Noncommutative sets of small doubling

A corollary of Kneser’s theorem, one sees that any finite non-empty subset $A$ of an abelian group $G=(G,+)$ with $\mathrm{\mid }A+A\mathrm{\mid }\le (2-\text{eps})\mathrm{\mid }A\mathrm{\mid }$ can be covered by at most $\frac{2}{\text{eps}}-1$ translates of a finite group $H$ of cardinality at most $(2-\text{eps})\mathrm{\mid }A\mathrm{\mid }$. 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 $\frac{2}{\text{eps}}\mathrm{\mid }A\mathrm{\mid }$, or can be covered by at most $\frac{2}{\text{eps}}-1$ right-cosets of a finite group $H$ of cardinality at most $\mathrm{\mid }A\mathrm{\mid }$. We also note some connections with some recent work of Sanders and of Petridis.