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Szemerédi’s theorem in the primes. (English) Zbl 1472.11061

For \(k\in \mathbb N\), suppose we have a nice (tending to \(0\), eventually continuous and strictly decreasing) upper bound \(\beta_k(N)\) on the density \(|A|/N\) of the largest subset of \(A\subseteq [N]:=\{1,2,\dots,N\}\) containing no \(k\)-term arithmetic progression. Such is the goal of quantitative refinements of Szemerédi’s theorem, and (at time of the publication of this paper), the state of the art is \(\beta_3(N)= (\log N)^{-1+o(1)}\) (due to T. F. Bloom [J. Lond. Math. Soc., II. Ser. 93, No. 3, 643–663 (2016; Zbl 1364.11024)]), \(\beta_4(N)=(\log N)^{-c}\) for some \(c>0\) (due to B. Green and T. Tao [Mathematika 63, No. 3, 944–1040 (2017; Zbl 1434.11037)]), and \(\beta_k(N)=(\log\log N)^{-c_k}\) for some \(c_k>0\) for \(k>4\) (due to [W. T. Gowers, Geom. Funct. Anal. 11, No. 3, 465–588 (2001); Erratum 11, No. 4, 869 (2001; Zbl 1028.11005)]).
Armed with such a density upper bound \(\beta_k(N)\), one can ask if this knowledge can be transferred to yield information about the relative density \(|A|/|\mathcal{P_N}|\) of the largest subset of \(A\subseteq \mathcal{P}_N:=\{p\in [N]: p \ \text{prime}\}\) containing no \(k\)-term arithmetic progression, which the authors denote \(\alpha_k(\mathcal{P}_N)\). Such results would constitute quantitative improvements of the Green-Tao theorem [B. Green and T. Tao, Ann. Math. (2) 167, No. 2, 481–547 (2008; Zbl 1191.11025)] on primes in arithmetic progression. In the case \(k=3\), the original result of [B. Green, Ann. Math. (2) 161, No. 3, 1609–1636 (2005; Zbl 1160.11307)] was improved by Helfgott and de Roton, and later Naslund, to establish \(\alpha_3(\mathcal{P}_N)\ll (\log \log N)^{-1+o(1)}\), generalized by K. Henriot [Proc. Edinb. Math. Soc., II. Ser. 60, No. 1, 133–163 (2017; Zbl 1419.11020)], see the concluding sentence of this review) to linear systems of complexity one. It is natural to ask if similar quantitative improvements can be made for \(k>3\).
Indeed, the central result of this paper achieves that goal in an elegant and satisfying way, showing that for every \(k\geq 4\), given \(\beta_k(N)\) as introduced in the opening paragraph, one has \[\alpha_k(\mathcal{P}_N) \ll_k \beta_k((\log\log N)^{c_k})\] for some \(c_k>0\) and all sufficiently large \(N\). In particular, using the best-known \(\beta_k(N)\) in each case, this yields triple-logarithmic decay for \(\alpha_4(\mathcal{P}_N)\), and quadruple logarithmic decay for \(\alpha_k(\mathcal{P}_N)\) for \(k>4\).
The main theorem is established by combining a quantified version of the relative Szemerédi theorem of D. Conlon et al. [Geom. Funct. Anal. 25, No. 3, 733–762 (2015; Zbl 1345.11008)] with estimates of K. Henriot [Proc. Edinb. Math. Soc., II. Ser. 60, No. 1, 133–163 (2017; Zbl 1419.11020)] on enveloping sieve weights. By “quantified” here we mean the establishment of explicit error terms, the details of which are provided in appendices. Modulo the careful quantification and bookkeeping of the appendices, the central argument is housed in a very pleasant few pages.
The paper is dedicated to the memory of Kevin Henriot, a promising young researcher in arithmetic combinatorics who passed away in 2016.

MSC:

11B30 Arithmetic combinatorics; higher degree uniformity
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References:

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