Kneser- and Jin-type inverse theorems in discrete abelian groups

Abstract

We characterize the pairs of sets $A, B$ in an arbitrary (countable or uncountable) discrete abelian group $\Gamma$ satisfying $\tilde{m}(A+B)<\tilde{m}(A)+\tilde{m}(B)$, where $\tilde{m}$ is an arbitrary finitely additive translation-invariant probability measure on $\Gamma$, extending M.~Kneser's theorem on Haar measure in compact abelian groups. We then characterize, for an arbitrary F{\o}lner sequence or F{\o}lner net $\mathbf F=(F_{i})_{i\in I}$ on $\Gamma$, those $A$, $B$ satisfying $\underline{d}_{\mathbf F}(A+B)<\underline{d}_{\mathbf F}(A)+\underline{d}_{\mathbf F}(B)$, where $\underline{d}_{\mathbf F}(C):=\liminf_{i\in I} |C\cap F_{i}|/|F_{i}|$. This extends Kneser's theorem on lower asymptotic density in $\mathbb N$. We also generalize theorems of Prerna Bihani and Renling Jin by characterizing pairs $A$, $B$ satisfying $d^{*}(A+B)<d^{*}(A)+d^{*}(B)$, where $d^{*}$ is upper Banach density on $\Gamma$.