Submission date: 15 June 2018

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 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.

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Full text arXiv 1806.06022: pdf, ps.