The positive-density k-fold sumset conjecture for abelian groups

Let $k\in\mathbb{N}$. Let $\Gamma$ be a countably infinite abelian group, and suppose

$$[\Gamma:k!\Gamma]<\infty.$$

For $A\subseteq\Gamma$, write $d^*(A)$ for its upper Banach density, and for a set $B\subseteq\Gamma$ let $B^{\oplus m}$ denote the $m$-fold restricted sumset. Positive-density k-fold sumset conjecture. If $d^*(A)>0$, then there exists $t\in\Gamma$ and an infinite set $B\subseteq\Gamma$ such that

$$B^{\oplus m}\subseteq A-t$$

for every $1\leq m\leq k$. The conjecture extends the positive-density k-fold sumset theorem from the integers to abelian groups; the stated finite-index condition excludes the counterexample mechanism established earlier in the paper, while the general assertion remains open.