A group version of stable regularity

TL;DR

This paper establishes a regularity lemma for stable subsets within finite groups, showing they can be approximated by unions of cosets of a normal subgroup with bounded index, generalizing previous results in vector spaces.

Contribution

It introduces a group version of the stable regularity lemma, extending prior work from vector spaces to arbitrary finite groups.

Findings

Existence of a normal subgroup with bounded index approximating the stable set

Stable sets are close to unions of cosets of a normal subgroup

Generalization of Terry and Wolf's work to finite groups

Abstract

We prove that, given $\epsilon>0$ and $k\geq 1$, there is an integer $n$ such that the following holds. Suppose $G$ is a finite group and $A\subseteq G$ is $k$-stable. Then there is a normal subgroup $H\leq G$ of index at most $n$, and a set $Y\subseteq G$, which is a union of cosets of $H$, such that $|A\vartriangle Y|\leq\epsilon|H|$. It follows that, for any coset $C$ of $H$, either $|C\cap A|\leq \epsilon|H|$ or $|C\setminus A|\leq \epsilon|H|$. This qualitatively generalizes recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$.