Bohr sets in sumsets II: countable abelian groups

Abstract

We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting G be a countable discrete abelian group and φ1, φ2, φ3 : G → G be commuting endomorphisms whose images have finite indices, we show that (1) If A ⊂ G has positive upper Banach density and φ1 + φ2 + φ3 = 0, then φ1(A) + φ2(A) + φ3(A) contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in Z and a recent result of the first author.(2) For any partition G = r i=1 Ai, there exists an i ∈ {1, . . . , r} such that φ1(Ai) + φ2(Ai) − φ2(Ai) contains a Bohr set. This generalizes a result of the second and third authors from Z to countable abelian groups.(3) If B, C ⊂ G have positive upper Banach density and G = r i=1 Ai is a partition, B + C + Ai contains a Bohr set for some i ∈ {1, . . . , r}. This is a strengthening of a theorem of Bergelson, Furstenberg, and Weiss. All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices [G : φj(G)], the upper Banach density of A (in (1)), or the number of sets in the given partition (in (2) and ( 3)).