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

TL;DR

This paper investigates the properties of non-lacunary semigroups of natural numbers, demonstrating their recurrence behavior and density in the torus, with implications for ergodic theory and number theory.

Contribution

It proves that certain non-lacunary semigroups are sets of Bohr recurrence and establishes density results for polynomial images of exponential sets, generalizing classical theorems.

Findings

The set \\{k! 2^m 3^n\\} is a Bohr recurrence set.

Polynomial images of exponential sets are dense in the torus when coefficients are irrational.

Generalizes results of Furstenberg and Weyl regarding recurrence and density.

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.