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 (1) If $A\subset G$ has positive upper Banach density and ${\phi }_1+{\phi }_2+{\phi }_3=0$, then ${\phi }_1(A)+{\phi }_2(A)+{\phi }_3(A)$ contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in $\mathbb{Z}$ and a recent result of the first author. (2) For any partition $G={\bigcup }_{i=1}^r{A}_i$, there exists an $i\in \{1,\dots ,r\}$ such that ${\phi }_1(A_i)+{\phi }_2(A_i)-{\phi }_2(A_i)$ contains a Bohr set. This generalizes a result of the second and third authors from $\mathbb{Z}$ to countable abelian groups. (3) If $B,C\subset G$ have positive upper Banach density and $G={\bigcup }_{i=1}^r{A}_i$ is a partition, $B+C+A_i$ contains a Bohr set for some $i\in \{1,\dots ,r\}$. This is a strengthening of a theorem of Bergelson, Furstenberg, and Weiss. These 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)).