Bohr topology and difference sets for some abelian groups

Abstract

For a fixed prime $p$, $\mathbb F_{p}$ denotes the field with $p$ elements, and $\mathbb F_{p}^{\omega}$ denotes the countable direct sum $\bigoplus_{n=1}^{\infty} \mathbb F_{p}$. Viewing $\mathbb F_{p}^{\omega}$ as a countable abelian group, we construct a set $A\subseteq \mathbb F_{p}^{\omega}$ having positive upper Banach density while the difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any $c\in \mathbb F_{p}^{\omega}$. For $p=2$ we obtain a stronger conclusion: $A-A$ does not contain a set of the form $g+(B-B)$, where $B$ is piecewise syndetic. This construction answers negatively a variant of the following question asked by several authors: if $A\subseteq \mathbb Z$ has positive upper Banach density, must $A-A$ contain a Bohr neighborhood of some $n\in \mathbb Z$? We also construct sets $S, A\subseteq \mathbb F_{p}^{\omega}$ such that $S$ is dense in the Bohr topology of $\mathbb F_{p}^{\omega}$, $A$ has positive upper Banach density, and $A+S$ is not piecewise Bohr. For $p=2$ we show that every translate of $S$ is a set of topological recurrence and $A+S$ is not piecewise syndetic. These constructions answer a variant of a question asked by the author.