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

TL;DR

This paper extends inverse theorems related to sumsets in discrete abelian groups, characterizing when sumset measures are less than the sum of individual measures across various measure types.

Contribution

It generalizes Kneser's theorem to arbitrary discrete abelian groups and different measure types, including finitely additive, F{46}lner, and Banach densities.

Findings

Characterization of sumsets with measure less than sum of measures in arbitrary groups.

Extension of Kneser's theorem to F{46}lner sequences and nets.

Generalization of Jin and Bihani's theorems to Banach density.

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 $\Gamma$.