Recent progress on bounds for sets with no three terms in arithmetic progression

TL;DR

This paper reviews recent advances in establishing bounds for sets lacking three-term arithmetic progressions, highlighting key contributions from multiple researchers and their impact on additive combinatorics.

Contribution

It synthesizes recent progress and techniques in bounding sets with no three-term arithmetic progression, emphasizing developments by Bloom, Sisask, Croot, Lev, Pach, Ellenberg, and Gijswijt.

Findings

Significant improvements in bounds for progression-free sets.

Introduction of new combinatorial and algebraic methods.

Enhanced understanding of structure in sets avoiding 3-term arithmetic progressions.

Abstract

This is the text accompanying my Bourbaki seminar on the work of Bloom and Sisask, Croot, Lev, and Pach, and Ellenberg and Gijswijt.