Stable functions and Følner's Theorem

We show that if $G$ is an amenable group and $A\subseteq G$ has positive upper Banach density, then there is an identity neighborhood $B$ in the Bohr topology on $G$ that is almost contained in $A{A}^{-1}$ in the sense that $B\mathrm{\backslash }A{A}^{-1}$ has upper Banach density $0$. This generalizes the abelian case (due to Følner) and the countable case (due to Beiglböck, Bergelson, and Fish). The proof is indirectly based on local stable group theory in continuous logic. The main ingredients are Grothendieck’s double-limit characterization of relatively weakly compact sets in spaces of continuous functions, along with results of Ellis and Nerurkar on the topological dynamics of weakly almost periodic flows.