Bohr sets in sumsets II: countable abelian groups

TL;DR

This paper establishes the existence of Bohr sets within sumsets in countable abelian groups, extending previous results from integers and providing quantitative bounds based on group indices and densities.

Contribution

It generalizes known theorems about Bohr sets in sumsets from integers to countable abelian groups, with explicit quantitative bounds.

Findings

Bohr sets exist in sumsets of positive density subsets under certain conditions.

Partition results guarantee Bohr sets in sumsets of partitioned groups.

Sumsets of positive density sets in partitions contain Bohr sets with bounds depending on densities and indices.

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 $\phi_1, \phi_2, \phi_3: G \to G$ 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. 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)).