  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.

Combinatorics;Logic