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Progression-free sets in \(\mathbb{Z}_4^n\) are exponentially small. (English) Zbl 1425.11019
This paper represents without a doubt one of the most important breakthroughs in discrete mathematics in 2016. This is despite (or possibly to some extent because of) its relative brevity and elementary nature.
Given a finite abelian group \(G\), let \(r_k(G)\) denote the cardinality of a largest subset of \(G\) containing no non-trivial \(k\)-term arithmetic progression. The case when \(k=3\) and \(G\) is an \(n\)-dimensional vector space over a finite field of small fixed characteristic \(p\) is considered a toy problem for Roth’s problem in the integers [B. Green, in: Surveys in combinatorics 2005, Lond. Math. Soc. Lect. Note Ser. 327, 1–27 (2005; Zbl 1155.11306)], but it is also of independent interest to the design theory and theoretical computer science communities, amongst others.
R. Meshulam [J. Comb. Theory, Ser. A 71, No. 1, 168–172 (1995; Zbl 0832.11006)] proved that \[ r_3(\mathbb F^n_3) \ll 3^n/n, \] using a Fourier iteration argument going back to [K. F. Roth, J. Lond. Math. Soc. 28, 104–109 (1953; Zbl 0050.04002)] which is now considered standard in the field. Nudging this upper bound toward the lower bound of \[ 3^{0.72485n} \approx 2.2174^n\] provided by a construction of Y. Edel [Des. Codes Cryptography 31, No. 1, 5–14 (2004; Zbl 1057.51005)] has proved surprisingly difficult. It was only in 2012 that M. D. Bateman and N. H. Katz [J. Am. Math. Soc. 25, No. 2, 585–613 (2012; Zbl 1262.11010)], in a paper that is by many considered a technical tour de force, managed to improve the exponent of the denominator from \(1\) to \(1+\varepsilon\) for some small but explicit \(\varepsilon >0\).
Finite abelian groups \(G\) of even order were first considered by V. F. Lev J. Number Theory 104, No. 1, 162–169 (2004; Zbl 1043.11022)], who adapted Meshulam’s proof to show that \[ r_3(G) < 2|G|/\mathrm{rank}(2G). \] T. Sanders [Anal. PDE 2, No. 2, 211–234 (2009; Zbl 1197.11017)] improved upon this for the specific group \(G=(\mathbb Z/4\mathbb Z)^n\), showing that \[ r_3((\mathbb Z/4\mathbb Z)^n) \ll 4n/(n\log^\varepsilon n) \] for some absolute constant \(\varepsilon >0\), using a more sophisticated (but still Fourier-analytic) iteration technique.
The main result of the present paper is the following.
Theorem. Let \(n\ge 1\) and suppose that \(A\subseteq(\mathbb Z/4\mathbb Z)^n\) contains no non-trivial arithmetic progression of length \(3\). Then \[ |A| \le 4^{\gamma n} \] for a constant \(0<\gamma<1\).
Specifically, the constant \(\gamma\approx 0.926\) is obtained as the maximum over \(0<\varepsilon <1/4\) of \(\tfrac12(H(0.5-\varepsilon)+H(2\varepsilon))\), where \(H\) is the binary entropy function.
This result is of significance for at least two reasons: first, it represents an exponential improvement over previous work, bringing the upper bound for the first time within reasonable reach of the lower bound; second, it spawned a flurry of further results in the immediate aftermath of its publication. Arguably the most important of these to date is the paper by J. S. Ellenberg and D. C. Gijswijt [Ann. Math. (2) 185, No. 1, 339–343 (2017; Zbl 1425.11020)], which reduces the upper bound on \(r_3(\mathbb F^n_3)\) to roughly \(2.756^n\), alongside a handful of others that had not been formally published at the time that this review was written.
The core contribution of Croot, Lev and Pach is the realisation that a version of the polynomial method can be used to tackle Roth-type problems in certain finite abelian groups. For a comprehensive survey on the polynomial method, its variants and their applications, see [T. C. Tao, EMS Surv. Math. Sci. 1, No. 1, 1–46 (2014; Zbl 1294.05044)]. One recent application that stands out – having surfaced as unexpectedly as the result of the present paper – is the resolution of the finite-field Kakeya conjecture by Z. Dvir [J. Am. Math. Soc. 22, No. 4, 1093–1097 (2009; Zbl 1202.52021)].
The key ingredient in the proof of the above theorem is the following simple linear-algebraic lemma, also used in the aforementioned subsequent work of Ellenberg and Gijswijt: If \(P\) is a multilinear polynomial in \(n\) variables of total degree at most \(d\) over a field \(F\) such that \(P(a-b)=0\) for all \(a\ne b\in A\), then \(P(-a)\) cannot be non-zero for too many elements \(a\in A\).
This lemma can be used to prove that if \(A\) is a progression-free subset of \(\mathbb Z/4\mathbb Z)^n\) then there are few \(F_n\)-cosets containing many elements of \(A\), where \(F_n\) denotes the subgroup of \(\mathbb Z/4\mathbb Z)^n\) generated by its involutions (which is isomorphic to \((\mathbb Z/4\mathbb Z)^n))\). From this the bound on the size of \(A\) follows essentially by averaging and the tensor-power trick.
Reviewer: Julia Wolf

11B25 Arithmetic progressions
11B13 Additive bases, including sumsets
Full Text: DOI
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