Bohr neighborhoods in generalized difference sets

Abstract

If $A$ is a set of integers having positive upper Banach density and $r,s,t$ are nonzero integers whose sum is zero, a theorem of Bergelson and Ruzsa says that the set $rA+sA+tA:=\{r{a}_1+s{a}_2+t{a}_3:a_i\in A\}$ contains a Bohr neighborhood of zero. We prove the natural generalization of this result for subsets of countable abelian groups and more summands.

17 pages