Quantitative structure of stable sets in finite abelian groups

TL;DR

This paper establishes a quantitative arithmetic regularity lemma for stable subsets in finite abelian groups, extending previous results from vector spaces over finite fields to a broader class of groups using combinatorial methods.

Contribution

It provides a new, highly quantitative proof of the regularity lemma for stable sets in finite abelian groups, generalizing prior qualitative results.

Findings

Proves an arithmetic regularity lemma for stable subsets in finite abelian groups.

Extends previous results from vector spaces over finite fields to general finite abelian groups.

Uses combinatorial techniques to achieve a highly quantitative version.

Abstract

We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A qualitative version of this generalisation was recently obtained by the first author in joint work with Conant and Pillay, using model-theoretic techniques. In contrast, the approach in the present paper is highly quantitative and relies on several key ingredients from arithmetic combinatorics.