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 $\hat{I}“$ satisfying $\tilde{m}(A+B)<\tilde{m}(A)+\tilde{m}(B)$, where $\tilde{m}$ is an arbitrary finitely additive translation-invariant probability measure on $\hat{I}“$, extending M.~Kneser's theorem on Haar measure in compact abelian groups. We then characterize, for an arbitrary Følner sequence or Følner net $\mathbf{F}=(F_i{)}_{i\in I}$ on $\hat{I}“$, those $A$, $B$ satisfying ${\underline{d}}_{\mathbf{F}}(A+B)<{\underline{d}}_{\mathbf{F}}(A)+{\underline{d}}_{\mathbf{F}}(B)$, where ${\underline{d}}_{\mathbf{F}}(C):={\mathrm{liminf}}_{i\in I}|C\cap F_i|/|F_i|$. 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 $d^{\ast }(A+B)<d^{\ast }(A)+d^{\ast }(B)$, where $d^{\ast }$ is upper Banach density on $\hat{I}“$.

43 pages