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arXiv · 2406.01353

Bohr recurrence and density of non-lacunary semigroups of $\mathbb{N}$

Nikos Frantzikinakis·Bernard Host·Bryna Kra
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Abstract

A subset $R$ of integers is a set of Bohr recurrence if every rotation on $\mathbb{T}^d$ returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!\, 2^m3^n\colon k,m,n\in \mathbb{N}\}$ is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if $P$ is a real polynomial with at least one non-constant irrational coefficient, then the set $\{P(2^m3^n)\colon m,n\in \mathbb{N}\}$ is dense in $\mathbb{T}$, thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl.

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Nikos Frantzikinakis, Bernard Host, Bryna Kra. 2024-06-03. Bohr recurrence and density of non-lacunary semigroups of $\mathbb{N}$. https://arxiv.org/abs/2406.01353

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