The Kelley--Meka bounds for sets free of three-term arithmetic progressions

TL;DR

This paper presents a simplified exposition of Kelley and Meka's recent bounds on the size of sets without three-term arithmetic progressions, improving understanding and extending applications to longer progressions.

Contribution

It offers a clearer proof of Kelley and Meka's bounds and introduces minor simplifications that broaden the method's applicability.

Findings

Sets without 3-term arithmetic progressions are significantly smaller than previously known.

The bounds imply improved lower bounds for long arithmetic progressions in sumsets.

Simplifications make the method more accessible and extendable.

Abstract

We give a self-contained exposition of the recent remarkable result of Kelley and Meka: if $A\subseteq \{1,\ldots,N\}$ has no non-trivial three-term arithmetic progressions then $\lvert A\rvert \leq \exp(-c(\log N)^{1/12})N$ for some constant $c>0$. Although our proof is identical to that of Kelley and Meka in all of the main ideas, we also incorporate some minor simplifications relating to Bohr sets. This eases some of the technical difficulties tackled by Kelley and Meka and widens the scope of their method. As a consequence, we improve the lower bounds for finding long arithmetic progressions in $A+A+A$, where $A\subseteq \{1,\ldots,N\}$.