Noncommutative sets of small doubling

TL;DR

This paper extends Kneser's theorem to noncommutative groups, showing that small doubling sets are either contained in a coset of a finite subgroup or covered by a few cosets, with bounds depending on epsilon.

Contribution

It provides the first noncommutative analogue of Kneser's theorem for small doubling sets, establishing new structural results.

Findings

Sets with small doubling are contained in a coset of a finite subgroup or covered by few cosets.

Bounds on subgroup size and number of cosets depend on epsilon.

Connections with recent work of Sanders and Petridis.

Abstract

A corollary of Kneser's theorem, one sees that any finite non-empty subset $A$ of an abelian group $G = (G,+)$ with $|A + A| \leq (2-\eps) |A|$ can be covered by at most $\frac{2}{\eps}-1$ translates of a finite group $H$ of cardinality at most $(2-\eps)|A|$. Using some arguments of Hamidoune, we establish an analogue in the noncommutative setting. Namely, if $A$ is a finite non-empty subset of a nonabelian group $G = (G,\cdot)$ such that $|A \cdot A| \leq (2-\eps) |A|$, then $A$ is either contained in a right-coset of a finite group $H$ of cardinality at most $\frac{2}{\eps}|A|$, or can be covered by at most $\frac{2}{\eps}-1$ right-cosets of a finite group $H$ of cardinality at most $|A|$. We also note some connections with some recent work of Sanders and of Petridis.