Bohr sets in sumsets II: countable abelian groups

We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting G be a countable discrete abelian group and $\phi _1, \phi _2, \phi _3: G \to G$ be commuting endomorphisms whose images have finite indices, we show that If has positive upper Banach density and , then contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in and a recent result of the first author. For any partition , there exists an such that contains a Bohr set. This generalizes a result of the second and third authors from to countable abelian groups. If have positive upper Banach density and is a partition, contains a Bohr set for some . 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 $[G:\phi _j(G)]$ , the upper Banach density of A (in (1)), or the number of sets in the given partition (in (2) and (3)). 2020 Mathematics Subject Classification : 37A45, 11B13, 43A07