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Improved Bounds for Szemerédi's Theorem

James Leng, Ashwin Sah, Mehtaab Sawhney

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.

Improved Bounds for Szemerédi's Theorem

Abstract

Let denote the size of the largest subset of with no -term arithmetic progression. We show that for , there exists such that Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers -norm as well as the density increment strategy of Heath-Brown and Szemerédi as reformulated by Green and Tao.
Paper Structure (10 sections, 9 theorems, 61 equations)

This paper contains 10 sections, 9 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Proof outline and techniques
  3. Local and global inverse theorems
  4. Schmidt-type decomposition problems
  5. Iterative Schmidt refinement
  6. Organization and notation
  7. Schmidt's problem for nilsequences
  8. Completing the proof
  9. Preliminaries for density increment
  10. Constructing factor approximation and density increment

Key Result

Theorem 1.1

Fix $k\ge 5$. There is $c_k\in(0,1)$ such that

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of Lemma \ref{['lem:schmidt-nilman']}
  • Definition 3.1
  • Theorem 3.2
  • ...and 8 more