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

Abstract

We prove Bogolyubov-Ruzsa-type results for finite subsets of groups with small tripling, $|A^3|\le O(|A|)$, or small alternation, $|A{A}^{\text{-}1}A|\le O(|A|)$. 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.

arXiv comment24 pages, final version