Bohr sets in sumsets II: countable abelian groups

John T. Griesmer ; Anh N. Le ; Thái Hoàng Lê

Forum of Mathematics, Sigma, Tome 11 (2023)

Résumé

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 $i \in \{1, \ldots , r\}$ 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 $i \in \{1, \ldots , r\}$. 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 $[G:\phi _j(G)]$, the upper Banach density of A (in (1)), or the number of sets in the given partition (in (2) and (3)).

Détail

DOI : 10.1017/fms.2023.49