Subset-Sum Density Realization in Locally Finite Abelian Groups

Let $G$ be a countable locally finite abelian group, and let $$G_1\le G_2\le \cdots ,\bigcup _{i\ge 1}{G}_i=G,$$ be any filtration of $G$ by finite subgroups. For $A\subseteq G$, let $\mathcal{P}(A)$ denote the set of all finite subset sums of elements of $A$, and let $2G:=\{2g:g\in G\}$. We prove that $\mathrm{\mid }2G\mathrm{\mid }=\mathrm{\infty }$ if and only if for every filtration and every interval $[\alpha ,\beta ]\subseteq [0,1]$, there exists $A\subseteq G$ such that the set of limit points of $${\left(\frac{\mathrm{\mid }\mathcal{P}(A)\cap G_i\mathrm{\mid }}{\mathrm{\mid }{G}_i\mathrm{\mid }}\right)}_{i\ge 1}$$ is exactly $[\alpha ,\beta ]$. We also prove a positive-density result that does not require $\mathrm{\mid }2G\mathrm{\mid }=\mathrm{\infty }$: if $G$ is infinite and $\mathcal{P}(A)$ has positive upper density along the given filtration, then there is an infinite set $B\subseteq \mathcal{P}(A)$ such that $B+B\subseteq \mathcal{P}(A)$. Combining this with the realization theorem, we show that, when $\mathrm{\mid }2G\mathrm{\mid }=\mathrm{\infty }$, every interval $[\alpha ,\beta ]$ with $0\le \alpha \le \beta \le 1$ and $\beta >0$ can be realized by a set $A$ with this additional property.