Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields

TL;DR

This paper advances the understanding of large subsets without three-term arithmetic progressions by providing new lower bounds in both integers and finite fields, surpassing classical constructions and longstanding open problems.

Contribution

It introduces improved lower bounds on the size of progression-free sets in integers and finite fields, representing the first quasipolynomial improvement over Behrend's classical construction.

Findings

First quasipolynomial improvement over Behrend's construction in integers.

Established a lower bound of $(cp)^n$ with $c > 1/2$ in finite fields.

Surpassed classical bounds, addressing a long-standing open problem.

Abstract

We prove new lower bounds on the maximum size of subsets $A\subseteq \{1,\dots,N\}$ or $A\subseteq \mathbb{F}_p^n$ not containing three-term arithmetic progressions. In the setting of $\{1,\dots,N\}$, this is the first improvement upon a classical construction of Behrend from 1946 beyond lower-order factors (in particular, it is the first quasipolynomial improvement). In the setting of $\mathbb{F}_p^n$ for a fixed prime $p$ and large $n$, we prove a lower bound of $(cp)^n$ for some absolute constant $c>1/2$ (for $c = 1/2$, such a bound can be obtained via classical constructions from the 1940s, but improving upon this has been a well-known open problem).