On finite sets of small tripling or small alternation in arbitrary groups. (30th November 2020) - Conant, Gabriel
Abstract: We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, | A 3 | ≤ O (| A |), or small alternation, | AA −1 A | ≤ 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.