Bohr sets in sumsets III: expanding difference sets and almost Bohr sets

TL;DR

This paper investigates the conditions under which sumsets and difference sets in abelian groups contain Bohr sets, identifying specific sets with this property and exploring implications for recurrence and structure in additive combinatorics.

Contribution

It characterizes sets that ensure sumsets or difference sets contain Bohr sets for all positive density subsets, extending previous results and answering open questions.

Findings

Certain polynomial and prime-related sets have the property that their sumsets with positive density sets contain Bohr sets.

Sets of the form A + S contain Bohr sets for all almost Bohr sets A, under specific conditions.

Results extend the understanding of recurrence properties and structure of sets in additive combinatorics.

Abstract

Let $G$ be a discrete abelian group. F{\o}lner showed that if $A \subseteq G$ has positive upper Banach density, then $A - A$ contains an almost Bohr set -- a set of the form $B \setminus E$ where $B$ is a Bohr set and $E$ has zero Banach density. We study the sets $S \subseteq G$ for which $A - A + S$ contains a Bohr set for every $A \subseteq G$ of positive upper Banach density. For $G = \mathbb{Z}$, we show that the sets $\{n^2: n \in \mathbb{N}\}$, $\{p - 1: p \text{ prime}\}$, and $\{ \lfloor n^c \rfloor: n \in \mathbb{N} \}$ with $c > 0$, have this property. We also study those sets $S$ such that $A + S$ contains a Bohr set for every almost Bohr set $A$. As applications, we prove: (i) If $\phi_1, \phi_2: G \to G$ are (not necessarily commuting) homomorphisms with finite indices $[G: \phi_i(G)]$, and $C \subseteq G$ is a central set, then $\phi_1(C) - \phi_1(C) + \phi_2(C)$ contains a Bohr set. This answers one of our questions in [35] and generalizes results in [44, 48]; (ii) Every set of pointwise recurrence in $\mathbb{Z}$ is a set of nice recurrence and a van der Corput set, extending known properties of sets of pointwise recurrence studied in [26, 27, 40].