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

TL;DR

The paper explicates Kelley and Meka's breakthrough bound for subsets $A\subseteq\{1,\dots,N\}$ with no nontrivial three-term arithmetic progressions, showing $|A| \le N/\exp\big(c(\log N)^{1/12}\big)$. It transfers the core ideas from the finite-field setting to the integers using Bohr sets, dependent random choice, and almost-periodicity to obtain efficient density increments on structured sets, culminating in Behrend-scale lower bounds for $A$-free progressions and improved lower bounds for $A+A+A$-based progressions. The key contributions include a self-contained exposition of KM's argument, minor Bohr-set refinements that widen applicability, and a quantified structural framework that yields near-optimal bounds within this approach. The results substantially advance the quantitative understanding of progression-free sets and offer a robust method with potential for further refinements toward Behrend-type bounds in additive combinatorics.

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 $A\subseteq \{1,\ldots,N\}$.

Theorems & Definitions (43)

• Theorem 1: Kelley-Meka
• Theorem 2: Kelley-Meka
• Theorem 3
• Theorem 4: Kelley-Meka
• Theorem 5
• Theorem 6: Kelley-Meka
• Lemma 7
• proof
• Lemma 8
• Corollary 9