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

Abstract

Let $G$ be a discrete abelian group. Følner 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. Moreover, we prove that there are sets $A,B\subseteq \mathbb{Z}$ such that $A$ is dense in the Bohr topology of $\mathbb{Z}$, $d^{\ast }(B)>0$, while $A+B$ is not piecewise Bohr, answering two questions of the second author in [31]. 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 $\ddot{I}{•}_1,\ddot{I}{•}_2:G\to G$ are (not necessarily commuting) homomorphisms with finite indices $[G:\ddot{I}{•}_i(G)]$, and $C\subseteq G$ is a central set, then $\ddot{I}{•}_1(C)-\ddot{I}{•}_1(C)+\ddot{I}{•}_2(C)$ 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].

43 pages, 1 figure. We added Theorem D which answers two questions of the second author in [31]