$\mathfrak{B}$-free integers in number fields and dynamics

TL;DR

This paper generalizes the study of square-free integers to $rak{B}$-free integers in number fields, analyzing their dynamical systems, measure-theoretic properties, and pattern structures, revealing connections to ergodic rotations and entropy characteristics.

Contribution

It extends the dynamical analysis of square-free integers to a broader class of $rak{B}$-free integers in number fields, establishing measure-theoretic and topological properties and their relations to ergodic rotations.

Findings

Characteristic function is generic for an invariant measure.

Dynamical system is isomorphic to an ergodic rotation on a compact Abelian group.

Topological entropy of the orbit closure is computed.

Abstract

In 2010, Sarnak initiated the study of the dynamics of the system determined by the square of the M\"obius function (the characteristic function of the square-free integers). We deal with his program in the more general context of $\mathfrak{B}$-free integers in number fields, suggested 5 years later by Baake and Huck. This setting encompasses the classical square-free case and its generalizations. Given a number field $K$, let $\mathfrak{B}$ be a family of pairwise coprime ideals in its ring of integers $\mathcal{O}_K$, such that $\sum_{\mathfrak{b}\in\mathfrak{B}}1/|\mathcal{O}_K / \mathfrak{b}|<\infty$. We study the dynamical system determined by the set $\mathcal{F}_\mathfrak{B}=\mathcal{O}_K\setminus \bigcup_{\mathfrak{b}\in\mathfrak{B}}\mathfrak{b}$ of $\mathfrak{B}$-free integers in $\mathcal{O}_K$. We show that the characteristic function $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$ of $\mathcal{F}_\mathfrak{B}$ is generic along the natural F\o{}lner sequence for a probability measure on $\{0,1\}^{\mathcal{O}_K}$, invariant under the multidimensional shift. The corresponding measure-theoretical dynamical system is proved to be isomorphic to an ergodic rotation on a compact Abelian group. In particular, it is of zero Kolmogorov entropy. Moreover, we provide a description of ``patterns'' appearing in $\mathcal{F}_\mathfrak{B}$ and compute the topological entropy of the orbit closure of $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$. Finally, we show that this topological dynamical system has a non-trivial topological joining with an ergodic rotation on a compact Abelian group.