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 $rA+sA+tA:=\{ra_1+sa_2+ta_3:a_i\in A\}$ contains a Bohr neighborhood of zero.  We prove a natural generalization of this result for subsets of countable abelian groups and more summands.