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 $\mathbb{Z}$, $d^*(B) > 0$, 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 $\phi_1, \phi_2: G \to G$ 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].
| Comments: | 43 pages, 1 figure. We added Theorem D which answers two questions of the second author in [31] |
| Subjects: | Dynamical Systems (math.DS); Combinatorics (math.CO) |
| MSC classes: | Primary: 37A45, Secondary: 11B13, 43A07 |
| Cite as: | arXiv:2603.11376 [math.DS] |
| (or arXiv:2603.11376v2 [math.DS] for this version) | |
| https://doi.org/10.48550/arXiv.2603.11376 arXiv-issued DOI via DataCite |
From: Anh N. Le [view email]
[v1]
Wed, 11 Mar 2026 23:31:16 UTC (63 KB)
[v2]
Mon, 25 May 2026 18:02:33 UTC (60 KB)
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