On a non-abelian Balog-Szemeredi-type lemma

TL;DR

This paper proves a non-abelian version of the Balog-Szemeredi lemma, demonstrating that small doubling sets in groups contain large symmetric neighborhoods with controlled powers, extending combinatorial group theory results.

Contribution

It introduces a non-abelian analogue of the Balog-Szemeredi lemma, providing bounds on symmetric neighborhoods within groups with small doubling properties.

Findings

Existence of symmetric neighborhoods with controlled powers

Quantitative bounds on the size of neighborhoods

Extension of additive combinatorics to non-abelian groups

Abstract

We show that if G is a group and A is a finite subset of G with |A^2| < K|A|, then for all k there is a symmetric neighbourhood of the identity S with S^k a subset of A^2A^{-2} and |S| > exp(-K^{O(k)})|A|.