A group version of stable regularity

Abstract

We prove that, given $\epsilon >0$ and $k\ge 1$, 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 $H\le G$ of index at most $n$, and a set $Y\subseteq G$, which is a union of cosets of $H$, such that $|A△Y|\le \epsilon |H|$. It follows that, for any coset $C$ of $H$, either $|C\cap A|\le \epsilon |H|$ or $|C\setminus A|\le \epsilon |H|$. This qualitatively generalizes recent work of Terry and Wolf on vector spaces over ${\mathbb{F}}_p$.

10 pages, final version