Kneser's Theorem in σ-finite Abelian groups

• Published in: Canadian mathematical bulletin Vol. 65; no. 4
• Main Authors: Bienvenu, Pierre-Yves, Hennecart, François
• Format: Journal Article
• Language: English
• Published: Cambridge University Press 10.01.2022
• Subjects: Group Theory; Mathematics; Number Theory
• ISSN: 0008-4395; 1496-4287
• DOI: 10.4153/S0008439521001053

Let $G$ be a $\sigma$-finite abelian group, i.e. $G=\bigcup_{n\geq 1} G_n$ where $(G_n)_{n\geq 1}$ is a non decreasing sequence of finite subgroups. For any $A\subset G$, let $\underline{\mathrm{d}}(A):=\liminf_{n\to\infty}\frac{|A\cap G_n|}{|G_n|}$ be its lower asymptotic density. We show that for any subsets $A$ and $B$ of $G$, whenever $\underline{\mathrm{d}}(A+B)< \underline{\mathrm{d}}(A)+\underline{\mathrm{d}}(B)$, the sumset $A+B$ must be periodic, that is, a union of translates of a subgroup $H\leq G$ of finite index. This is exactly analogous to Kneser's theorem regarding the density of infinite sets of integers. Further,we show similar statements for the upper asymptotic density in the case where $A=\pm B$.An analagous statement had already been proven by Griesmerin the very general context of countable abelian groups,but the present paper provides a much simpler argument specifically tailored for the setting of $\sigma$-finite abelian groups.This argument relies on an appeal to another theorem of Kneser, namely the one regarding finite sumsets in an abelian group.