C%5E2%29%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28%5Csum_%7Bj%3D1%7D%5EJ+%7CA_j%7C%5E2%29%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(\sum_{j=1}^J |A_j|^2)^{1/2}}" class="latex" />, or trying to understand the precise circumstances in which a “large values” scenario <img src="https://s0.wp.com/latex.php?latex=%7B%7CA_j%7C+%5Cgg+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA_j%7C+%5Cgg+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA_j%7C+%5Cgg+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A_j| \gg X}" class="latex" /> occurs, and how these scenarios interact with each other for different <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" />, but these are beyond the scope of this post, as they are rarely needed when dealing with sums or integrals of elementary functions.)

If one is faced with the task of estimating a doubly infinite sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D-%5Cinfty%7D%5E%5Cinfty+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D-%5Cinfty%7D%5E%5Cinfty+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D-%5Cinfty%7D%5E%5Cinfty+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{j=-\infty}^\infty A_j}" class="latex" />, it can often be useful to first think about how one would proceed in estimating <img src="https://s0.wp.com/latex.php?latex=%7BA_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_j}" class="latex" /> when <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" /> is very large and positive, and how one would proceed when <img src="https://s0.wp.com/latex.php?latex=%7BA_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_j}" class="latex" /> is very large and negative. In many cases, one can simply decompose the sum into two pieces such as <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D1%7D%5E%5Cinfty+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D1%7D%5E%5Cinfty+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D1%7D%5E%5Cinfty+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{j=1}^\infty A_j}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D-%5Cinfty%7D%5E%7B-1%7D+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D-%5Cinfty%7D%5E%7B-1%7D+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D-%5Cinfty%7D%5E%7B-1%7D+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{j=-\infty}^{-1} A_j}" class="latex" /> and use whatever methods you came up with to handle the two extreme cases; in some cases one also needs a third argument to handle the case when <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" /> is of bounded (or somewhat bounded) size, in which case one may need to divide into three pieces such as <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3DJ_%2B%7D%5E%5Cinfty+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3DJ_%2B%7D%5E%5Cinfty+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3DJ_%2B%7D%5E%5Cinfty+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{j=J_+}^\infty A_j}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D-%5Cinfty%7D%5E%7BJ_-%7D+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D-%5Cinfty%7D%5E%7BJ_-%7D+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3D-%5Cinfty%7D%5E%7BJ_-%7D+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{j=-\infty}^{J_-} A_j}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3DJ_-%2B1%7D%5E%7BJ_%2B-1%7D+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3DJ_-%2B1%7D%5E%7BJ_%2B-1%7D+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bj%3DJ_-%2B1%7D%5E%7BJ_%2B-1%7D+A_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{j=J_-+1}^{J_+-1} A_j}" class="latex" />. Sometimes there will be a natural candidate for the places <img src="https://s0.wp.com/latex.php?latex=%7BJ_-%2C+J_%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BJ_-%2C+J_%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BJ_-%2C+J_%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{J_-, J_+}" class="latex" /> where one is cutting the sum, but in other situations it may be best to just leave these cut points as unspecified parameters initially, obtain bounds that depend on these parameters, and optimize at the end. (Typically, the optimization proceeds by trying to balance the magnitude of a term that is increasing with respect to a parameter, with one that is decreasing. For instance, if one ends up with a bound such as <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Clambda+%2B+B%2F%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Clambda+%2B+B%2F%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Clambda+%2B+B%2F%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \lambda + B/\lambda}" class="latex" /> for some parameter <img src="https://s0.wp.com/latex.php?latex=%7B%5Clambda%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clambda%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clambda%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\lambda>0}" class="latex" /> and quantities <img src="https://s0.wp.com/latex.php?latex=%7BA%2CB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2CB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2CB%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A,B>0}" class="latex" />, it makes sense to select <img src="https://s0.wp.com/latex.php?latex=%7B%5Clambda+%3D+%5Csqrt%7BB%2FA%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clambda+%3D+%5Csqrt%7BB%2FA%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clambda+%3D+%5Csqrt%7BB%2FA%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\lambda = \sqrt{B/A}}" class="latex" /> to balance the two terms. Or, if faced with something like <img src="https://s0.wp.com/latex.php?latex=%7BA+e%5E%7B-%5Clambda%7D+%2B+%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+e%5E%7B-%5Clambda%7D+%2B+%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+e%5E%7B-%5Clambda%7D+%2B+%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A e^{-\lambda} + \lambda}" class="latex" /> for some <img src="https://s0.wp.com/latex.php?latex=%7BA+%3E+2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%3E+2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%3E+2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A > 2}" class="latex" />, then something like <img src="https://s0.wp.com/latex.php?latex=%7B%5Clambda+%3D+%5Clog+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clambda+%3D+%5Clog+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clambda+%3D+%5Clog+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\lambda = \log A}" class="latex" /> would be close to the optimal choice of parameter. And so forth.)

 — 1.1. Psychological distinctions between exact and asymptotic arithmetic — 

The adoption of the “divide and conquer” strategy requires a certain mental shift from the “simplify, simplify” strategy that one is taught in high school algebra. In the latter strategy, one tries to collect terms in an expression make them as short as possible, for instance by working with a common denominator, with the idea that unified and elegant-looking expressions are “simpler” than sprawling expressions with many terms. In contrast, the divide and conquer strategy is intentionally extremely willing to greatly increase the total length of the expressions to be estimated, so long as each individual component of the expressions appears easier to estimate than the original one. Both strategies are still trying to reduce the original problem to a simpler problem (or collection of simpler sub-problems), but the metric by which one judges whether the problem has become simpler is rather different.

A related mental shift that one needs to adopt in analysis is to move away from the exact identities that are so prized in algebra (and in undergraduate calculus), as the precision they offer is often unnecessary and distracting for the task at hand, and often fail to generalize to more complicated contexts in which exact identities are no longer available. As a simple example, consider the task of estimating the expression <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^a \frac{dx}{1+x^2}" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7Ba+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a > 0}" class="latex" /> is a parameter. With a trigonometric substitution, one can evaluate this expression exactly as <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Barctan%7D%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Barctan%7D%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Barctan%7D%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mathrm{arctan}(a)}" class="latex" />, however the presence of the arctangent can be inconvenient if one has to do further estimation tasks (for instance, if <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" /> depends in a complicated fashion on other parameters, which one then also wants to sum or integrate over). Instead, by observing the trivial bounds <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Cleq+%5Cint_0%5Ea%5C+dx+%3D+a&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Cleq+%5Cint_0%5Ea%5C+dx+%3D+a&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Cleq+%5Cint_0%5Ea%5C+dx+%3D+a&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^a \frac{dx}{1+x^2} \leq \int_0^a\ dx = a" class="latex" />
and <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Cleq+%5Cint_0%5E%5Cinfty%5C+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%3D+%5Cfrac%7B%5Cpi%7D%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Cleq+%5Cint_0%5E%5Cinfty%5C+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%3D+%5Cfrac%7B%5Cpi%7D%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Cleq+%5Cint_0%5E%5Cinfty%5C+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%3D+%5Cfrac%7B%5Cpi%7D%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^a \frac{dx}{1+x^2} \leq \int_0^\infty\ \frac{dx}{1+x^2} = \frac{\pi}{2}" class="latex" />
one can combine them using <a href="#min">(2)</a> to obtain the upper bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Cleq+%5Cmin%28+a%2C+%5Cfrac%7B%5Cpi%7D%7B2%7D+%29+%5Casymp+%5Cmin%28a%2C1%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Cleq+%5Cmin%28+a%2C+%5Cfrac%7B%5Cpi%7D%7B2%7D+%29+%5Casymp+%5Cmin%28a%2C1%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Cleq+%5Cmin%28+a%2C+%5Cfrac%7B%5Cpi%7D%7B2%7D+%29+%5Casymp+%5Cmin%28a%2C1%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^a \frac{dx}{1+x^2} \leq \min( a, \frac{\pi}{2} ) \asymp \min(a,1)" class="latex" />
and similar arguments also give the matching lower bound, thus <a name="xai"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Casymp+%5Cmin%28a%2C1%29.+%5C+%5C+%5C+%5C+%5C+%289%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Casymp+%5Cmin%28a%2C1%29.+%5C+%5C+%5C+%5C+%5C+%289%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D+%5Casymp+%5Cmin%28a%2C1%29.+%5C+%5C+%5C+%5C+%5C+%289%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^a \frac{dx}{1+x^2} \asymp \min(a,1). \ \ \ \ \ (9)" class="latex" />
</a> This bound, while cruder than the exact answer of <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Barctan%7D%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Barctan%7D%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Barctan%7D%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mathrm{arctan}(a)}" class="latex" />, is often good enough for many applications (particularly in situations where one is willing to concede constants in the bounds), and can be more tractible to work with than the exact answer. Furthermore, these arguments can be adapted without difficulty to treat similar expressions such as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B%281%2Bx%5E2%29%5E%5Calpha%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B%281%2Bx%5E2%29%5E%5Calpha%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5Ea+%5Cfrac%7Bdx%7D%7B%281%2Bx%5E2%29%5E%5Calpha%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^a \frac{dx}{(1+x^2)^\alpha}" class="latex" />
for any fixed exponent <img src="https://s0.wp.com/latex.php?latex=%7B%5Calpha%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Calpha%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Calpha%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\alpha>0}" class="latex" />, which need not have closed form exact expressions in terms of elementary functions such as the arctangent when <img src="https://s0.wp.com/latex.php?latex=%7B%5Calpha%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Calpha%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Calpha%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\alpha}" class="latex" /> is non-integer.

As a general rule, instead of relying exclusively on exact formulae, one should seek approximations that are valid up to the degree of precision that one seeks in the final estimate. For instance, suppose one one wishes to establish the bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csec%28x%29+-+%5Ccos%28x%29+%3D+x%5E2+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csec%28x%29+-+%5Ccos%28x%29+%3D+x%5E2+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csec%28x%29+-+%5Ccos%28x%29+%3D+x%5E2+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sec(x) - \cos(x) = x^2 + O(x^3)" class="latex" />
for all sufficiently small <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" />. If one was clinging to the exact identity mindset, one could try to look for some trigonometric identity to simplify the left-hand side exactly, but the quicker (and more robust) way to proceed is just to use Taylor expansion up to the specified accuracy <img src="https://s0.wp.com/latex.php?latex=%7BO%28x%5E3%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28x%5E3%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28x%5E3%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(x^3)}" class="latex" /> to obtain <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Ccos%28x%29+%3D+1+-+%5Cfrac%7Bx%5E2%7D%7B2%7D+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Ccos%28x%29+%3D+1+-+%5Cfrac%7Bx%5E2%7D%7B2%7D+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Ccos%28x%29+%3D+1+-+%5Cfrac%7Bx%5E2%7D%7B2%7D+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \cos(x) = 1 - \frac{x^2}{2} + O(x^3)" class="latex" />
which one can invert using the geometric series formula <img src="https://s0.wp.com/latex.php?latex=%7B%281-y%29%5E%7B-1%7D+%3D+1+%2B+y+%2B+y%5E2+%2B+%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%281-y%29%5E%7B-1%7D+%3D+1+%2B+y+%2B+y%5E2+%2B+%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%281-y%29%5E%7B-1%7D+%3D+1+%2B+y+%2B+y%5E2+%2B+%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(1-y)^{-1} = 1 + y + y^2 + \dots}" class="latex" /> to obtain <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csec%28x%29+%3D+1+%2B+%5Cfrac%7Bx%5E2%7D%7B2%7D+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csec%28x%29+%3D+1+%2B+%5Cfrac%7Bx%5E2%7D%7B2%7D+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csec%28x%29+%3D+1+%2B+%5Cfrac%7Bx%5E2%7D%7B2%7D+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sec(x) = 1 + \frac{x^2}{2} + O(x^3)" class="latex" />
from which the claim follows. (One could also have computed the Taylor expansion of <img src="https://s0.wp.com/latex.php?latex=%7B%5Csec%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csec%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csec%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sec(x)}" class="latex" /> by repeatedly differentiating the secant function, but as this is a series that is usually not memorized, this can take a little bit more time than just computing it directly to the required accuracy as indicated above.) Note that the notion of “specified accuracy” may have to be interpreted in a relative sense if one is planning to multiply or divide several estimates together. For instance, if one wishes to establsh the bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csin%28x%29+%5Ccos%28x%29+%3D+x+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csin%28x%29+%5Ccos%28x%29+%3D+x+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csin%28x%29+%5Ccos%28x%29+%3D+x+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sin(x) \cos(x) = x + O(x^3)" class="latex" />
for small <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" />, one needs an approximation <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csin%28x%29+%3D+x+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csin%28x%29+%3D+x+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csin%28x%29+%3D+x+%2B+O%28x%5E3%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sin(x) = x + O(x^3)" class="latex" />
to the sine function that is accurate to order <img src="https://s0.wp.com/latex.php?latex=%7BO%28x%5E3%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28x%5E3%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28x%5E3%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(x^3)}" class="latex" />, but one only needs an approximation <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Ccos%28x%29+%3D+1+%2B+O%28x%5E2%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Ccos%28x%29+%3D+1+%2B+O%28x%5E2%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Ccos%28x%29+%3D+1+%2B+O%28x%5E2%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \cos(x) = 1 + O(x^2)" class="latex" />
to the cosine function that is accurate to order <img src="https://s0.wp.com/latex.php?latex=%7BO%28x%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28x%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28x%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(x^2)}" class="latex" />, because the cosine is to be multiplied by <img src="https://s0.wp.com/latex.php?latex=%7B%5Csin%28x%29%3D+O%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csin%28x%29%3D+O%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csin%28x%29%3D+O%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sin(x)= O(x)}" class="latex" />. Here the key is to obtain estimates that have a relative error of <img src="https://s0.wp.com/latex.php?latex=%7BO%28x%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28x%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28x%5E2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(x^2)}" class="latex" />, compared to the main term (which is <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> for cosine, and <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" /> for sine).

The following table lists some common approximations that can be used to simplify expressions when one is only interested in order of magnitude bounds (with <img src="https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c>0}" class="latex" /> an arbitrary small constant):

<table align="center"><tr><td align="left"> The quantity… </td><td align="left"> has magnitude comparable to … </td><td align="left"> provided that… </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X+Y}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+Y+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+Y+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%5Cleq+Y+%5Cll+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 \leq Y \ll X}" class="latex" /> or <img src="https://s0.wp.com/latex.php?latex=%7B%7CY%7C+%5Cleq+%281-c%29X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CY%7C+%5Cleq+%281-c%29X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CY%7C+%5Cleq+%281-c%29X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|Y| \leq (1-c)X}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X+Y}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmax%28X%2CY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmax%28X%2CY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmax%28X%2CY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\max(X,Y)}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7BX%2CY+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2CY+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2CY+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X,Y \geq 0}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Csin+z%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csin+z%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csin+z%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sin z}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5Ctan+z%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ctan+z%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ctan+z%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\tan z}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7Biz%7D-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7Biz%7D-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7Biz%7D-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{iz}-1}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cz%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|z|}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D+-+c%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D+-+c%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D+-+c%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|z| \leq \frac{\pi}{2} - c}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Ccos+z%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ccos+z%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ccos+z%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\cos z}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cpi%2F2+-+c%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cpi%2F2+-+c%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cpi%2F2+-+c%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|z| \leq \pi/2 - c}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Csin+x%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csin+x%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csin+x%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sin x}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Bdist%7D%28x%2C+%5Cpi+%7B%5Cbf+Z%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Bdist%7D%28x%2C+%5Cpi+%7B%5Cbf+Z%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Bdist%7D%28x%2C+%5Cpi+%7B%5Cbf+Z%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mathrm{dist}(x, \pi {\bf Z})}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" /> real </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7Bix%7D-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7Bix%7D-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7Bix%7D-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{ix}-1}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Bdist%7D%28x%2C+2%5Cpi+%7B%5Cbf+Z%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Bdist%7D%28x%2C+2%5Cpi+%7B%5Cbf+Z%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Bdist%7D%28x%2C+2%5Cpi+%7B%5Cbf+Z%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mathrm{dist}(x, 2\pi {\bf Z})}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" /> real </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Barcsin%7D+x%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Barcsin%7D+x%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Barcsin%7D+x%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mathrm{arcsin} x}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cx%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cx%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cx%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|x|}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cx%7C+%5Cleq+1-c%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cx%7C+%5Cleq+1-c%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cx%7C+%5Cleq+1-c%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|x| \leq 1-c}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Clog%281%2Bz%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clog%281%2Bz%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clog%281%2Bz%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\log(1+z)}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cz%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|z|}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+1-c%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+1-c%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+1-c%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|z| \leq 1-c}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Be%5Ez-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5Ez-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5Ez-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^z-1}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5Csinh+z%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csinh+z%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csinh+z%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sinh z}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5Ctanh+z%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ctanh+z%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ctanh+z%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\tanh z}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cz%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|z|}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D-c%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D-c%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D-c%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|z| \leq \frac{\pi}{2}-c}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Ccosh+z%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ccosh+z%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ccosh+z%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\cosh z}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D-c%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D-c%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D-c%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|z| \leq \frac{\pi}{2}-c}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Csinh+x%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csinh+x%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csinh+x%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sinh x}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5Ccosh+x%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ccosh+x%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ccosh+x%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\cosh x}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Be%5Ex%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5Ex%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5Ex%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^x}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cx%7C+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cx%7C+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cx%7C+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|x| \gg 1}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Ctanh+x%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ctanh+x%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ctanh+x%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\tanh x}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmin%28%7Cx%7C%2C+1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmin%28%7Cx%7C%2C+1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmin%28%7Cx%7C%2C+1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\min(|x|, 1)}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" /> real </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%281%2Bx%29%5Ea-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%281%2Bx%29%5Ea-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%281%2Bx%29%5Ea-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(1+x)^a-1}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Ba%7Cx%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7Cx%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7Cx%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a|x|}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Ba+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a \gg 1}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Ba+%7Cx%7C+%5Cll+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba+%7Cx%7C+%5Cll+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba+%7Cx%7C+%5Cll+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a |x| \ll 1}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Bn%21%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%21%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%21%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n!}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Bn%5En+e%5E%7B-n%7D+%5Csqrt%7Bn%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%5En+e%5E%7B-n%7D+%5Csqrt%7Bn%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%5En+e%5E%7B-n%7D+%5Csqrt%7Bn%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n^n e^{-n} \sqrt{n}}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \geq 1}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5CGamma%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CGamma%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CGamma%28s%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Gamma(s)}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cs%5Es+e%5E%7B-s%7D%7C+%2F+%7Cs%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cs%5Es+e%5E%7B-s%7D%7C+%2F+%7Cs%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cs%5Es+e%5E%7B-s%7D%7C+%2F+%7Cs%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|s^s e^{-s}| / |s|^{1/2}}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cz%7C+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|z| \gg 1}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%7C%5Cmathrm%7Barg%7D+z%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D+-+c%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7C%5Cmathrm%7Barg%7D+z%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D+-+c%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7C%5Cmathrm%7Barg%7D+z%7C+%5Cleq+%5Cfrac%7B%5Cpi%7D%7B2%7D+-+c%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|\mathrm{arg} z| \leq \frac{\pi}{2} - c}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5CGamma%28%5Csigma%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CGamma%28%5Csigma%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CGamma%28%5Csigma%2Bit%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Gamma(\sigma+it)}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%7Ct%7C%5E%7B%5Csigma-1%2F2%7D+e%5E%7B-%5Cpi+%7Ct%7C%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Ct%7C%5E%7B%5Csigma-1%2F2%7D+e%5E%7B-%5Cpi+%7Ct%7C%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Ct%7C%5E%7B%5Csigma-1%2F2%7D+e%5E%7B-%5Cpi+%7Ct%7C%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|t|^{\sigma-1/2} e^{-\pi |t|/2}}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Csigma+%3D+O%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csigma+%3D+O%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csigma+%3D+O%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sigma = O(1)}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%7Ct%7C+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Ct%7C+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Ct%7C+%5Cgg+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|t| \gg 1}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bm%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bm%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bm%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\binom{n}{m}}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7Bn+%28p+%5Clog+%5Cfrac%7B1%7D%7Bp%7D+%2B+%281-p%29+%5Clog+%5Cfrac%7B1%7D%7B1-p%7D%29%7D+%2F+n%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7Bn+%28p+%5Clog+%5Cfrac%7B1%7D%7Bp%7D+%2B+%281-p%29+%5Clog+%5Cfrac%7B1%7D%7B1-p%7D%29%7D+%2F+n%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7Bn+%28p+%5Clog+%5Cfrac%7B1%7D%7Bp%7D+%2B+%281-p%29+%5Clog+%5Cfrac%7B1%7D%7B1-p%7D%29%7D+%2F+n%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{n (p \log \frac{1}{p} + (1-p) \log \frac{1}{1-p})} / n^{1/2}}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Bm%3Dpn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3Dpn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3Dpn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=pn}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Bc+%3C+p+%3C+1-c%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc+%3C+p+%3C+1-c%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc+%3C+p+%3C+1-c%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bm%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bm%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bm%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\binom{n}{m}}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B2%5En+e%5E%7B-2%28m-n%2F2%29%5E2%7D+%2F+n%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5En+e%5E%7B-2%28m-n%2F2%29%5E2%7D+%2F+n%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5En+e%5E%7B-2%28m-n%2F2%29%5E2%7D+%2F+n%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^n e^{-2(m-n/2)^2} / n^{1/2}}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Bm+%3D+n%2F2+%2B+O%28n%5E%7B2%2F3%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm+%3D+n%2F2+%2B+O%28n%5E%7B2%2F3%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm+%3D+n%2F2+%2B+O%28n%5E%7B2%2F3%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m = n/2 + O(n^{2/3})}" class="latex" /> </td></tr><tr><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bm%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bm%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bm%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\binom{n}{m}}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Bn%5Em%2Fm%21%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%5Em%2Fm%21%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%5Em%2Fm%21%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n^m/m!}" class="latex" /> </td><td align="left"> <img src="https://s0.wp.com/latex.php?latex=%7Bm+%5Cll+%5Csqrt%7Bn%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm+%5Cll+%5Csqrt%7Bn%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm+%5Cll+%5Csqrt%7Bn%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m \ll \sqrt{n}}" class="latex" /> </td></tr></table>

On the other hand, some exact formulae are still very useful, particularly if the end result of that formula is clean and tractable to work with (as opposed to involving somewhat exotic functions such as the arctangent). The geometric series formula, for instance, is an extremely handy exact formula, so much so that it is often desirable to control summands by a geometric series purely to use this formula (we already saw an example of this in <a href="#2cj">(7)</a>). Exact integral identities, such as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B1%7D%7Ba%7D+%3D+%5Cint_0%5E%5Cinfty+e%5E%7B-at%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B1%7D%7Ba%7D+%3D+%5Cint_0%5E%5Cinfty+e%5E%7B-at%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B1%7D%7Ba%7D+%3D+%5Cint_0%5E%5Cinfty+e%5E%7B-at%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{1}{a} = \int_0^\infty e^{-at}\ dt" class="latex" />
or more generally <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B%5CGamma%28s%29%7D%7Ba%5Es%7D+%3D+%5Cint_0%5E%5Cinfty+e%5E%7B-at%7D+t%5E%7Bs-1%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B%5CGamma%28s%29%7D%7Ba%5Es%7D+%3D+%5Cint_0%5E%5Cinfty+e%5E%7B-at%7D+t%5E%7Bs-1%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B%5CGamma%28s%29%7D%7Ba%5Es%7D+%3D+%5Cint_0%5E%5Cinfty+e%5E%7B-at%7D+t%5E%7Bs-1%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{\Gamma(s)}{a^s} = \int_0^\infty e^{-at} t^{s-1}\ dt" class="latex" />
for <img src="https://s0.wp.com/latex.php?latex=%7Ba%2Cs%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%2Cs%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%2Cs%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a,s>0}" class="latex" /> (where <img src="https://s0.wp.com/latex.php?latex=%7B%5CGamma%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CGamma%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CGamma%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Gamma}" class="latex" /> is the <a href="https://en.wikipedia.org/wiki/Gamma_function">Gamma function</a>) are also quite commonly used, and fundamental exact integration rules such as the change of variables formula, the Fubini-Tonelli theorem or integration by parts are all esssential tools for an analyst trying to prove estimates. Because of this, it is often desirable to estimate a sum by an integral. The <a href="https://en.wikipedia.org/wiki/Integral_test_for_convergence">integral test</a> is a classic example of this principle in action: a more quantitative versions of this test is the bound <a name="integral"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7Ba%7D%5E%7Bb%2B1%7D+f%28t%29%5C+dt+%5Cleq+%5Csum_%7Bn%3Da%7D%5Eb+f%28n%29+%5Cleq+%5Cint_%7Ba-1%7D%5Eb+f%28t%29%5C+dt+%5C+%5C+%5C+%5C+%5C+%2810%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7Ba%7D%5E%7Bb%2B1%7D+f%28t%29%5C+dt+%5Cleq+%5Csum_%7Bn%3Da%7D%5Eb+f%28n%29+%5Cleq+%5Cint_%7Ba-1%7D%5Eb+f%28t%29%5C+dt+%5C+%5C+%5C+%5C+%5C+%2810%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7Ba%7D%5E%7Bb%2B1%7D+f%28t%29%5C+dt+%5Cleq+%5Csum_%7Bn%3Da%7D%5Eb+f%28n%29+%5Cleq+%5Cint_%7Ba-1%7D%5Eb+f%28t%29%5C+dt+%5C+%5C+%5C+%5C+%5C+%2810%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{a}^{b+1} f(t)\ dt \leq \sum_{n=a}^b f(n) \leq \int_{a-1}^b f(t)\ dt \ \ \ \ \ (10)" class="latex" />
</a> whenever <img src="https://s0.wp.com/latex.php?latex=%7Ba+%5Cleq+b%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba+%5Cleq+b%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba+%5Cleq+b%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a \leq b}" class="latex" /> are integers and <img src="https://s0.wp.com/latex.php?latex=%7Bf%3A+%5Ba-1%2Cb%2B1%5D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%3A+%5Ba-1%2Cb%2B1%5D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%3A+%5Ba-1%2Cb%2B1%5D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f: [a-1,b+1] \rightarrow {\bf R}}" class="latex" /> is monotone decreasing, or the closely related bound <a name="integral-2"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Ba+%5Cleq+n+%5Cleq+b%7D+f%28n%29+%3D+%5Cint_a%5Eb+f%28t%29%5C+dt+%2B+O%28+%7Cf%28a%29%7C+%2B+%7Cf%28b%29%7C+%29+%5C+%5C+%5C+%5C+%5C+%2811%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Ba+%5Cleq+n+%5Cleq+b%7D+f%28n%29+%3D+%5Cint_a%5Eb+f%28t%29%5C+dt+%2B+O%28+%7Cf%28a%29%7C+%2B+%7Cf%28b%29%7C+%29+%5C+%5C+%5C+%5C+%5C+%2811%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Ba+%5Cleq+n+%5Cleq+b%7D+f%28n%29+%3D+%5Cint_a%5Eb+f%28t%29%5C+dt+%2B+O%28+%7Cf%28a%29%7C+%2B+%7Cf%28b%29%7C+%29+%5C+%5C+%5C+%5C+%5C+%2811%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{a \leq n \leq b} f(n) = \int_a^b f(t)\ dt + O( |f(a)| + |f(b)| ) \ \ \ \ \ (11)" class="latex" />
</a> whenever <img src="https://s0.wp.com/latex.php?latex=%7Ba+%5Cgeq+b%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba+%5Cgeq+b%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba+%5Cgeq+b%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a \geq b}" class="latex" /> are reals and <img src="https://s0.wp.com/latex.php?latex=%7Bf%3A+%5Ba%2Cb%5D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%3A+%5Ba%2Cb%5D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%3A+%5Ba%2Cb%5D+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f: [a,b] \rightarrow {\bf R}}" class="latex" /> is monotone (either increasing or decreasing); see Lemma 2 of <a href="https://terrytao.wordpress.com/2014/11/23/254a-notes-1-elementary-multiplicative-number-theory/">this previous post</a>. Such bounds allow one to switch back and forth quite easily between sums and integrals as long as the summand or integrand behaves in a mostly monotone fashion (for instance, if it is monotone increasing on one portion of the domain and monotone decreasing on the other). For more precision, one could turn to more advanced relationships between sums and integrals, such as the <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula">Euler-Maclaurin formula</a> or the <a href="https://en.wikipedia.org/wiki/Poisson_summation_formula">Poisson summation formula</a>, but these are beyond the scope of this post.

<blockquote>Exercise 1 <a name="quasi"></a> Suppose <img src="https://s0.wp.com/latex.php?latex=%7Bf%3A+%7B%5Cbf+R%7D+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%3A+%7B%5Cbf+R%7D+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%3A+%7B%5Cbf+R%7D+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f: {\bf R} \rightarrow {\bf R}^+}" class="latex" /> obeys the quasi-monotonicity property <img src="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%5Cll+f%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%5Cll+f%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28x%29+%5Cll+f%28y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(x) \ll f(y)}" class="latex" /> whenever <img src="https://s0.wp.com/latex.php?latex=%7By-1+%5Cleq+x+%5Cleq+y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By-1+%5Cleq+x+%5Cleq+y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By-1+%5Cleq+x+%5Cleq+y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y-1 \leq x \leq y}" class="latex" />. Show that <img src="https://s0.wp.com/latex.php?latex=%7B%5Cint_a%5E%7Bb-1%7D+f%28t%29%5C+dt+%5Cll+%5Csum_%7Bn%3Da%7D%5Eb+f%28n%29+%5Cll+%5Cint_a%5E%7Bb%2B1%7D+f%28t%29%5C+dt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cint_a%5E%7Bb-1%7D+f%28t%29%5C+dt+%5Cll+%5Csum_%7Bn%3Da%7D%5Eb+f%28n%29+%5Cll+%5Cint_a%5E%7Bb%2B1%7D+f%28t%29%5C+dt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cint_a%5E%7Bb-1%7D+f%28t%29%5C+dt+%5Cll+%5Csum_%7Bn%3Da%7D%5Eb+f%28n%29+%5Cll+%5Cint_a%5E%7Bb%2B1%7D+f%28t%29%5C+dt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\int_a^{b-1} f(t)\ dt \ll \sum_{n=a}^b f(n) \ll \int_a^{b+1} f(t)\ dt}" class="latex" /> for any integers <img src="https://s0.wp.com/latex.php?latex=%7Ba+%3C+b%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba+%3C+b%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba+%3C+b%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a < b}" class="latex" />. </blockquote>


<blockquote>Exercise 2 <a name="Stirling"></a> Use <a href="#integral-2">(11)</a> to obtain the “cheap <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation">Stirling approximation</a>” <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+%5Cexp%28+n+%5Clog+n+-+n+%2B+O%28%5Clog+n%29+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+%5Cexp%28+n+%5Clog+n+-+n+%2B+O%28%5Clog+n%29+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+%5Cexp%28+n+%5Clog+n+-+n+%2B+O%28%5Clog+n%29+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle n! = \exp( n \log n - n + O(\log n) )" class="latex" />
for any natural number <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \geq 2}" class="latex" />. (Hint: take logarithms to convert the product <img src="https://s0.wp.com/latex.php?latex=%7Bn%21+%3D+1+%5Ctimes+2+%5Ctimes+%5Cdots+%5Ctimes+n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%21+%3D+1+%5Ctimes+2+%5Ctimes+%5Cdots+%5Ctimes+n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%21+%3D+1+%5Ctimes+2+%5Ctimes+%5Cdots+%5Ctimes+n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n! = 1 \times 2 \times \dots \times n}" class="latex" /> into a sum.) </blockquote>


With practice, you will be able to identify any term in a computation which is already “negligible” or “acceptable” in the sense that its contribution is always going to lead to an error that is smaller than the desired accuracy of the final estimate. One can then work “modulo” these negligible terms and discard them as soon as they appear. This can help remove a lot of clutter in one’s arguments. For instance, if one wishes to establish an asymptotic of the form <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%3D+X+%2B+O%28Y%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%3D+X+%2B+O%28Y%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%3D+X+%2B+O%28Y%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A = X + O(Y)" class="latex" />
for some main term <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> and lower order error <img src="https://s0.wp.com/latex.php?latex=%7BO%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(Y)}" class="latex" />, any component of <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> that one can already identify to be of size <img src="https://s0.wp.com/latex.php?latex=%7BO%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28Y%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(Y)}" class="latex" /> is negligible and can be removed from <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> “for free”. Conversely, it can be useful to add negligible terms to an expression, if it makes the expression easier to work with. For instance, suppose one wants to estimate the expression <a name="psum"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D.+%5C+%5C+%5C+%5C+%5C+%2812%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D.+%5C+%5C+%5C+%5C+%5C+%2812%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D.+%5C+%5C+%5C+%5C+%5C+%2812%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N \frac{1}{n^2}. \ \ \ \ \ (12)" class="latex" />
</a> This is a partial sum for the zeta function <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Czeta%282%29+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Czeta%282%29+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Czeta%282%29+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} = \zeta(2) = \frac{\pi^2}{6}" class="latex" />
so it can make sense to add and subtract the tail <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n=N+1}^\infty \frac{1}{n^2}}" class="latex" /> to the expression <a href="#psum">(12)</a> to rewrite it as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+-+%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+-+%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+-+%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{\pi^2}{6} - \sum_{n=N+1}^\infty \frac{1}{n^2}." class="latex" />
To deal with the tail, we switch from a sum to the integral using <a href="#integral">(10)</a> to bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%5Cll+%5Cint_N%5E%5Cinfty+%5Cfrac%7B1%7D%7Bt%5E2%7D%5C+dt+%3D+%5Cfrac%7B1%7D%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%5Cll+%5Cint_N%5E%5Cinfty+%5Cfrac%7B1%7D%7Bt%5E2%7D%5C+dt+%3D+%5Cfrac%7B1%7D%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3DN%2B1%7D%5E%5Cinfty+%5Cfrac%7B1%7D%7Bn%5E2%7D+%5Cll+%5Cint_N%5E%5Cinfty+%5Cfrac%7B1%7D%7Bt%5E2%7D%5C+dt+%3D+%5Cfrac%7B1%7D%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=N+1}^\infty \frac{1}{n^2} \ll \int_N^\infty \frac{1}{t^2}\ dt = \frac{1}{N}" class="latex" />
giving us the reasonably accurate bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+-+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+-+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5EN+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D+-+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^N \frac{1}{n^2} = \frac{\pi^2}{6} - O(\frac{1}{N})." class="latex" />
One can sharpen this approximation somewhat using <a href="#integral-2">(11)</a> or the Euler–Maclaurin formula; we leave this to the interested reader.

Another psychological shift when switching from algebraic simplification problems to estimation problems is that one has to be prepared to let go of constraints in an expression that complicate the analysis. Suppose for instance we now wish to estimate the variant <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7B1+%5Cleq+n+%5Cleq+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7B1+%5Cleq+n+%5Cleq+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7B1+%5Cleq+n+%5Cleq+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{1 \leq n \leq N, \hbox{ square-free}} \frac{1}{n^2}" class="latex" />
of <a href="#psum">(12)</a>, where we are now restricting <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" /> to be <a href="https://en.wikipedia.org/wiki/Square-free_integer">square-free</a>. An identity from analytic number theory (the <a href="https://en.wikipedia.org/wiki/Euler_product">Euler product identity</a>) lets us calculate the exact sum <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cgeq+1%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Czeta%282%29%7D%7B%5Czeta%284%29%7D+%3D+%5Cfrac%7B15%7D%7B%5Cpi%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cgeq+1%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Czeta%282%29%7D%7B%5Czeta%284%29%7D+%3D+%5Cfrac%7B15%7D%7B%5Cpi%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cgeq+1%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B%5Czeta%282%29%7D%7B%5Czeta%284%29%7D+%3D+%5Cfrac%7B15%7D%7B%5Cpi%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n \geq 1, \hbox{ square-free}} \frac{1}{n^2} = \frac{\zeta(2)}{\zeta(4)} = \frac{15}{\pi^2}" class="latex" />
so as before we can write the desired expression as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B15%7D%7B%5Cpi%5E2%7D+-+%5Csum_%7Bn+%3E+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B15%7D%7B%5Cpi%5E2%7D+-+%5Csum_%7Bn+%3E+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B15%7D%7B%5Cpi%5E2%7D+-+%5Csum_%7Bn+%3E+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{15}{\pi^2} - \sum_{n > N, \hbox{ square-free}} \frac{1}{n^2}." class="latex" />
Previously, we applied the integral test <a href="#integral">(10)</a>, but this time we cannot do so, because the restriction to square-free integers destroys the monotonicity. But we can simply remove this restriction: <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%3E+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D+%5Cleq+%5Csum_%7Bn+%3E+N%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%3E+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D+%5Cleq+%5Csum_%7Bn+%3E+N%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%3E+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D+%5Cleq+%5Csum_%7Bn+%3E+N%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n > N, \hbox{ square-free}} \frac{1}{n^2} \leq \sum_{n > N} \frac{1}{n^2}." class="latex" />
Heuristically at least, this move only “costs us a constant”, since a positive fraction (<img src="https://s0.wp.com/latex.php?latex=%7B1%2F%5Czeta%282%29%3D+6%2F%5Cpi%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2F%5Czeta%282%29%3D+6%2F%5Cpi%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2F%5Czeta%282%29%3D+6%2F%5Cpi%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1/\zeta(2)= 6/\pi^2}" class="latex" />, in fact) of all integers are square-free. Now that this constraint has been removed, we can use the integral test as before and obtain the reasonably accurate asymptotic <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7B1+%5Cleq+n+%5Cleq+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B15%7D%7B%5Cpi%5E2%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7B1+%5Cleq+n+%5Cleq+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B15%7D%7B%5Cpi%5E2%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7B1+%5Cleq+n+%5Cleq+N%2C+%5Chbox%7B+square-free%7D%7D+%5Cfrac%7B1%7D%7Bn%5E2%7D+%3D+%5Cfrac%7B15%7D%7B%5Cpi%5E2%7D+%2B+O%28%5Cfrac%7B1%7D%7BN%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{1 \leq n \leq N, \hbox{ square-free}} \frac{1}{n^2} = \frac{15}{\pi^2} + O(\frac{1}{N})." class="latex" />

 — 2. More on decomposition — 

The way in which one decomposes a sum or integral such as <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn+%5Cin+A%7D+f%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn+%5Cin+A%7D+f%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bn+%5Cin+A%7D+f%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{n \in A} f(n)}" class="latex" /> or <img src="https://s0.wp.com/latex.php?latex=%7B%5Cint_A+f%28x%29%5C+dx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cint_A+f%28x%29%5C+dx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cint_A+f%28x%29%5C+dx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\int_A f(x)\ dx}" class="latex" /> is often guided by the “geometry” of <img src="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f}" class="latex" />, and in particular where <img src="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f}" class="latex" /> is large or small (or whether various component terms in <img src="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f}" class="latex" /> are large or small relative to each other). For instance, if <img src="https://s0.wp.com/latex.php?latex=%7Bf%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28x%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(x)}" class="latex" /> comes close to a maximum at some point <img src="https://s0.wp.com/latex.php?latex=%7Bx%3Dx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%3Dx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%3Dx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x=x_0}" class="latex" />, then it may make sense to decompose based on the distance <img src="https://s0.wp.com/latex.php?latex=%7B%7Cx-x_0%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cx-x_0%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cx-x_0%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|x-x_0|}" class="latex" /> to <img src="https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_0}" class="latex" />, or perhaps to treat the cases <img src="https://s0.wp.com/latex.php?latex=%7Bx+%5Cleq+x_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx+%5Cleq+x_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx+%5Cleq+x_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x \leq x_0}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bx%3Ex_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%3Ex_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%3Ex_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x>x_0}" class="latex" /> separately. (Note that <img src="https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_0}" class="latex" /> does not literally have to be the maximum in order for this to be a reasonable decomposition; if it is in “within reasonable distance” of the maximum, this could still be a good move. As such, it is often not worthwhile to try to compute the maximum of <img src="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f}" class="latex" /> exactly, especially if this exact formula ends up being too complicated to be useful.)

If an expression involves a distance <img src="https://s0.wp.com/latex.php?latex=%7B%7CX-Y%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CX-Y%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CX-Y%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|X-Y|}" class="latex" /> between two quantities <img src="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X,Y}" class="latex" />, it is sometimes useful to split into the case <img src="https://s0.wp.com/latex.php?latex=%7B%7CX%7C+%5Cleq+%7CY%7C%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CX%7C+%5Cleq+%7CY%7C%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CX%7C+%5Cleq+%7CY%7C%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|X| \leq |Y|/2}" class="latex" /> where <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> is much smaller than <img src="https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y}" class="latex" /> (so that <img src="https://s0.wp.com/latex.php?latex=%7B%7CX-Y%7C+%5Casymp+%7CY%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CX-Y%7C+%5Casymp+%7CY%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CX-Y%7C+%5Casymp+%7CY%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|X-Y| \asymp |Y|}" class="latex" />), the case <img src="https://s0.wp.com/latex.php?latex=%7B%7CY%7C+%5Cleq+%7CX%7C%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CY%7C+%5Cleq+%7CX%7C%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CY%7C+%5Cleq+%7CX%7C%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|Y| \leq |X|/2}" class="latex" /> where <img src="https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y}" class="latex" /> is much smaller than <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> (so that <img src="https://s0.wp.com/latex.php?latex=%7B%7CX-Y%7C+%5Casymp+%7CX%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CX-Y%7C+%5Casymp+%7CX%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CX-Y%7C+%5Casymp+%7CX%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|X-Y| \asymp |X|}" class="latex" />), or the case when neither of the two previous cases apply (so that <img src="https://s0.wp.com/latex.php?latex=%7B%7CX%7C+%5Casymp+%7CY%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CX%7C+%5Casymp+%7CY%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CX%7C+%5Casymp+%7CY%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|X| \asymp |Y|}" class="latex" />). The factors of <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" /> here are not of critical importance; the point is that in each of these three cases, one has some hope of simplifying the expression into something more tractable. For instance, suppose one wants to estimate the expression <a name="xeq"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29+%281%2B%28x-b%29%5E2%29%7D+%5C+%5C+%5C+%5C+%5C+%2813%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29+%281%2B%28x-b%29%5E2%29%7D+%5C+%5C+%5C+%5C+%5C+%2813%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29+%281%2B%28x-b%29%5E2%29%7D+%5C+%5C+%5C+%5C+%5C+%2813%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{-\infty}^\infty \frac{dx}{(1+(x-a)^2) (1+(x-b)^2)} \ \ \ \ \ (13)" class="latex" />
</a> in terms of the two real parameters <img src="https://s0.wp.com/latex.php?latex=%7Ba%2C+b%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%2C+b%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%2C+b%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a, b}" class="latex" />, which we will take to be distinct for sake of this discussion. This particular integral is simple enough that it can be evaluated exactly (for instance using contour integration techniques), but in the spirit of Principle 1, let us avoid doing so and instead try to decompose this expression into simpler pieces. A graph of the integrand reveals that it peaks when <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" /> is near <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" /> or near <img src="https://s0.wp.com/latex.php?latex=%7Bb%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b}" class="latex" />. Inspired by this, one can decompose the region of integration into three pieces:

<ul> <li>(i) The region where <img src="https://s0.wp.com/latex.php?latex=%7B%7Cx-a%7C+%5Cleq+%5Cfrac%7B%7Ca-b%7C%7D%7B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cx-a%7C+%5Cleq+%5Cfrac%7B%7Ca-b%7C%7D%7B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cx-a%7C+%5Cleq+%5Cfrac%7B%7Ca-b%7C%7D%7B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|x-a| \leq \frac{|a-b|}{2}}" class="latex" />. </li><li>(ii) The region where <img src="https://s0.wp.com/latex.php?latex=%7B%7Cx-b%7C+%5Cleq+%5Cfrac%7B%7Ca-b%7C%7D%7B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cx-b%7C+%5Cleq+%5Cfrac%7B%7Ca-b%7C%7D%7B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cx-b%7C+%5Cleq+%5Cfrac%7B%7Ca-b%7C%7D%7B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|x-b| \leq \frac{|a-b|}{2}}" class="latex" />. </li><li>(iii) The region where <img src="https://s0.wp.com/latex.php?latex=%7B%7Cx-a%7C%2C+%7Cx-b%7C+%3E+%5Cfrac%7B%7Ca-b%7C%7D%7B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cx-a%7C%2C+%7Cx-b%7C+%3E+%5Cfrac%7B%7Ca-b%7C%7D%7B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cx-a%7C%2C+%7Cx-b%7C+%3E+%5Cfrac%7B%7Ca-b%7C%7D%7B2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|x-a|, |x-b| > \frac{|a-b|}{2}}" class="latex" />.
</li></ul>

(This is not the only way to cut up the integral, but it will suffice. Often there is no “canonical” or “elegant” way to perform the decomposition; one should just try to find a decomposition that is convenient for the problem at hand.)

The reason why we want to perform such a decomposition is that in each of the three cases, one can simplify how the integrand depends on <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" />. For instance, in region (i), we see from the triangle inequality that <img src="https://s0.wp.com/latex.php?latex=%7B%7Cx-b%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cx-b%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cx-b%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|x-b|}" class="latex" /> is now comparable to <img src="https://s0.wp.com/latex.php?latex=%7B%7Ca-b%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Ca-b%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Ca-b%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|a-b|}" class="latex" />, so that this contribution to <a href="#xeq">(13)</a> is comparable to <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cint_%7B%7Cx-a%7C+%5Cleq+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29+%281%2B%28a-b%29%5E2%29%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cint_%7B%7Cx-a%7C+%5Cleq+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29+%281%2B%28a-b%29%5E2%29%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cint_%7B%7Cx-a%7C+%5Cleq+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29+%281%2B%28a-b%29%5E2%29%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \asymp \int_{|x-a| \leq |a-b|/2} \frac{dx}{(1+(x-a)^2) (1+(a-b)^2)}." class="latex" />
Using a variant of <a href="#xai">(9)</a>, this expression is comparable to <a name="top"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cmin%28+1%2C+%7Ca-b%7C%2F2%29+%5Cfrac%7B1%7D%7B1%2B%28a-b%29%5E2%7D+%5Casymp+%5Cfrac%7B%5Cmin%281%2C+%7Ca-b%7C%29%7D%7B1%2B%28a-b%29%5E2%7D.+%5C+%5C+%5C+%5C+%5C+%2814%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cmin%28+1%2C+%7Ca-b%7C%2F2%29+%5Cfrac%7B1%7D%7B1%2B%28a-b%29%5E2%7D+%5Casymp+%5Cfrac%7B%5Cmin%281%2C+%7Ca-b%7C%29%7D%7B1%2B%28a-b%29%5E2%7D.+%5C+%5C+%5C+%5C+%5C+%2814%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cmin%28+1%2C+%7Ca-b%7C%2F2%29+%5Cfrac%7B1%7D%7B1%2B%28a-b%29%5E2%7D+%5Casymp+%5Cfrac%7B%5Cmin%281%2C+%7Ca-b%7C%29%7D%7B1%2B%28a-b%29%5E2%7D.+%5C+%5C+%5C+%5C+%5C+%2814%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \asymp \min( 1, |a-b|/2) \frac{1}{1+(a-b)^2} \asymp \frac{\min(1, |a-b|)}{1+(a-b)^2}. \ \ \ \ \ (14)" class="latex" />
</a> The contribution of region (ii) can be handled similarly, and is also comparable to <a href="#top">(14)</a>. Finally, in region (iii), we see from the triangle inequality that <img src="https://s0.wp.com/latex.php?latex=%7B%7Cx-a%7C%2C+%7Cx-b%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cx-a%7C%2C+%7Cx-b%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cx-a%7C%2C+%7Cx-b%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|x-a|, |x-b|}" class="latex" /> are now comparable to each other, and so the contribution of this region is comparable to <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cint_%7B%7Cx-a%7C%2C+%7Cx-b%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cint_%7B%7Cx-a%7C%2C+%7Cx-b%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cint_%7B%7Cx-a%7C%2C+%7Cx-b%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \asymp \int_{|x-a|, |x-b| > |a-b|/2} \frac{dx}{(1+(x-a)^2)^2}." class="latex" />
Now that we have centered the integral around <img src="https://s0.wp.com/latex.php?latex=%7Bx%3Da%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%3Da%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%3Da%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x=a}" class="latex" />, we will discard the <img src="https://s0.wp.com/latex.php?latex=%7B%7Cx-b%7C+%3E+%7Ca-b%7C%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Cx-b%7C+%3E+%7Ca-b%7C%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Cx-b%7C+%3E+%7Ca-b%7C%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|x-b| > |a-b|/2}" class="latex" /> constraint, upper bounding this integral by <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cint_%7B%7Cx-a%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cint_%7B%7Cx-a%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cint_%7B%7Cx-a%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \asymp \int_{|x-a| > |a-b|/2} \frac{dx}{(1+(x-a)^2)^2}." class="latex" />
On the one hand this integral is bounded by <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D+%3D+%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2Bx%5E2%29%5E2%7D+%5Casymp+1&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D+%3D+%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2Bx%5E2%29%5E2%7D+%5Casymp+1&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D+%3D+%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2Bx%5E2%29%5E2%7D+%5Casymp+1&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{-\infty}^\infty \frac{dx}{(1+(x-a)^2)^2} = \int_{-\infty}^\infty \frac{dx}{(1+x^2)^2} \asymp 1" class="latex" />
and on the other hand we can bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%7Cx-a%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D+%5Cleq+%5Cint_%7B%7Cx-a%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%28x-a%29%5E4%7D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%7Cx-a%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D+%5Cleq+%5Cint_%7B%7Cx-a%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%28x-a%29%5E4%7D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%7Cx-a%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29%5E2%7D+%5Cleq+%5Cint_%7B%7Cx-a%7C+%3E+%7Ca-b%7C%2F2%7D+%5Cfrac%7Bdx%7D%7B%28x-a%29%5E4%7D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{|x-a| > |a-b|/2} \frac{dx}{(1+(x-a)^2)^2} \leq \int_{|x-a| > |a-b|/2} \frac{dx}{(x-a)^4} " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Casymp+%7Ca-b%7C%5E%7B-3%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Casymp+%7Ca-b%7C%5E%7B-3%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Casymp+%7Ca-b%7C%5E%7B-3%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \asymp |a-b|^{-3}" class="latex" />
and so we can bound the contribution of (iii) by <img src="https://s0.wp.com/latex.php?latex=%7BO%28+%5Cmin%28+1%2C+%7Ca-b%7C%5E%7B-3%7D+%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28+%5Cmin%28+1%2C+%7Ca-b%7C%5E%7B-3%7D+%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28+%5Cmin%28+1%2C+%7Ca-b%7C%5E%7B-3%7D+%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O( \min( 1, |a-b|^{-3} ))}" class="latex" />. Putting all this together, and dividing into the cases <img src="https://s0.wp.com/latex.php?latex=%7B%7Ca-b%7C+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Ca-b%7C+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Ca-b%7C+%5Cleq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|a-b| \leq 1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B%7Ca-b%7C+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Ca-b%7C+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Ca-b%7C+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|a-b| > 1}" class="latex" />, one can soon obtain a total bound of <img src="https://s0.wp.com/latex.php?latex=%7BO%28%5Cmin%28+1%2C+%7Ca-b%7C%5E%7B-2%7D%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28%5Cmin%28+1%2C+%7Ca-b%7C%5E%7B-2%7D%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28%5Cmin%28+1%2C+%7Ca-b%7C%5E%7B-2%7D%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(\min( 1, |a-b|^{-2}))}" class="latex" /> for the entire integral. One can also adapt this argument to show that this bound is sharp up to constants, thus <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29+%281%2B%28x-b%29%5E2%29%7D+%5Casymp+%5Cmin%28+1%2C+%7Ca-b%7C%5E%7B-2%7D%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29+%281%2B%28x-b%29%5E2%29%7D+%5Casymp+%5Cmin%28+1%2C+%7Ca-b%7C%5E%7B-2%7D%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cfrac%7Bdx%7D%7B%281%2B%28x-a%29%5E2%29+%281%2B%28x-b%29%5E2%29%7D+%5Casymp+%5Cmin%28+1%2C+%7Ca-b%7C%5E%7B-2%7D%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{-\infty}^\infty \frac{dx}{(1+(x-a)^2) (1+(x-b)^2)} \asymp \min( 1, |a-b|^{-2})" class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cfrac%7B1%7D%7B1%2B%7Ca-b%7C%5E2%7D.+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cfrac%7B1%7D%7B1%2B%7Ca-b%7C%5E2%7D.+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Cfrac%7B1%7D%7B1%2B%7Ca-b%7C%5E2%7D.+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \asymp \frac{1}{1+|a-b|^2}. " class="latex" />

A powerful and common type of decomposition is dyadic decomposition. If the summand or integrand involves some quantity <img src="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q}" class="latex" /> in a key way, it is often useful to break up into dyadic regions such as <img src="https://s0.wp.com/latex.php?latex=%7B2%5E%7Bj-1%7D+%5Cleq+Q+%3C+2%5E%7Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5E%7Bj-1%7D+%5Cleq+Q+%3C+2%5E%7Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5E%7Bj-1%7D+%5Cleq+Q+%3C+2%5E%7Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^{j-1} \leq Q < 2^{j}}" class="latex" />, so that <img src="https://s0.wp.com/latex.php?latex=%7BQ+%5Csim+2%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ+%5Csim+2%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ+%5Csim+2%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q \sim 2^j}" class="latex" />, and then sum over <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" />. (One can tweak the dyadic range <img src="https://s0.wp.com/latex.php?latex=%7B2%5E%7Bj-1%7D+%5Cleq+Q+%3C+2%5E%7Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5E%7Bj-1%7D+%5Cleq+Q+%3C+2%5E%7Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5E%7Bj-1%7D+%5Cleq+Q+%3C+2%5E%7Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^{j-1} \leq Q < 2^{j}}" class="latex" /> here with minor variants such as <img src="https://s0.wp.com/latex.php?latex=%7B2%5E%7Bj%7D+%3C+Q+%5Cleq+2%5E%7Bj%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5E%7Bj%7D+%3C+Q+%5Cleq+2%5E%7Bj%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5E%7Bj%7D+%3C+Q+%5Cleq+2%5E%7Bj%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^{j} < Q \leq 2^{j+1}}" class="latex" />, or replace the base <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" /> by some other base, but these modifications mostly have a minor aesthetic impact on the arguments at best.) For instance, one could break up a sum <a name="fsum"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+f%28n%29+%5C+%5C+%5C+%5C+%5C+%2815%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+f%28n%29+%5C+%5C+%5C+%5C+%5C+%2815%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+f%28n%29+%5C+%5C+%5C+%5C+%5C+%2815%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n=1}^{\infty} f(n) \ \ \ \ \ (15)" class="latex" />
</a> into dyadic pieces <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bj%3D1%7D%5E%5Cinfty+%5Csum_%7B2%5E%7Bj-1%7D+%5Cleq+n+%3C+2%5E%7Bj%7D%7D+f%28n%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bj%3D1%7D%5E%5Cinfty+%5Csum_%7B2%5E%7Bj-1%7D+%5Cleq+n+%3C+2%5E%7Bj%7D%7D+f%28n%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bj%3D1%7D%5E%5Cinfty+%5Csum_%7B2%5E%7Bj-1%7D+%5Cleq+n+%3C+2%5E%7Bj%7D%7D+f%28n%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{j=1}^\infty \sum_{2^{j-1} \leq n < 2^{j}} f(n)" class="latex" />
and then seek to estimate each dyadic block <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7B2%5E%7Bj-1%7D+%5Cleq+n+%3C+2%5E%7Bj%7D%7D+f%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7B2%5E%7Bj-1%7D+%5Cleq+n+%3C+2%5E%7Bj%7D%7D+f%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7B2%5E%7Bj-1%7D+%5Cleq+n+%3C+2%5E%7Bj%7D%7D+f%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{2^{j-1} \leq n < 2^{j}} f(n)}" class="latex" /> separately (hoping to get some exponential or polynomial decay in <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" />). The classical technique of <a href="https://en.wikipedia.org/wiki/Cauchy_condensation_test">Cauchy condensation</a> is a basic example of this strategy. But one can also dyadically decompose other quantities than <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" />. For instance one can perform a “vertical” dyadic decomposition (in contrast to the “horizontal” one just performed) by rewriting <a href="#fsum">(15)</a> as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk+%5Cin+%7B%5Cbf+Z%7D%7D+%5Csum_%7Bn+%5Cgeq+1%3A+2%5E%7Bk-1%7D+%5Cleq+f%28n%29+%3C+2%5Ek%7D+f%28n%29%3B&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk+%5Cin+%7B%5Cbf+Z%7D%7D+%5Csum_%7Bn+%5Cgeq+1%3A+2%5E%7Bk-1%7D+%5Cleq+f%28n%29+%3C+2%5Ek%7D+f%28n%29%3B&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk+%5Cin+%7B%5Cbf+Z%7D%7D+%5Csum_%7Bn+%5Cgeq+1%3A+2%5E%7Bk-1%7D+%5Cleq+f%28n%29+%3C+2%5Ek%7D+f%28n%29%3B&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{k \in {\bf Z}} \sum_{n \geq 1: 2^{k-1} \leq f(n) < 2^k} f(n);" class="latex" />
since the summand <img src="https://s0.wp.com/latex.php?latex=%7Bf%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(n)}" class="latex" /> is <img src="https://s0.wp.com/latex.php?latex=%7B%5Casymp+2%5Ek%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Casymp+2%5Ek%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Casymp+2%5Ek%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\asymp 2^k}" class="latex" />, we may simplify this to <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Csum_%7Bk+%5Cin+%7B%5Cbf+Z%7D%7D+2%5Ek+%5C%23+%5C%7B+n+%5Cgeq+1%3A+2%5E%7Bk-1%7D+%5Cleq+f%28n%29+%3C+2%5Ek%5C%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Csum_%7Bk+%5Cin+%7B%5Cbf+Z%7D%7D+2%5Ek+%5C%23+%5C%7B+n+%5Cgeq+1%3A+2%5E%7Bk-1%7D+%5Cleq+f%28n%29+%3C+2%5Ek%5C%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Casymp+%5Csum_%7Bk+%5Cin+%7B%5Cbf+Z%7D%7D+2%5Ek+%5C%23+%5C%7B+n+%5Cgeq+1%3A+2%5E%7Bk-1%7D+%5Cleq+f%28n%29+%3C+2%5Ek%5C%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \asymp \sum_{k \in {\bf Z}} 2^k \# \{ n \geq 1: 2^{k-1} \leq f(n) < 2^k\}." class="latex" />
This now converts the problem of estimating the sum <a href="#fsum">(15)</a> to the more combinatorial problem of estimating the size of the dyadic level sets <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7B+n+%5Cgeq+1%3A+2%5E%7Bk-1%7D+%5Cleq+f%28n%29+%3C+2%5Ek%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7B+n+%5Cgeq+1%3A+2%5E%7Bk-1%7D+%5Cleq+f%28n%29+%3C+2%5Ek%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7B+n+%5Cgeq+1%3A+2%5E%7Bk-1%7D+%5Cleq+f%28n%29+%3C+2%5Ek%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\{ n \geq 1: 2^{k-1} \leq f(n) < 2^k\}}" class="latex" /> for various <img src="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k}" class="latex" />. In a similar spirit, we have <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+%5Casymp+%5Csum_%7Bk+%5Cin+%7B%5Cbf+Z%7D%7D+2%5Ek+%7C+%5C%7B+x+%5Cin+A%3A+2%5E%7Bk-1%7D+%5Cleq+f%28x%29+%3C+2%5Ek+%5C%7D%7C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+%5Casymp+%5Csum_%7Bk+%5Cin+%7B%5Cbf+Z%7D%7D+2%5Ek+%7C+%5C%7B+x+%5Cin+A%3A+2%5E%7Bk-1%7D+%5Cleq+f%28x%29+%3C+2%5Ek+%5C%7D%7C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+f%28x%29%5C+dx+%5Casymp+%5Csum_%7Bk+%5Cin+%7B%5Cbf+Z%7D%7D+2%5Ek+%7C+%5C%7B+x+%5Cin+A%3A+2%5E%7Bk-1%7D+%5Cleq+f%28x%29+%3C+2%5Ek+%5C%7D%7C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_A f(x)\ dx \asymp \sum_{k \in {\bf Z}} 2^k | \{ x \in A: 2^{k-1} \leq f(x) < 2^k \}|" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7B%7CE%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CE%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CE%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|E|}" class="latex" /> denotes the Lebesgue measure of a set <img src="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E}" class="latex" />, and now we are faced with a geometric problem of estimating the measure of some explicit set. This allows one to use geometric intuition to solve the problem, instead of multivariable calculus:

<blockquote>Exercise 3 Let <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" /> be a smooth compact submanifold of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+R%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+R%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+R%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf R}^d}" class="latex" />. Establish the bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7BB%280%2CC%29%7D+%5Cfrac%7Bdx%7D%7B%5Cvarepsilon%5E2+%2B+%5Cmathrm%7Bdist%7D%28x%2CS%29%5E2%7D+%5Cll+%5Cvarepsilon%5E%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7BB%280%2CC%29%7D+%5Cfrac%7Bdx%7D%7B%5Cvarepsilon%5E2+%2B+%5Cmathrm%7Bdist%7D%28x%2CS%29%5E2%7D+%5Cll+%5Cvarepsilon%5E%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7BB%280%2CC%29%7D+%5Cfrac%7Bdx%7D%7B%5Cvarepsilon%5E2+%2B+%5Cmathrm%7Bdist%7D%28x%2CS%29%5E2%7D+%5Cll+%5Cvarepsilon%5E%7B-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{B(0,C)} \frac{dx}{\varepsilon^2 + \mathrm{dist}(x,S)^2} \ll \varepsilon^{-1}" class="latex" />
for all <img src="https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 < \varepsilon < C}" class="latex" />, where the implied constants are allowed to depend on <img src="https://s0.wp.com/latex.php?latex=%7BC%2C+d%2C+S%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BC%2C+d%2C+S%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BC%2C+d%2C+S%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{C, d, S}" class="latex" />. (This can be accomplished either by a vertical dyadic decomposition, or a dyadic decomposition of the quantity <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Bdist%7D%28x%2CS%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Bdist%7D%28x%2CS%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7Bdist%7D%28x%2CS%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mathrm{dist}(x,S)}" class="latex" />.) </blockquote>


<blockquote>Exercise 4 Solve problem (ii) from the introduction to this post by dyadically decomposing in the <img src="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d}" class="latex" /> variable. </blockquote>


<blockquote>Remark 5 By such tools as <a href="#integral">(10)</a>, <a href="#integral-2">(11)</a>, or Exercise <a href="#quasi">1</a>, one could convert the dyadic sums one obtains from dyadic decomposition into integral variants. However, if one wished, one could “cut out the middle-man” and work with continuous dyadic decompositions rather than discrete ones. Indeed, from the integral identity <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+1_%7B%5Clambda+%3C+Q+%5Cleq+2%5Clambda%7D+%5Cfrac%7Bd%5Clambda%7D%7B%5Clambda%7D+%3D+%5Clog+2&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+1_%7B%5Clambda+%3C+Q+%5Cleq+2%5Clambda%7D+%5Cfrac%7Bd%5Clambda%7D%7B%5Clambda%7D+%3D+%5Clog+2&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+1_%7B%5Clambda+%3C+Q+%5Cleq+2%5Clambda%7D+%5Cfrac%7Bd%5Clambda%7D%7B%5Clambda%7D+%3D+%5Clog+2&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^\infty 1_{\lambda < Q \leq 2\lambda} \frac{d\lambda}{\lambda} = \log 2" class="latex" />
for any <img src="https://s0.wp.com/latex.php?latex=%7BQ%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q>0}" class="latex" />, together with the Fubini–Tonelli theorem, we obtain the continuous dyadic decomposition <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%3D+%5Cint_0%5E%5Cinfty+%5Csum_%7Bn+%5Cin+A%3A+%5Clambda+%5Cleq+Q%28n%29+%3C+2%5Clambda%7D+f%28n%29%5C+%5Cfrac%7Bd%5Clambda%7D%7B%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%3D+%5Cint_0%5E%5Cinfty+%5Csum_%7Bn+%5Cin+A%3A+%5Clambda+%5Cleq+Q%28n%29+%3C+2%5Clambda%7D+f%28n%29%5C+%5Cfrac%7Bd%5Clambda%7D%7B%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+f%28n%29+%3D+%5Cint_0%5E%5Cinfty+%5Csum_%7Bn+%5Cin+A%3A+%5Clambda+%5Cleq+Q%28n%29+%3C+2%5Clambda%7D+f%28n%29%5C+%5Cfrac%7Bd%5Clambda%7D%7B%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n \in A} f(n) = \int_0^\infty \sum_{n \in A: \lambda \leq Q(n) < 2\lambda} f(n)\ \frac{d\lambda}{\lambda}" class="latex" />
for any quantity <img src="https://s0.wp.com/latex.php?latex=%7BQ%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q(n)}" class="latex" /> that is positive whenever <img src="https://s0.wp.com/latex.php?latex=%7Bf%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%28n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f(n)}" class="latex" /> is positive. Similarly if we work with integrals <img src="https://s0.wp.com/latex.php?latex=%7B%5Cint_A+f%28x%29%5C+dx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cint_A+f%28x%29%5C+dx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cint_A+f%28x%29%5C+dx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\int_A f(x)\ dx}" class="latex" /> rather than sums. This version of dyadic decomposition is occasionally a little more convenient to work with, particularly if one then wants to perform various changes of variables in the <img src="https://s0.wp.com/latex.php?latex=%7B%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clambda%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\lambda}" class="latex" /> parameter which would be tricky to execute if this were a discrete variable. </blockquote>


 — 3. Exponential weights — 

Many sums involve expressions that are “exponentially large” or “exponentially small” in some parameter. A basic rule of thumb is that any quantity that is “exponentially small” will likely give a negligible contribution when compared against quantities that are not exponentially small. For instance, if an expression involves a term of the form <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-Q%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-Q%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7B-Q%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{-Q}}" class="latex" /> for some non-negative quantity <img src="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q}" class="latex" />, which can be bounded on at least one portion of the domain of summation or integration, then one expects the region where <img src="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q}" class="latex" /> is bounded to provide the dominant contribution. For instance, if one wishes to estimate the integral <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^\infty e^{-\varepsilon x} \frac{dx}{1+x}" class="latex" />
for some <img src="https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 < \varepsilon < 1/2}" class="latex" />, this heuristic suggests that the dominant contribution should come from the region <img src="https://s0.wp.com/latex.php?latex=%7Bx+%3D+O%281%2F%5Cvarepsilon%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx+%3D+O%281%2F%5Cvarepsilon%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx+%3D+O%281%2F%5Cvarepsilon%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x = O(1/\varepsilon)}" class="latex" />, in which one can bound <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cvarepsilon+x%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cvarepsilon+x%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cvarepsilon+x%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{-\varepsilon x}}" class="latex" /> simply by <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> and obtain an upper bound of <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cll+%5Cint_%7Bx+%3D+O%281%2F%5Cvarepsilon%29%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D+%5Cll+%5Clog+%5Cfrac%7B1%7D%7B%5Cvarepsilon%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cll+%5Cint_%7Bx+%3D+O%281%2F%5Cvarepsilon%29%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D+%5Cll+%5Clog+%5Cfrac%7B1%7D%7B%5Cvarepsilon%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cll+%5Cint_%7Bx+%3D+O%281%2F%5Cvarepsilon%29%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D+%5Cll+%5Clog+%5Cfrac%7B1%7D%7B%5Cvarepsilon%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \ll \int_{x = O(1/\varepsilon)} \frac{dx}{1+x} \ll \log \frac{1}{\varepsilon}." class="latex" />

To make such a heuristic precise, one can perform a dyadic decomposition in the exponential weight <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cvarepsilon+x%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cvarepsilon+x%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cvarepsilon+x%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{-\varepsilon x}}" class="latex" />, or equivalently perform an additive decomposition in the exponent <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon+x%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon+x%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon+x%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon x}" class="latex" />, for instance writing <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D+%3D+%5Csum_%7Bj%3D1%7D%5E%5Cinfty+%5Cint_%7Bj-1+%5Cleq+%5Cvarepsilon+x+%3C+j%7D+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D+%3D+%5Csum_%7Bj%3D1%7D%5E%5Cinfty+%5Cint_%7Bj-1+%5Cleq+%5Cvarepsilon+x+%3C+j%7D+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D+%3D+%5Csum_%7Bj%3D1%7D%5E%5Cinfty+%5Cint_%7Bj-1+%5Cleq+%5Cvarepsilon+x+%3C+j%7D+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^\infty e^{-\varepsilon x} \frac{dx}{1+x} = \sum_{j=1}^\infty \int_{j-1 \leq \varepsilon x < j} e^{-\varepsilon x} \frac{dx}{1+x}." class="latex" />

<blockquote>Exercise 6 Use this decomposition to rigorously establish the bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D+%5Cll+%5Clog+%5Cfrac%7B1%7D%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D+%5Cll+%5Clog+%5Cfrac%7B1%7D%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cvarepsilon+x%7D+%5Cfrac%7Bdx%7D%7B1%2Bx%7D+%5Cll+%5Clog+%5Cfrac%7B1%7D%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^\infty e^{-\varepsilon x} \frac{dx}{1+x} \ll \log \frac{1}{\varepsilon}" class="latex" />
for any <img src="https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 < \varepsilon < 1/2}" class="latex" />. </blockquote>


<blockquote>Exercise 7 Solve problem (i) from the introduction to this post. </blockquote>


More generally, if one is working with a sum or integral such as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+e%5E%7B%5Cphi%28n%29%7D+%5Cpsi%28n%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+e%5E%7B%5Cphi%28n%29%7D+%5Cpsi%28n%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bn+%5Cin+A%7D+e%5E%7B%5Cphi%28n%29%7D+%5Cpsi%28n%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{n \in A} e^{\phi(n)} \psi(n)" class="latex" />
or <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+e%5E%7B%5Cphi%28x%29%7D+%5Cpsi%28x%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+e%5E%7B%5Cphi%28x%29%7D+%5Cpsi%28x%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_A+e%5E%7B%5Cphi%28x%29%7D+%5Cpsi%28x%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_A e^{\phi(x)} \psi(x)\ dx" class="latex" />
with some exponential weight <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^\phi}" class="latex" /> and a lower order amplitude <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpsi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpsi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpsi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\psi}" class="latex" />, then one typically expects the dominant contribution to come from the region where <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi}" class="latex" /> comes close to attaining its maximal value. If this maximum is attained on the boundary, then one typically has geometric series behavior away from the boundary, and one can often get a good estimate by obtaining geometric series type behavior. For instance, suppose one wants to estimate the error function <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%3D+%5Cfrac%7B2%7D%7B%5Csqrt%7B%5Cpi%7D%7D+%5Cint_0%5Ez+e%5E%7B-t%5E2%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%3D+%5Cfrac%7B2%7D%7B%5Csqrt%7B%5Cpi%7D%7D+%5Cint_0%5Ez+e%5E%7B-t%5E2%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%3D+%5Cfrac%7B2%7D%7B%5Csqrt%7B%5Cpi%7D%7D+%5Cint_0%5Ez+e%5E%7B-t%5E2%7D%5C+dt&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \mathrm{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\ dt" class="latex" />
for <img src="https://s0.wp.com/latex.php?latex=%7Bz+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bz+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bz+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{z \geq 1}" class="latex" />. In view of the complete integral <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-t%5E2%7D%5C+dt+%3D+%5Cfrac%7B%5Csqrt%7B%5Cpi%7D%7D%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-t%5E2%7D%5C+dt+%3D+%5Cfrac%7B%5Csqrt%7B%5Cpi%7D%7D%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-t%5E2%7D%5C+dt+%3D+%5Cfrac%7B%5Csqrt%7B%5Cpi%7D%7D%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^\infty e^{-t^2}\ dt = \frac{\sqrt{\pi}}{2}" class="latex" />
we can rewrite this as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%3D+1+-+%5Cfrac%7B2%7D%7B%5Csqrt%7B%5Cpi%7D%7D+%5Cint_z%5E%5Cinfty+e%5E%7B-t%5E2%7D%5C+dt.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%3D+1+-+%5Cfrac%7B2%7D%7B%5Csqrt%7B%5Cpi%7D%7D+%5Cint_z%5E%5Cinfty+e%5E%7B-t%5E2%7D%5C+dt.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%3D+1+-+%5Cfrac%7B2%7D%7B%5Csqrt%7B%5Cpi%7D%7D+%5Cint_z%5E%5Cinfty+e%5E%7B-t%5E2%7D%5C+dt.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \mathrm{erf}(z) = 1 - \frac{2}{\sqrt{\pi}} \int_z^\infty e^{-t^2}\ dt." class="latex" />
The exponential weight <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-t%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-t%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7B-t%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{-t^2}}" class="latex" /> attains its maximum at the left endpoint <img src="https://s0.wp.com/latex.php?latex=%7Bt%3Dz%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%3Dz%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%3Dz%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t=z}" class="latex" /> and decays quickly away from that endpoint. One could estimate this by dyadic decomposition of <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-t%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-t%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7B-t%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{-t^2}}" class="latex" /> as discussed previously, but a slicker way to proceed here is to use the convexity of <img src="https://s0.wp.com/latex.php?latex=%7Bt%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t^2}" class="latex" /> to obtain a geometric series upper bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++e%5E%7B-t%5E2%7D+%5Cleq+e%5E%7B-z%5E2+-+2+z+%28t-z%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++e%5E%7B-t%5E2%7D+%5Cleq+e%5E%7B-z%5E2+-+2+z+%28t-z%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++e%5E%7B-t%5E2%7D+%5Cleq+e%5E%7B-z%5E2+-+2+z+%28t-z%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle e^{-t^2} \leq e^{-z^2 - 2 z (t-z)}" class="latex" />
for <img src="https://s0.wp.com/latex.php?latex=%7Bt+%5Cgeq+z%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt+%5Cgeq+z%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt+%5Cgeq+z%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t \geq z}" class="latex" />, which on integration gives <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_z%5E%5Cinfty+e%5E%7B-t%5E2%7D%5C+dt+%5Cleq+%5Cint_z%5E%5Cinfty+e%5E%7B-z%5E2+-+2+z+%28t-z%29%7D%5C+dt+%3D+%5Cfrac%7Be%5E%7B-z%5E2%7D%7D%7B2z%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_z%5E%5Cinfty+e%5E%7B-t%5E2%7D%5C+dt+%5Cleq+%5Cint_z%5E%5Cinfty+e%5E%7B-z%5E2+-+2+z+%28t-z%29%7D%5C+dt+%3D+%5Cfrac%7Be%5E%7B-z%5E2%7D%7D%7B2z%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_z%5E%5Cinfty+e%5E%7B-t%5E2%7D%5C+dt+%5Cleq+%5Cint_z%5E%5Cinfty+e%5E%7B-z%5E2+-+2+z+%28t-z%29%7D%5C+dt+%3D+%5Cfrac%7Be%5E%7B-z%5E2%7D%7D%7B2z%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_z^\infty e^{-t^2}\ dt \leq \int_z^\infty e^{-z^2 - 2 z (t-z)}\ dt = \frac{e^{-z^2}}{2z}" class="latex" />
giving the asymptotic <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%3D+1+-+O%28+%5Cfrac%7Be%5E%7B-z%5E2%7D%7D%7Bz%7D%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%3D+1+-+O%28+%5Cfrac%7Be%5E%7B-z%5E2%7D%7D%7Bz%7D%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%3D+1+-+O%28+%5Cfrac%7Be%5E%7B-z%5E2%7D%7D%7Bz%7D%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \mathrm{erf}(z) = 1 - O( \frac{e^{-z^2}}{z}) " class="latex" />
for <img src="https://s0.wp.com/latex.php?latex=%7Bz+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bz+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bz+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{z \geq 1}" class="latex" />.

<blockquote>Exercise 8 In the converse direction, establish the upper bound <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%5Cleq+1+-+c+%5Cfrac%7Be%5E%7B-z%5E2%7D%7D%7Bz%7D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%5Cleq+1+-+c+%5Cfrac%7Be%5E%7B-z%5E2%7D%7D%7Bz%7D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathrm%7Berf%7D%28z%29+%5Cleq+1+-+c+%5Cfrac%7Be%5E%7B-z%5E2%7D%7D%7Bz%7D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \mathrm{erf}(z) \leq 1 - c \frac{e^{-z^2}}{z} " class="latex" />
for some absolute constant <img src="https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c>0}" class="latex" /> and all <img src="https://s0.wp.com/latex.php?latex=%7Bz+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bz+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bz+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{z \geq 1}" class="latex" />. </blockquote>


<blockquote>Exercise 9 If <img src="https://s0.wp.com/latex.php?latex=%7B%5Ctheta+n+%5Cleq+m+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ctheta+n+%5Cleq+m+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ctheta+n+%5Cleq+m+%5Cleq+n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\theta n \leq m \leq n}" class="latex" /> for some <img src="https://s0.wp.com/latex.php?latex=%7B1%2F2+%3C+%5Ctheta+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2F2+%3C+%5Ctheta+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2F2+%3C+%5Ctheta+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1/2 < \theta < 1}" class="latex" />, show that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk%3Dm%7D%5En+%5Cbinom%7Bn%7D%7Bk%7D+%5Cll+%5Cfrac%7B1%7D%7B2%5Ctheta-1%7D+%5Cbinom%7Bn%7D%7Bm%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk%3Dm%7D%5En+%5Cbinom%7Bn%7D%7Bk%7D+%5Cll+%5Cfrac%7B1%7D%7B2%5Ctheta-1%7D+%5Cbinom%7Bn%7D%7Bm%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk%3Dm%7D%5En+%5Cbinom%7Bn%7D%7Bk%7D+%5Cll+%5Cfrac%7B1%7D%7B2%5Ctheta-1%7D+%5Cbinom%7Bn%7D%7Bm%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{k=m}^n \binom{n}{k} \ll \frac{1}{2\theta-1} \binom{n}{m}." class="latex" />
(Hint: estimate the ratio between consecutive binomial coefficients <img src="https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bk%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bk%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cbinom%7Bn%7D%7Bk%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\binom{n}{k}}" class="latex" /> and then control the sum by a geometric series). </blockquote>


When the maximum of the exponent <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi}" class="latex" /> occurs in the interior of the region of summation or integration, then one can get good results by some version of <a href="https://en.wikipedia.org/wiki/Laplace%27s_method">Laplace’s method</a>. For simplicity we will discuss this method in the context of one-dimensional integrals <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_a%5Eb+e%5E%7B%5Cphi%28x%29%7D+%5Cpsi%28x%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_a%5Eb+e%5E%7B%5Cphi%28x%29%7D+%5Cpsi%28x%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_a%5Eb+e%5E%7B%5Cphi%28x%29%7D+%5Cpsi%28x%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_a^b e^{\phi(x)} \psi(x)\ dx" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi}" class="latex" /> attains a non-degenerate global maximum at some interior point <img src="https://s0.wp.com/latex.php?latex=%7Bx+%3D+x_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx+%3D+x_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx+%3D+x_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x = x_0}" class="latex" />. The rule of thumb here is that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_a%5Eb+e%5E%7B%5Cphi%28x%29%7D+%5Cpsi%28x%29%5C+dx+%5Capprox+%5Csqrt%7B%5Cfrac%7B2%5Cpi%7D%7B%7C%5Cphi%27%27%28x_0%29%7C%7D%7D+e%5E%7B%5Cphi%28x_0%29%7D+%5Cpsi%28x_0%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_a%5Eb+e%5E%7B%5Cphi%28x%29%7D+%5Cpsi%28x%29%5C+dx+%5Capprox+%5Csqrt%7B%5Cfrac%7B2%5Cpi%7D%7B%7C%5Cphi%27%27%28x_0%29%7C%7D%7D+e%5E%7B%5Cphi%28x_0%29%7D+%5Cpsi%28x_0%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_a%5Eb+e%5E%7B%5Cphi%28x%29%7D+%5Cpsi%28x%29%5C+dx+%5Capprox+%5Csqrt%7B%5Cfrac%7B2%5Cpi%7D%7B%7C%5Cphi%27%27%28x_0%29%7C%7D%7D+e%5E%7B%5Cphi%28x_0%29%7D+%5Cpsi%28x_0%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_a^b e^{\phi(x)} \psi(x)\ dx \approx \sqrt{\frac{2\pi}{|\phi''(x_0)|}} e^{\phi(x_0)} \psi(x_0)." class="latex" />
The heuristic justification is as follows. The main contribution should be when <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" /> is close to <img src="https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_0}" class="latex" />. Here we can perform a Taylor expansion <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%5Capprox+%5Cphi%28x_0%29+-+%5Cfrac%7B1%7D%7B2%7D+%7C%5Cphi%27%27%28x_0%29%7C+%28x-x_0%29%5E2&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%5Capprox+%5Cphi%28x_0%29+-+%5Cfrac%7B1%7D%7B2%7D+%7C%5Cphi%27%27%28x_0%29%7C+%28x-x_0%29%5E2&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%5Capprox+%5Cphi%28x_0%29+-+%5Cfrac%7B1%7D%7B2%7D+%7C%5Cphi%27%27%28x_0%29%7C+%28x-x_0%29%5E2&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \phi(x) \approx \phi(x_0) - \frac{1}{2} |\phi''(x_0)| (x-x_0)^2" class="latex" />
since at a non-degenerate maximum we have <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%28x_0%29%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%28x_0%29%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%28x_0%29%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi'(x_0)=0}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%27%28x_0%29+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%27%28x_0%29+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%27%28x_0%29+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi''(x_0) > 0}" class="latex" />. Also, if <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpsi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpsi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpsi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\psi}" class="latex" /> is continuous, then <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpsi%28x%29+%5Capprox+%5Cpsi%28x_0%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpsi%28x%29+%5Capprox+%5Cpsi%28x_0%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpsi%28x%29+%5Capprox+%5Cpsi%28x_0%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\psi(x) \approx \psi(x_0)}" class="latex" /> when <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" /> is close to <img src="https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_0}" class="latex" />. Thus we should be able to estimate the above integral by the gaussian integral <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5Cbf+R%7D+e%5E%7B%5Cphi%28x_0%29+-+%5Cfrac%7B1%7D%7B2%7D+%7C%5Cphi%27%27%28x_0%29%7C+%28x-x_0%29%5E2%7D+%5Cpsi%28x_0%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5Cbf+R%7D+e%5E%7B%5Cphi%28x_0%29+-+%5Cfrac%7B1%7D%7B2%7D+%7C%5Cphi%27%27%28x_0%29%7C+%28x-x_0%29%5E2%7D+%5Cpsi%28x_0%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5Cbf+R%7D+e%5E%7B%5Cphi%28x_0%29+-+%5Cfrac%7B1%7D%7B2%7D+%7C%5Cphi%27%27%28x_0%29%7C+%28x-x_0%29%5E2%7D+%5Cpsi%28x_0%29%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{\bf R} e^{\phi(x_0) - \frac{1}{2} |\phi''(x_0)| (x-x_0)^2} \psi(x_0)\ dx" class="latex" />
which can be computed to equal <img src="https://s0.wp.com/latex.php?latex=%7B%5Csqrt%7B%5Cfrac%7B2%5Cpi%7D%7B%7C%5Cphi%27%27%28x_0%29%7C%7D%7D+e%5E%7B%5Cphi%28x_0%29%7D+%5Cpsi%28x_0%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csqrt%7B%5Cfrac%7B2%5Cpi%7D%7B%7C%5Cphi%27%27%28x_0%29%7C%7D%7D+e%5E%7B%5Cphi%28x_0%29%7D+%5Cpsi%28x_0%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csqrt%7B%5Cfrac%7B2%5Cpi%7D%7B%7C%5Cphi%27%27%28x_0%29%7C%7D%7D+e%5E%7B%5Cphi%28x_0%29%7D+%5Cpsi%28x_0%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sqrt{\frac{2\pi}{|\phi''(x_0)|}} e^{\phi(x_0)} \psi(x_0)}" class="latex" /> as desired.

Let us illustrate how this argument can be made rigorous by considering the task of estimating the factorial <img src="https://s0.wp.com/latex.php?latex=%7Bn%21%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%21%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%21%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n!}" class="latex" /> of a large number. In contrast to what we did in Exercise <a href="#Stirling">2</a>, we will proceed using a version of Laplace’s method, relying on the integral representation <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+%5CGamma%28n%2B1%29+%3D+%5Cint_0%5E%5Cinfty+x%5En+e%5E%7B-x%7D%5C+dx.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+%5CGamma%28n%2B1%29+%3D+%5Cint_0%5E%5Cinfty+x%5En+e%5E%7B-x%7D%5C+dx.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+%5CGamma%28n%2B1%29+%3D+%5Cint_0%5E%5Cinfty+x%5En+e%5E%7B-x%7D%5C+dx.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle n! = \Gamma(n+1) = \int_0^\infty x^n e^{-x}\ dx." class="latex" />
As <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" /> is large, we will consider <img src="https://s0.wp.com/latex.php?latex=%7Bx%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x^n}" class="latex" /> to be part of the exponential weight rather than the amplitude, writing this expression as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cphi%28x%29%7D%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cphi%28x%29%7D%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%5Cinfty+e%5E%7B-%5Cphi%28x%29%7D%5C+dx&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^\infty e^{-\phi(x)}\ dx" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%3D+x+-+n+%5Clog+x.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%3D+x+-+n+%5Clog+x.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%3D+x+-+n+%5Clog+x.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \phi(x) = x - n \log x." class="latex" />
The function <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi}" class="latex" /> attains a global maximum at <img src="https://s0.wp.com/latex.php?latex=%7Bx_0+%3D+n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_0+%3D+n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_0+%3D+n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_0 = n}" class="latex" />, with <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%28n%29+%3D+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%28n%29+%3D+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%28n%29+%3D+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi'(n) = 0}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%27%28n%29+%3D+1%2Fn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%27%28n%29+%3D+1%2Fn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%27%28n%29+%3D+1%2Fn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi''(n) = 1/n}" class="latex" />. We will therefore decompose this integral into three pieces <a name="onr"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%7Bn-R%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%2B+%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%2B+%5Cint_%7Bn%2BR%7D%5E%5Cinfty+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%5C+%5C+%5C+%5C+%5C+%2816%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%7Bn-R%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%2B+%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%2B+%5Cint_%7Bn%2BR%7D%5E%5Cinfty+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%5C+%5C+%5C+%5C+%5C+%2816%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%7Bn-R%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%2B+%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%2B+%5Cint_%7Bn%2BR%7D%5E%5Cinfty+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%5C+%5C+%5C+%5C+%5C+%2816%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^{n-R} e^{-\phi(x)}\ dx + \int_{n-R}^{n+R} e^{-\phi(x)}\ dx + \int_{n+R}^\infty e^{-\phi(x)}\ dx \ \ \ \ \ (16)" class="latex" />
</a> where <img src="https://s0.wp.com/latex.php?latex=%7B0+%3C+R+%3C+n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%3C+R+%3C+n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%3C+R+%3C+n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 < R < n}" class="latex" /> is a radius parameter which we will choose later, as it is not immediately obvious for now what the optimal value of this parameter is (although the previous heuristics do suggest that <img src="https://s0.wp.com/latex.php?latex=%7BR+%5Capprox+1+%2F+%7C%5Cphi%27%27%28x_0%29%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BR+%5Capprox+1+%2F+%7C%5Cphi%27%27%28x_0%29%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BR+%5Capprox+1+%2F+%7C%5Cphi%27%27%28x_0%29%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{R \approx 1 / |\phi''(x_0)|^{1/2}}" class="latex" /> might be a reasonable choice).

The main term is expected to be the middle term, so we shall use crude methods to bound the other two terms. For the first part where <img src="https://s0.wp.com/latex.php?latex=%7B0+%3C+x+%5Cleq+n-R%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%3C+x+%5Cleq+n-R%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%3C+x+%5Cleq+n-R%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 < x \leq n-R}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi}" class="latex" /> is increasing so we can crudely bound <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28x%29%7D+%5Cleq+e%5E%7B-%5Cphi%28n-R%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28x%29%7D+%5Cleq+e%5E%7B-%5Cphi%28n-R%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28x%29%7D+%5Cleq+e%5E%7B-%5Cphi%28n-R%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{-\phi(x)} \leq e^{-\phi(n-R)}}" class="latex" /> and thus <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%7Bn-R%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%5Cleq+%28n-R%29+e%5E%7B-%5Cphi%28n-R%29%7D+%5Cleq+n+e%5E%7B-%5Cphi%28n-R%29%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%7Bn-R%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%5Cleq+%28n-R%29+e%5E%7B-%5Cphi%28n-R%29%7D+%5Cleq+n+e%5E%7B-%5Cphi%28n-R%29%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_0%5E%7Bn-R%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%5Cleq+%28n-R%29+e%5E%7B-%5Cphi%28n-R%29%7D+%5Cleq+n+e%5E%7B-%5Cphi%28n-R%29%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_0^{n-R} e^{-\phi(x)}\ dx \leq (n-R) e^{-\phi(n-R)} \leq n e^{-\phi(n-R)}." class="latex" />
(We expect <img src="https://s0.wp.com/latex.php?latex=%7BR%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BR%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BR%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{R}" class="latex" /> to be much smaller than <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" />, so there is not much point to saving the tiny <img src="https://s0.wp.com/latex.php?latex=%7B-R%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B-R%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B-R%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{-R}" class="latex" /> term in the <img src="https://s0.wp.com/latex.php?latex=%7Bn-R%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn-R%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn-R%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n-R}" class="latex" /> factor.) For the third part where <img src="https://s0.wp.com/latex.php?latex=%7Bx+%5Cgeq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx+%5Cgeq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx+%5Cgeq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x \geq n+R}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi}" class="latex" /> is decreasing, but bounding <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28x%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28x%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28x%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{-\phi(x)}}" class="latex" /> by <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28n%2BR%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28n%2BR%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28n%2BR%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{-\phi(n+R)}}" class="latex" /> would not work because of the unbounded nature of <img src="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x}" class="latex" />; some additional decay is needed. Fortunately, we have a strict increase <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%27%28x%29+%3D+1+-+%5Cfrac%7Bn%7D%7Bx%7D+%5Cgeq+1+-+%5Cfrac%7Bn%7D%7Bn%2BR%7D+%3D+%5Cfrac%7BR%7D%7Bn%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%27%28x%29+%3D+1+-+%5Cfrac%7Bn%7D%7Bx%7D+%5Cgeq+1+-+%5Cfrac%7Bn%7D%7Bn%2BR%7D+%3D+%5Cfrac%7BR%7D%7Bn%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%27%28x%29+%3D+1+-+%5Cfrac%7Bn%7D%7Bx%7D+%5Cgeq+1+-+%5Cfrac%7Bn%7D%7Bn%2BR%7D+%3D+%5Cfrac%7BR%7D%7Bn%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \phi'(x) = 1 - \frac{n}{x} \geq 1 - \frac{n}{n+R} = \frac{R}{n+R}" class="latex" />
for <img src="https://s0.wp.com/latex.php?latex=%7Bx+%5Cgeq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx+%5Cgeq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx+%5Cgeq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x \geq n+R}" class="latex" />, so by the intermediate value theorem we have <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%5Cgeq+%5Cphi%28n%2BR%29+%2B+%5Cfrac%7BR%7D%7Bn%2BR%7D+%28x-n-R%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%5Cgeq+%5Cphi%28n%2BR%29+%2B+%5Cfrac%7BR%7D%7Bn%2BR%7D+%28x-n-R%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%5Cgeq+%5Cphi%28n%2BR%29+%2B+%5Cfrac%7BR%7D%7Bn%2BR%7D+%28x-n-R%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \phi(x) \geq \phi(n+R) + \frac{R}{n+R} (x-n-R)" class="latex" />
and after a short calculation this gives <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7Bn%2BR%7D%5E%5Cinfty+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%5Cleq+%5Cfrac%7Bn%2BR%7D%7BR%7D+e%5E%7B-%5Cphi%28n%2BR%29%7D+%5Cll+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-%5Cphi%28n%2BR%29%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7Bn%2BR%7D%5E%5Cinfty+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%5Cleq+%5Cfrac%7Bn%2BR%7D%7BR%7D+e%5E%7B-%5Cphi%28n%2BR%29%7D+%5Cll+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-%5Cphi%28n%2BR%29%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7Bn%2BR%7D%5E%5Cinfty+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%5Cleq+%5Cfrac%7Bn%2BR%7D%7BR%7D+e%5E%7B-%5Cphi%28n%2BR%29%7D+%5Cll+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-%5Cphi%28n%2BR%29%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{n+R}^\infty e^{-\phi(x)}\ dx \leq \frac{n+R}{R} e^{-\phi(n+R)} \ll \frac{n}{R} e^{-\phi(n+R)}." class="latex" />
Now we turn to the important middle term. If we assume <img src="https://s0.wp.com/latex.php?latex=%7BR+%5Cleq+n%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BR+%5Cleq+n%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BR+%5Cleq+n%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{R \leq n/2}" class="latex" />, then we will have <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%27%27%28x%29+%3D+O%28+1%2Fn%5E2+%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%27%27%28x%29+%3D+O%28+1%2Fn%5E2+%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%27%27%27%28x%29+%3D+O%28+1%2Fn%5E2+%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi'''(x) = O( 1/n^2 )}" class="latex" /> in the region <img src="https://s0.wp.com/latex.php?latex=%7Bn-R+%5Cleq+x+%5Cleq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn-R+%5Cleq+x+%5Cleq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn-R+%5Cleq+x+%5Cleq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n-R \leq x \leq n+R}" class="latex" />, so by Taylor’s theorem with remainder <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%3D+%5Cphi%28n%29+%2B+%5Cphi%27%28n%29+%28x-n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Cphi%27%27%28n%29+%28x-n%29%5E2+%2B+O%28+%5Cfrac%7B%7Cx-n%7C%5E3%7D%7Bn%5E2%7D+%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%3D+%5Cphi%28n%29+%2B+%5Cphi%27%28n%29+%28x-n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Cphi%27%27%28n%29+%28x-n%29%5E2+%2B+O%28+%5Cfrac%7B%7Cx-n%7C%5E3%7D%7Bn%5E2%7D+%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cphi%28x%29+%3D+%5Cphi%28n%29+%2B+%5Cphi%27%28n%29+%28x-n%29+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Cphi%27%27%28n%29+%28x-n%29%5E2+%2B+O%28+%5Cfrac%7B%7Cx-n%7C%5E3%7D%7Bn%5E2%7D+%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \phi(x) = \phi(n) + \phi'(n) (x-n) + \frac{1}{2} \phi''(n) (x-n)^2 + O( \frac{|x-n|^3}{n^2} ) " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%5Cphi%28n%29+%2B+%5Cfrac%7B%28x-n%29%5E2%7D%7B2n%7D+%2B+O%28+%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%5Cphi%28n%29+%2B+%5Cfrac%7B%28x-n%29%5E2%7D%7B2n%7D+%2B+O%28+%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%5Cphi%28n%29+%2B+%5Cfrac%7B%28x-n%29%5E2%7D%7B2n%7D+%2B+O%28+%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle = \phi(n) + \frac{(x-n)^2}{2n} + O( \frac{R^3}{n^2} )." class="latex" />
If we assume that <img src="https://s0.wp.com/latex.php?latex=%7BR+%3D+O%28n%5E%7B2%2F3%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BR+%3D+O%28n%5E%7B2%2F3%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BR+%3D+O%28n%5E%7B2%2F3%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{R = O(n^{2/3})}" class="latex" />, then the error term is bounded and we can exponentiate to obtain <a name="ephi"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++e%5E%7B-%5Cphi%28x%29%7D+%3D+%281+%2B+O%28%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D%29%29+e%5E%7B-%5Cphi%28n%29+-+%5Cfrac%7B%28x-n%29%5E2%7D%7B2n%7D%7D+%5C+%5C+%5C+%5C+%5C+%2817%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++e%5E%7B-%5Cphi%28x%29%7D+%3D+%281+%2B+O%28%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D%29%29+e%5E%7B-%5Cphi%28n%29+-+%5Cfrac%7B%28x-n%29%5E2%7D%7B2n%7D%7D+%5C+%5C+%5C+%5C+%5C+%2817%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++e%5E%7B-%5Cphi%28x%29%7D+%3D+%281+%2B+O%28%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D%29%29+e%5E%7B-%5Cphi%28n%29+-+%5Cfrac%7B%28x-n%29%5E2%7D%7B2n%7D%7D+%5C+%5C+%5C+%5C+%5C+%2817%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle e^{-\phi(x)} = (1 + O(\frac{R^3}{n^2})) e^{-\phi(n) - \frac{(x-n)^2}{2n}} \ \ \ \ \ (17)" class="latex" />
</a> for <img src="https://s0.wp.com/latex.php?latex=%7Bn-R+%5Cleq+x+%5Cleq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn-R+%5Cleq+x+%5Cleq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn-R+%5Cleq+x+%5Cleq+n%2BR%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n-R \leq x \leq n+R}" class="latex" /> and hence <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%3D+%281+%2B+O%28%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D%29%29+e%5E%7B-%5Cphi%28n%29%7D+%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%28x-n%29%5E2%2F2n%7D%5C+dx.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%3D+%281+%2B+O%28%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D%29%29+e%5E%7B-%5Cphi%28n%29%7D+%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%28x-n%29%5E2%2F2n%7D%5C+dx.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%5Cphi%28x%29%7D%5C+dx+%3D+%281+%2B+O%28%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D%29%29+e%5E%7B-%5Cphi%28n%29%7D+%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%28x-n%29%5E2%2F2n%7D%5C+dx.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{n-R}^{n+R} e^{-\phi(x)}\ dx = (1 + O(\frac{R^3}{n^2})) e^{-\phi(n)} \int_{n-R}^{n+R} e^{-(x-n)^2/2n}\ dx." class="latex" />
If we also assume that <img src="https://s0.wp.com/latex.php?latex=%7BR+%5Cgg+%5Csqrt%7Bn%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BR+%5Cgg+%5Csqrt%7Bn%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BR+%5Cgg+%5Csqrt%7Bn%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{R \gg \sqrt{n}}" class="latex" />, we can use the error function type estimates from before to estimate <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%28x-n%29%5E2%2F2n%7D%5C+dx+%3D+%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-R%5E2%2F2n%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%28x-n%29%5E2%2F2n%7D%5C+dx+%3D+%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-R%5E2%2F2n%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7Bn-R%7D%5E%7Bn%2BR%7D+e%5E%7B-%28x-n%29%5E2%2F2n%7D%5C+dx+%3D+%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-R%5E2%2F2n%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_{n-R}^{n+R} e^{-(x-n)^2/2n}\ dx = \sqrt{2\pi n} + O( \frac{n}{R} e^{-R^2/2n} )." class="latex" />
Putting all this together, and using <a href="#ephi">(17)</a> to estimate <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28n+%5Cpm+R%29%7D+%5Cll+e%5E%7B-%5Cphi%28n%29+-+%5Cfrac%7BR%5E2%7D%7B2n%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28n+%5Cpm+R%29%7D+%5Cll+e%5E%7B-%5Cphi%28n%29+-+%5Cfrac%7BR%5E2%7D%7B2n%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7B-%5Cphi%28n+%5Cpm+R%29%7D+%5Cll+e%5E%7B-%5Cphi%28n%29+-+%5Cfrac%7BR%5E2%7D%7B2n%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{-\phi(n \pm R)} \ll e^{-\phi(n) - \frac{R^2}{2n}}}" class="latex" />, we conclude that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+e%5E%7B-%5Cphi%28n%29%7D+%28+%281+%2B+O%28%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D%29%29+%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-R%5E2%2F2n%7D%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+e%5E%7B-%5Cphi%28n%29%7D+%28+%281+%2B+O%28%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D%29%29+%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-R%5E2%2F2n%7D%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+e%5E%7B-%5Cphi%28n%29%7D+%28+%281+%2B+O%28%5Cfrac%7BR%5E3%7D%7Bn%5E2%7D%29%29+%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-R%5E2%2F2n%7D%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle n! = e^{-\phi(n)} ( (1 + O(\frac{R^3}{n^2})) \sqrt{2\pi n} + O( \frac{n}{R} e^{-R^2/2n}) " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+O%28+n+e%5E%7B-R%5E2%2F2n%7D+%29+%2B+O%28+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-R%5E2%2F2n%7D+%29+%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+O%28+n+e%5E%7B-R%5E2%2F2n%7D+%29+%2B+O%28+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-R%5E2%2F2n%7D+%29+%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+O%28+n+e%5E%7B-R%5E2%2F2n%7D+%29+%2B+O%28+%5Cfrac%7Bn%7D%7BR%7D+e%5E%7B-R%5E2%2F2n%7D+%29+%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + O( n e^{-R^2/2n} ) + O( \frac{n}{R} e^{-R^2/2n} ) )" class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+e%5E%7B-n%2Bn+%5Clog+n%7D+%28%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+%5Cfrac%7BR%5E2%7D%7Bn%7D+%2B+n+e%5E%7B-R%5E2%2F2n%7D+%29%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+e%5E%7B-n%2Bn+%5Clog+n%7D+%28%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+%5Cfrac%7BR%5E2%7D%7Bn%7D+%2B+n+e%5E%7B-R%5E2%2F2n%7D+%29%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+e%5E%7B-n%2Bn+%5Clog+n%7D+%28%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+%5Cfrac%7BR%5E2%7D%7Bn%7D+%2B+n+e%5E%7B-R%5E2%2F2n%7D+%29%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle = e^{-n+n \log n} (\sqrt{2\pi n} + O( \frac{R^2}{n} + n e^{-R^2/2n} ))" class="latex" />

so if we select <img src="https://s0.wp.com/latex.php?latex=%7BR%3Dn%5E%7B2%2F3%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BR%3Dn%5E%7B2%2F3%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BR%3Dn%5E%7B2%2F3%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{R=n^{2/3}}" class="latex" /> for instance, we obtain the <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation">Stirling approximation</a> <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+%5Cfrac%7Bn%5En%7D%7Be%5En%7D+%28%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+n%5E%7B1%2F3%7D%29+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+%5Cfrac%7Bn%5En%7D%7Be%5En%7D+%28%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+n%5E%7B1%2F3%7D%29+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++n%21+%3D+%5Cfrac%7Bn%5En%7D%7Be%5En%7D+%28%5Csqrt%7B2%5Cpi+n%7D+%2B+O%28+n%5E%7B1%2F3%7D%29+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle n! = \frac{n^n}{e^n} (\sqrt{2\pi n} + O( n^{1/3}) )." class="latex" />
One can improve the error term by a finer decomposition than <a href="#onr">(16)</a>; we leave this as an exercise to the interested reader.

<blockquote>Remark 10 It can be convenient to do some initial rescalings to this analysis to achieve a nice normalization; see <a href="https://terrytao.wordpress.com/2010/01/02/254a-notes-0a-stirlings-formula/">this previous blog post</a> for details. </blockquote>


<blockquote>Exercise 11 Solve problem (iii) from the introduction. (Hint: extract out the term <img src="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7Bk%5E%7B2n-4k%7D%7D%7B%28n-k%29%5E%7B2n-4k%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7Bk%5E%7B2n-4k%7D%7D%7B%28n-k%29%5E%7B2n-4k%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7Bk%5E%7B2n-4k%7D%7D%7B%28n-k%29%5E%7B2n-4k%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\frac{k^{2n-4k}}{(n-k)^{2n-4k}}}" class="latex" /> to write as the exponential factor <img src="https://s0.wp.com/latex.php?latex=%7Be%5E%7B%5Cphi%28k%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%5E%7B%5Cphi%28k%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%5E%7B%5Cphi%28k%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{e^{\phi(k)}}" class="latex" />, placing all the other terms (which are of polynomial size) in the amplitude function <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpsi%28k%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpsi%28k%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpsi%28k%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\psi(k)}" class="latex" />. The function <img src="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cphi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\phi}" class="latex" /> will then attain a maximum at <img src="https://s0.wp.com/latex.php?latex=%7Bk%3Dn%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%3Dn%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%3Dn%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k=n/2}" class="latex" />; perform a Taylor expansion and mimic the arguments above.) </blockquote>


]]></content:encoded>
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<slash:comments>29</slash:comments>
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<media:title type="html">Terry</media:title>
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</item>
<item>
<title>Undecidability of translational monotilings</title>
<link>https://terrytao.wordpress.com/2023/09/18/undecidability-of-translational-monotilings/</link>
<comments>https://terrytao.wordpress.com/2023/09/18/undecidability-of-translational-monotilings/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Tue, 19 Sep 2023 03:17:04 +0000</pubDate>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[math.LO]]></category>
<category><![CDATA[paper]]></category>
<category><![CDATA[Rachel Greenfeld]]></category>
<category><![CDATA[tiling]]></category>
<category><![CDATA[undecidability]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=13968</guid>
<description><![CDATA[Rachel Greenfeld and I have just uploaded to the arXiv our paper “Undecidability of translational monotilings“. This is a sequel to our previous paper in which we constructed a translational monotiling of a high-dimensional lattice (thus the monotile is a finite set and the translates , of partition ) which was aperiodic (there is no […]]]></description>
<content:encoded><![CDATA[
<a href="https://www.math.ias.edu/~rgreenfeld/">Rachel Greenfeld</a> and I have just uploaded to the arXiv our paper “<a href="https://arxiv.org/abs/2309.09504">Undecidability of translational monotilings</a>“. This is a sequel to <a href="https://terrytao.wordpress.com/2022/11/29/a-counterexample-to-the-periodic-tiling-conjecture-2/">our previous paper</a> in which we constructed a translational monotiling <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Coplus+F+%3D+%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Coplus+F+%3D+%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Coplus+F+%3D+%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \oplus F = {\bf Z}^d}" class="latex" /> of a high-dimensional lattice <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" /> (thus the monotile <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> is a finite set and the translates <img src="https://s0.wp.com/latex.php?latex=%7Ba%2BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%2BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%2BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a+F}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Ba+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a \in A}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> partition <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />) which was aperiodic (there is no way to “repair” this tiling into a periodic tiling <img src="https://s0.wp.com/latex.php?latex=%7BA%27+%5Coplus+F+%3D+%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%27+%5Coplus+F+%3D+%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%27+%5Coplus+F+%3D+%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A' \oplus F = {\bf Z}^d}" class="latex" />, in which <img src="https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A'}" class="latex" /> is now periodic with respect to a finite index subgroup of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />). This disproved the periodic tiling conjecture of <a href="https://zbmath.org/0284.20048">Stein</a>, <a href="https://zbmath.org/0746.52001">Grunbaum-Shephard</a> and <a href="https://zbmath.org/0847.05037">Lagarias-Wang</a>, which asserted that such aperiodic translational monotilings do not exist. (Compare with the “<a href="https://arxiv.org/abs/2303.10798">hat monotile</a>“, which is a recently discovered aperiodic isometric monotile for of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+R%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+R%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+R%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf R}^2}" class="latex" />, where one is now allowed to use rotations and reflections as well as translations, or the even more recent “<a href="https://arxiv.org/abs/2305.17743">spectre monotile</a>“, which is similar except that no reflections are needed.)

One of the motivations of this conjecture was the observation of <a href="https://zbmath.org/0137.01001">Hao Wang</a> that if the periodic tiling conjecture were true, then the translational monotiling problem is (algorithmically) decidable: there is a Turing machine which, when given a dimension <img src="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d}" class="latex" /> and a finite subset <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />, can determine in finite time whether <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> can tile <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />. This is because if a periodic tiling exists, it can be found by computer search; and if no tiling exists at all, then (by the compactness theorem) there exists some finite subset of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" /> that cannot be covered by disjoint translates of <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" />, and this can also be discovered by computer search. The periodic tiling conjecture asserts that these are the only two possible scenarios, thus giving the decidability.


On the other hand, Wang’s argument is not known to be reversible: the failure of the periodic tiling conjecture does not automatically imply the undecidability of the translational monotiling problem, as it does not rule out the existence of some other algorithm to determine tiling that does not rely on the existence of a periodic tiling. (For instance, even with the newly discovered hat and spectre tiles, it remains an open question whether the isometric monotiling problem for (say) polygons with rational coefficients in <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+R%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+R%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+R%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf R}^2}" class="latex" /> is decidable, with or without reflections.)


The main result of this paper settles this question (with one caveat):

<blockquote class="wp-block-quote is-layout-flow wp-block-quote-is-layout-flow">
Theorem 1 There does not exist any algorithm which, given a dimension <img src="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d}" class="latex" />, a periodic subset <img src="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />, and a finite subset <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />, determines in finite time whether there is a translational tiling <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Coplus+F+%3D+E%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Coplus+F+%3D+E%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Coplus+F+%3D+E%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \oplus F = E}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E}" class="latex" /> by <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" />.
</blockquote>

The caveat is that we have to work with periodic subsets <img src="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />, rather than all of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />; we believe this is largely a technical restriction of our method, and it is likely that can be removed with additional effort and creativity. We also remark that when <img src="https://s0.wp.com/latex.php?latex=%7Bd%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d=2}" class="latex" />, the periodic tiling conjecture was established <a href="https://zbmath.org/1454.11125">by Bhattacharya</a>, and so the problem is decidable in the <img src="https://s0.wp.com/latex.php?latex=%7Bd%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d=2}" class="latex" /> case. It remains open whether the tiling problem is decidable for any fixed value of <img src="https://s0.wp.com/latex.php?latex=%7Bd%3E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%3E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%3E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d>2}" class="latex" /> (note in the above result that the dimension <img src="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d}" class="latex" /> is not fixed, but is part of the input).


Because of a well known link between algorithmic undecidability and logical undecidability (also known as <a href="https://en.wikipedia.org/wiki/Independence_(mathematical_logic)">logical independence</a>), the main theorem also implies the existence of an (in principle explicitly describable) dimension <img src="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d}" class="latex" />, periodic subset <img src="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />, and a finite subset <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />, such that the assertion that <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> tiles <img src="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E}" class="latex" /> by translation cannot be proven or disproven in ZFC set theory (assuming of course that this theory is consistent).


As a consequence of our method, we can also replace <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" /> here by “virtually two-dimensional” groups <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5E2+%5Ctimes+G_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5E2+%5Ctimes+G_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5E2+%5Ctimes+G_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^2 \times G_0}" class="latex" />, with <img src="https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G_0}" class="latex" /> a finite abelian group (which now becomes part of the input, in place of the dimension <img src="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d}" class="latex" />).


We now describe some of the main ideas of the proof. It is a common technique to show that a given problem is undecidable by demonstrating that some other problem that was already known to be undecidable can be “encoded” within the original problem, so that any algorithm for deciding the original problem would also decide the embedded problem. Accordingly, we will encode the <a href="https://en.wikipedia.org/wiki/Wang_tile">Wang tiling problem</a> as a monotiling problem in <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />:

<blockquote class="wp-block-quote is-layout-flow wp-block-quote-is-layout-flow">
Problem 2 (Wang tiling problem) Given a finite collection <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+W%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+W%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+W%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal W}}" class="latex" /> of Wang tiles (unit squares with each side assigned some color from a finite palette), is it possible to tile the plane with translates of these tiles along the standard lattice <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^2}" class="latex" />, such that adjacent tiles have matching colors along their common edge?
</blockquote>

It is a famous <a href="https://zbmath.org/0199.30802">result of Berger</a> that this problem is undecidable. The embedding of this problem into the higher-dimensional translational monotiling problem proceeds through some intermediate problems. Firstly, it is an easy matter to embed the Wang tiling problem into a similar problem which we call the domino problem:

<blockquote class="wp-block-quote is-layout-flow wp-block-quote-is-layout-flow">
Problem 3 (Domino problem) Given a finite collection <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal R}_1}" class="latex" /> (resp. <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal R}_2}" class="latex" />) of horizontal (resp. vertical) dominoes – pairs of adjacent unit squares, each of which is decorated with an element of a finite set <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+W%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+W%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+W%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal W}}" class="latex" /> of “pips”, is it possible to assign a pip to each unit square in the standard lattice tiling of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^2}" class="latex" />, such that every horizontal (resp. vertical) pair of squares in this tiling is decorated using a domino from <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal R}_1}" class="latex" /> (resp. <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal R}_2}" class="latex" />)?
</blockquote>

Indeed, one just has to interpet each Wang tile as a separate “pip”, and define the domino sets <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal R}_1}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal R}_2}" class="latex" /> to be the pairs of horizontally or vertically adjacent Wang tiles with matching colors along their edge.


Next, we embed the domino problem into a Sudoku problem:

<blockquote class="wp-block-quote is-layout-flow wp-block-quote-is-layout-flow">
Problem 4 (Sudoku problem) Given a column width <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" />, a digit set <img src="https://s0.wp.com/latex.php?latex=%7B%5CSigma%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CSigma%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CSigma%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Sigma}" class="latex" />, a collection <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+S%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+S%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+S%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal S}}" class="latex" /> of functions <img src="https://s0.wp.com/latex.php?latex=%7Bg%3A+%5C%7B0%2C%5Cdots%2CN-1%5C%7D+%5Crightarrow+%5CSigma%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bg%3A+%5C%7B0%2C%5Cdots%2CN-1%5C%7D+%5Crightarrow+%5CSigma%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bg%3A+%5C%7B0%2C%5Cdots%2CN-1%5C%7D+%5Crightarrow+%5CSigma%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{g: \{0,\dots,N-1\} \rightarrow \Sigma}" class="latex" />, and an “initial condition” <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal C}}" class="latex" /> (which we will not detail here, as it is a little technical), is it possible to assign a digit <img src="https://s0.wp.com/latex.php?latex=%7BF%28n%2Cm%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%28n%2Cm%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%28n%2Cm%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F(n,m)}" class="latex" /> to each cell <img src="https://s0.wp.com/latex.php?latex=%7B%28n%2Cm%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28n%2Cm%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28n%2Cm%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(n,m)}" class="latex" /> in the “Sudoku board” <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7B0%2C1%2C%5Cdots%2CN-1%5C%7D+%5Ctimes+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7B0%2C1%2C%5Cdots%2CN-1%5C%7D+%5Ctimes+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7B0%2C1%2C%5Cdots%2CN-1%5C%7D+%5Ctimes+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\{0,1,\dots,N-1\} \times {\bf Z}}" class="latex" /> such that for any slope <img src="https://s0.wp.com/latex.php?latex=%7Bj+%5Cin+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj+%5Cin+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj+%5Cin+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j \in {\bf Z}}" class="latex" /> and intercept <img src="https://s0.wp.com/latex.php?latex=%7Bi+%5Cin+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi+%5Cin+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi+%5Cin+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i \in {\bf Z}}" class="latex" />, the digits <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cmapsto+F%28n%2Cjn%2Bi%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cmapsto+F%28n%2Cjn%2Bi%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cmapsto+F%28n%2Cjn%2Bi%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \mapsto F(n,jn+i)}" class="latex" /> along the line <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7B%28n%2Cjn%2Bi%29%3A+0+%5Cleq+n+%5Cleq+N-1%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7B%28n%2Cjn%2Bi%29%3A+0+%5Cleq+n+%5Cleq+N-1%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7B%28n%2Cjn%2Bi%29%3A+0+%5Cleq+n+%5Cleq+N-1%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\{(n,jn+i): 0 \leq n \leq N-1\}}" class="latex" /> lie in <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+S%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+S%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+S%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal S}}" class="latex" /> (and also that <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> obeys the initial condition <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal C}}" class="latex" />)?
</blockquote>

The most novel part of the paper is the demonstration that the domino problem can indeed be embedded into the Sudoku problem. The embedding of the Sudoku problem into the monotiling problem follows from a modification of the methods in <a href="https://terrytao.wordpress.com/2021/08/19/undecidable-translational-tilings-with-only-two-tiles-or-one-nonabelian-tile/">our previous</a> <a href="https://terrytao.wordpress.com/2022/11/29/a-counterexample-to-the-periodic-tiling-conjecture-2/">papers</a>, which had also introduced versions of the Sudoku problem, and created a “tiling language” which could be used to “program” various problems, including the Sudoku problem, as monotiling problems.


To encode the domino problem into the Sudoku problem, we need to take a domino function <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%3A+%7B%5Cbf+Z%7D%5E2+%5Crightarrow+%7B%5Cmathcal+W%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%3A+%7B%5Cbf+Z%7D%5E2+%5Crightarrow+%7B%5Cmathcal+W%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%3A+%7B%5Cbf+Z%7D%5E2+%5Crightarrow+%7B%5Cmathcal+W%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal T}: {\bf Z}^2 \rightarrow {\mathcal W}}" class="latex" /> (obeying the domino constraints associated to some domino sets <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%2C+%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%2C+%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+R%7D_1%2C+%7B%5Cmathcal+R%7D_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal R}_1, {\mathcal R}_2}" class="latex" />) and use it to build a Sudoku function <img src="https://s0.wp.com/latex.php?latex=%7BF%3A+%5C%7B0%2C%5Cdots%2CN-1%5C%7D+%5Ctimes+%7B%5Cbf+Z%7D+%5Crightarrow+%5CSigma%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%3A+%5C%7B0%2C%5Cdots%2CN-1%5C%7D+%5Ctimes+%7B%5Cbf+Z%7D+%5Crightarrow+%5CSigma%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%3A+%5C%7B0%2C%5Cdots%2CN-1%5C%7D+%5Ctimes+%7B%5Cbf+Z%7D+%5Crightarrow+%5CSigma%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F: \{0,\dots,N-1\} \times {\bf Z} \rightarrow \Sigma}" class="latex" /> (obeying some Sudoku constraints relating to the domino sets); conversely, every Sudoku function obeying the rules of our Sudoku puzzle has to arise somehow from a domino function. The route to doing so was not immediately obvious, but after a helpful tip from <a href="https://members.loria.fr/EJeandel/">Emmanuel Jeandel</a>, we were able to adapt some ideas <a href="https://zbmath.org/0301.02042">of Aanderaa and Lewis</a>, in which certain hierarchical structures were used to encode one problem in another. Here, we interpret hierarchical structure <img src="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p}" class="latex" />-adically (using two different primes due to the two-dimensionality of the domino problem). The Sudoku function <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> that will exemplify our embedding is then built from <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal T}}" class="latex" /> by the formula <a name="f-targ"></a>
<a name="f-targ"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++F%28n%2Cm%29+%3A%3D+%28+f_%7Bp_1%7D%28m%29%2C+f_%7Bp_2%7D%28m%29%2C+%7B%5Cmathcal+T%7D%28%5Cnu_%7Bp_1%7D%28m%29%2C+%5Cnu_%7Bp_2%7D%28m%29%29+%29+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++F%28n%2Cm%29+%3A%3D+%28+f_%7Bp_1%7D%28m%29%2C+f_%7Bp_2%7D%28m%29%2C+%7B%5Cmathcal+T%7D%28%5Cnu_%7Bp_1%7D%28m%29%2C+%5Cnu_%7Bp_2%7D%28m%29%29+%29+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++F%28n%2Cm%29+%3A%3D+%28+f_%7Bp_1%7D%28m%29%2C+f_%7Bp_2%7D%28m%29%2C+%7B%5Cmathcal+T%7D%28%5Cnu_%7Bp_1%7D%28m%29%2C+%5Cnu_%7Bp_2%7D%28m%29%29+%29+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle F(n,m) := ( f_{p_1}(m), f_{p_2}(m), {\mathcal T}(\nu_{p_1}(m), \nu_{p_2}(m)) ) \ \ \ \ \ (1)" class="latex" /> </a> where <img src="https://s0.wp.com/latex.php?latex=%7Bp_1%2Cp_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp_1%2Cp_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp_1%2Cp_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p_1,p_2}" class="latex" /> are two large distinct primes (for instance one can take <img src="https://s0.wp.com/latex.php?latex=%7Bp_1%3D53%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp_1%3D53%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp_1%3D53%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p_1=53}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Bp_2%3D59%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp_2%3D59%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp_2%3D59%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p_2=59}" class="latex" /> for concreteness), <img src="https://s0.wp.com/latex.php?latex=%7B%5Cnu_p%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cnu_p%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cnu_p%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\nu_p(m)}" class="latex" /> denotes the number of times <img src="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p}" class="latex" /> divides <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%7Bf_p%28m%29+%5Cin+%7B%5Cbf+Z%7D%2Fp%7B%5Cbf+Z%7D+%5Cbackslash+%5C%7B0%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_p%28m%29+%5Cin+%7B%5Cbf+Z%7D%2Fp%7B%5Cbf+Z%7D+%5Cbackslash+%5C%7B0%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_p%28m%29+%5Cin+%7B%5Cbf+Z%7D%2Fp%7B%5Cbf+Z%7D+%5Cbackslash+%5C%7B0%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_p(m) \in {\bf Z}/p{\bf Z} \backslash \{0\}}" class="latex" /> is the last non-zero digit in the base <img src="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p}" class="latex" /> expansion of <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" />:
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++f_p%28m%29+%3A%3D+%5Cfrac%7Bm%7D%7Bp%5E%7B%5Cnu_p%28m%29%7D%7D+%5Chbox%7B+mod+%7D+p&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++f_p%28m%29+%3A%3D+%5Cfrac%7Bm%7D%7Bp%5E%7B%5Cnu_p%28m%29%7D%7D+%5Chbox%7B+mod+%7D+p&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++f_p%28m%29+%3A%3D+%5Cfrac%7Bm%7D%7Bp%5E%7B%5Cnu_p%28m%29%7D%7D+%5Chbox%7B+mod+%7D+p&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle f_p(m) := \frac{m}{p^{\nu_p(m)}} \hbox{ mod } p" class="latex" />
(with the conventions <img src="https://s0.wp.com/latex.php?latex=%7B%5Cnu_p%280%29%3D%2B%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cnu_p%280%29%3D%2B%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cnu_p%280%29%3D%2B%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\nu_p(0)=+\infty}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bf_p%280%29%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_p%280%29%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_p%280%29%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_p(0)=1}" class="latex" />). In the case <img src="https://s0.wp.com/latex.php?latex=%7Bp_1%3D3%2C+p_2%3D5%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp_1%3D3%2C+p_2%3D5%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp_1%3D3%2C+p_2%3D5%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p_1=3, p_2=5}" class="latex" />, the first component of <a href="#f-targ">(1)</a> looks like this:

<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/09/p3.png"><img loading="lazy" width="615" height="1024" data-attachment-id="13962" data-permalink="https://terrytao.wordpress.com/about/test-page/p3/" data-orig-file="https://terrytao.files.wordpress.com/2023/09/p3.png" data-orig-size="2562,4267" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="p3" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/09/p3.png?w=180" data-large-file="https://terrytao.files.wordpress.com/2023/09/p3.png?w=490" src="https://terrytao.files.wordpress.com/2023/09/p3.png?w=615" alt="" class="wp-image-13962" srcset="https://terrytao.files.wordpress.com/2023/09/p3.png?w=615 615w, https://terrytao.files.wordpress.com/2023/09/p3.png?w=1230 1230w, https://terrytao.files.wordpress.com/2023/09/p3.png?w=90 90w, https://terrytao.files.wordpress.com/2023/09/p3.png?w=180 180w, https://terrytao.files.wordpress.com/2023/09/p3.png?w=768 768w" sizes="(max-width: 615px) 100vw, 615px" /></a></figure>

and a typical instance of the final component <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%28%5Cnu_%7Bp_1%7D%28m%29%2C+%5Cnu_%7Bp_2%7D%28m%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%28%5Cnu_%7Bp_1%7D%28m%29%2C+%5Cnu_%7Bp_2%7D%28m%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%28%5Cnu_%7Bp_1%7D%28m%29%2C+%5Cnu_%7Bp_2%7D%28m%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal T}(\nu_{p_1}(m), \nu_{p_2}(m))}" class="latex" /> looks like this:

<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/09/domino.png"><img loading="lazy" width="1024" height="906" data-attachment-id="13959" data-permalink="https://terrytao.wordpress.com/about/test-page/domino/" data-orig-file="https://terrytao.files.wordpress.com/2023/09/domino.png" data-orig-size="5022,4448" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="domino" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/09/domino.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/09/domino.png?w=490" src="https://terrytao.files.wordpress.com/2023/09/domino.png?w=1024" alt="" class="wp-image-13959" srcset="https://terrytao.files.wordpress.com/2023/09/domino.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/09/domino.png?w=2046 2046w, https://terrytao.files.wordpress.com/2023/09/domino.png?w=150 150w, https://terrytao.files.wordpress.com/2023/09/domino.png?w=300 300w, https://terrytao.files.wordpress.com/2023/09/domino.png?w=768 768w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>

Amusingly, the decoration here is essentially following the rules of the children’s game “<a href="https://en.wikipedia.org/wiki/Fizz_buzz">Fizz buzz</a>“.


To demonstrate the embedding, we thus need to produce a specific Sudoku rule <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+S%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+S%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+S%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal S}}" class="latex" /> (as well as a more technical initial condition <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal C}}" class="latex" />, which is basically required to exclude degenerate Sudoku solutions such as a constant solution) that can “capture” the target function <a href="#f-targ">(1)</a>, in the sense that the only solutions to this specific Sudoku puzzle are given by variants of <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> (e.g., <img src="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BF%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{F}" class="latex" /> composed with various linear transformations). In <a href="https://terrytao.wordpress.com/2022/11/29/a-counterexample-to-the-periodic-tiling-conjecture-2/">our previous paper</a> we were able to build a Sudoku puzzle that could similarly capture either of the first two components <img src="https://s0.wp.com/latex.php?latex=%7Bf_%7Bp_1%7D%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_%7Bp_1%7D%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_%7Bp_1%7D%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_{p_1}(m)}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Bf_%7Bp_2%7D%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_%7Bp_2%7D%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_%7Bp_2%7D%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_{p_2}(m)}" class="latex" /> of our target function <a href="#f-targ">(1)</a> (up to linear transformations), by a procedure very akin to solving an actual Sudoku puzzle (combined with iterative use of a “Tetris” move in which we eliminate rows of the puzzle that we have fully solved, to focus on the remaining unsolved rows). Our previous paper treated the case when <img src="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p}" class="latex" /> was replaced with a power of <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" />, as this was the only case that we know how to embed in a monotiling problem of the entirety of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" /> (as opposed to a periodic subset <img src="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+Z%7D%5Ed%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf Z}^d}" class="latex" />), but the analysis is in fact easier when <img src="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p}" class="latex" /> is a large odd prime, instead of a power of <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" />. Once the first two components <img src="https://s0.wp.com/latex.php?latex=%7Bf_%7Bp_1%7D%28m%29%2C+f_%7Bp_2%7D%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_%7Bp_1%7D%28m%29%2C+f_%7Bp_2%7D%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_%7Bp_1%7D%28m%29%2C+f_%7Bp_2%7D%28m%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_{p_1}(m), f_{p_2}(m)}" class="latex" /> have been solved for, it is a relatively routine matter to design an additional constraint in the Sudoku rule that then constrains the third component to be of the desired form <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%28%5Cnu_%7Bp_1%7D%28m%29%2C+%5Cnu_%7Bp_2%7D%28m%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%28%5Cnu_%7Bp_1%7D%28m%29%2C+%5Cnu_%7Bp_2%7D%28m%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%28%5Cnu_%7Bp_1%7D%28m%29%2C+%5Cnu_%7Bp_2%7D%28m%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal T}(\nu_{p_1}(m), \nu_{p_2}(m))}" class="latex" />, with <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+T%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal T}}" class="latex" /> obeying the domino constraints.

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