Improved Bounds for 3-Progressions

TL;DR

This paper establishes a new upper bound on the size of subsets of integers lacking nontrivial three-term arithmetic progressions, using advanced combinatorial and harmonic analysis techniques.

Contribution

It introduces an improved bound for 3-progressions by combining an iterated sifting argument with an enhanced bootstrapping approach for almost-periodicity.

Findings

Bound: |A| ≤ exp(-c log(N)^{1/6} / log log(N)) N for progression-free sets.

Utilizes an iterated sifting argument and an improved almost-periodicity bootstrap.

Provides a tighter quantitative bound compared to previous results.

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.