Noncommutative sets of small doubling

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.

8 pages, no figures. To appear, European Journal of Combinatorics. This is the final version, incorporating the referee corrections

References in corpus (2)

• New Proofs of Plünnecke-type Estimates for Product Sets in Groups
• A quantitative version of the non-abelian idempotent theorem