Paper detail

Kneser's Theorem in $σ$-finite Abelian groups

Let $G$ be a $σ$-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&#39;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 Griesmer in the very general context of countable abelian groups, but the present paper provides a much simpler argument specifically tailored for the setting of $σ$-finite abelian groups. This argument relies on an appeal to another theorem of Kneser, namely the one regarding finite sumsets in an abelian group.