On finite sets of small tripling or small alternation in arbitrary groups

TL;DR

This paper establishes Bogolyubov-Ruzsa-type results for finite groups with small tripling or alternation, leading to new regularity lemmas and structural insights in non-abelian group settings.

Contribution

It extends classical additive combinatorics results to arbitrary finite groups, providing new qualitative and quantitative structural theorems.

Findings

A Bogolyubov-type lemma for dense sets in finite groups.

A quantitative arithmetic regularity lemma for sets with bounded VC-dimension.

Generalization of abelian case results to non-abelian finite groups.

Abstract

We prove Bogolyubov-Ruzsa-type results for finite subsets of groups with small tripling, $|A^3|\leq O(|A|)$, or small alternation, $|AA^{\text{-}1} A|\leq O(|A|)$. As applications, we obtain a qualitative analog of Bogolyubov's Lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox, and Zhao, and gives a quantitative version of previous work of the author, Pillay, and Terry.