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| ≤ O(|A|), or small alternation, |AA −1A| ≤ O(|A|). As applications, we obtain a qualitative analogue 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.