Bohr topology and difference sets for some abelian groups

TL;DR

This paper constructs specific subsets within certain abelian groups demonstrating that positive density does not imply the presence of Bohr neighborhoods in difference sets, challenging existing assumptions in additive combinatorics.

Contribution

It provides explicit examples of sets with positive density whose difference sets lack Bohr neighborhoods, answering negatively a question about the structure of difference sets in abelian groups.

Findings

Constructed sets with positive upper Banach density lacking Bohr neighborhoods in their difference sets.

Showed that for p=2, the difference set does not contain sets of the form g+(B-B) with B piecewise syndetic.

Demonstrated that certain dense sets in the Bohr topology lead to non-piecewise Bohr sums, countering previous conjectures.

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.