Submission date: 18 September 2020

Abstract:

We extend previous work on Hrushovski's stabilizer's theorem and prove a measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types. This generalizes results of Gowers and of Nikolov-Pyber, on products of three sets and yields model-theoretic proofs of existing asymptotic results for quasirandom groups. Furthermore, we bound the number of solutions to certain equations, such as x^n y^m=z^k$ for n+m=k, in subsets of small tripling in groups. In particular, we show the existence of lower bounds on the number of arithmetic progressions of length 3 for subsets of small doubling without involutions in arbitrary abelian groups.

Mathematics Subject Classification: 03C45, 11B30

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Full text arXiv 2009.08967: pdf, ps.