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On finite sets of small tripling or small alternation in arbitrary groups

Part of: Model theory Representation theory of groups Sequences and sets

Published online by Cambridge University Press:  30 June 2020

Gabriel Conant
Gabriel Conant*
Affiliation:
Department of Pure Mathematics & Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK
*
Email: gconant@maths.cam.ac.uk
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Abstract

We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A3| ≤ 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.

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems
Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity 03C20: Ultraproducts and related constructions
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 6 , November 2020 , pp. 807 - 829
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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