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

Aurelia Bartnicka, Joanna Kułaga-Przymus

Published 2015-07-03Version 1

Recently, 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 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 for a~probability measure on $\{0,1\}^{\mathcal{O}_K}$, invariant under the corresponding 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 topological system given by the closure of the orbit of $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$. Finally, we show that this topological dynamical system is proximal and therefore has no non-trivial equicontinuous factor, but has a~non-trivial topological joining with an ergodic rotation on a~compact Abelian group.