Finite field models in arithmetic combinatorics -- twenty years on

TL;DR

This survey reviews a decade of progress in finite field models within additive combinatorics, highlighting key results, developments, and future research directions since the influential earlier surveys.

Contribution

It provides an updated comprehensive overview of recent advances in finite field models in additive combinatorics over the past ten years.

Findings

Significant progress on central open problems in finite field additive combinatorics.

New techniques and results in both finite field and integer settings.

Identification of promising future research directions.

Abstract

About twenty years ago, Green wrote a survey article on the utility of looking at toy versions over finite fields of problems in additive combinatorics. This article was extremely influential, and the rapid development of additive combinatorics necessitated a follow-up survey ten years later, which was written by Wolf. Since the publication of Wolf's article, an immense amount of progress has been made on several central open problems in additive combinatorics in both the finite field model and integer settings. This survey, written to accompany my talk at the 2024 British Combinatorial Conference, covers some of the most significant results of the past ten years and suggests future directions.