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Improved Bounds for 3-Progressions

Rushil Raghavan

Abstract

We prove that if $A\subset \{1,\dots,N\}$ has no nontrivial three-term arithmetic progressions, then $|A|\leq \exp(-c\log(N)^{1/6}\log\log(N)^{-1})N$ for some absolute constant $c>0$. To obtain this bound, we use an iterated variant of the sifting argument of Kelley and Meka, as well as an improved bootstrapping argument for Croot-Sisask almost-periodicity due to Bloom and Sisask.

Improved Bounds for 3-Progressions

Abstract

We prove that if has no nontrivial three-term arithmetic progressions, then for some absolute constant . To obtain this bound, we use an iterated variant of the sifting argument of Kelley and Meka, as well as an improved bootstrapping argument for Croot-Sisask almost-periodicity due to Bloom and Sisask.

Paper Structure

This paper contains 7 sections, 26 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Notation and Strategy of Proof
  3. Definitions and Notation
  4. Strategy of Proof
  5. The Finite Field Case
  6. The Integer Case
  7. Bohr Sets

Key Result

Theorem 1.2

If $A\subseteq\{1,\dots,N\}$ contains no nontrivial three-term arithmetic progressions, then $|A|\leq \exp(-\Omega((\log N)^{1/12}))N$.

Theorems & Definitions (46)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: Expectations, Inner Products, Measures, $L^p$ norms
  • Definition 2.2: Density, Relative Density
  • Definition 2.3: Fourier Transform
  • Proposition 2.4
  • Definition 2.5
  • Lemma 2.6: Sifting
  • ...and 36 more