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arXiv · 2402.17995

Improved Bounds for Szemerédi's Theorem

James Leng·Ashwin Sah·Mehtaab Sawhney
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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 Szemerédi as reformulated by Green and Tao.

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James Leng, Ashwin Sah, Mehtaab Sawhney. 2024-02-29. Improved Bounds for Szemerédi's Theorem. https://arxiv.org/abs/2402.17995

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