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Improved Bounds for Szemer\'{e}di's Theorem

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arxiv 2402.17995 v2 pith:DQ5VEVMB submitted 2024-02-28 math.CO math.NT

James Leng , Ashwin Sah , Mehtaab Sawhney This is my paper · ORCID
classification math.COmath.NT
keywords boundsszemertheoremarithmeticconsequencedenotedensityexists
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abstract

Let $r_k(N)$ denote the size of the largest subset of $[N] = \{1,\ldots,N\}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that \[r_k(N)\ll N\exp(-(\log\log N)^{c_k}).\] Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm as well as the density increment strategy of Heath-Brown and Szemer\'{e}di as reformulated by Green and Tao.

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Cited by 6 Pith papers

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