On finite sets of small tripling or small alternation in arbitrary groups. (30th November 2020)

• Record Type: Journal Article
• Authors: Conant, Gabriel
• Is Part Of: Combinatorics, probability and computing. Volume 29:Number 6(2020)
• Page Start: 807
• Page End: 829
• Publication Date: 2020-11-30
• Subjects: 20D60, -- 11B30, -- 03C20; Combinatorial analysis -- Periodicals; Probabilities -- Periodicals; Computer science -- Mathematics -- Periodicals; 511.6
• DOI: 10.1017/S0963548320000176
• Languages: English
• ISSNs: 0963-5483

Abstract

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.