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 ${\mathbb{F}}_p$.

arXiv comment10 pages, final version

References in corpus (2)

• A Szemeredi-type regularity lemma in abelian groups, with applications
• Stable arithmetic regularity in the finite-field model

Cited by in corpus (8)

• Stable arithmetic regularity in the finite-field model
• Structure and regularity for subsets of groups with finite VC-dimension
• The étale-open topology and the stable fields conjecture
• Approximate subgroups with bounded VC-dimension
• Automata and tame expansions of $(\mathbb{Z},+)$
• Complete type amalgamation for non-standard finite groups
• Higher-order generalizations of stability and arithmetic regularity
• Remarks on recognizable subsets and local rank