$\mathfrak{B}$-free integers in number fields and dynamics
Francisco Araújo, Aurelia Dymek, Joanna Kułaga-Przymus
TL;DR
The paper generalizes Sarnak's dynamics for square-freeness to $\mathfrak{B}$-free integers in number fields, showing that the indicator of $\mathfrak{B}$-free integers is generic along Følner sequences and that the resulting system is measure-theoretically isomorphic to a rotation on a compact Abelian group, hence of zero entropy. It provides a precise topological entropy formula $h_{top}(X_{\mathfrak{B}},(S_a))=\prod_{\ell\ge 1}(1-1/N(\mathfrak{b}_\ell))$, characterizes admissible patterns via $X_{\mathfrak{B}}$, proves proximality with a trivial maximal equicontinuous factor, and demonstrates a non-trivial topological joining with the rotation system. The results extend known cases for square-free, $k$-free, and lattice-point settings and show the $\mathfrak{B}$-free framework captures and unifies these dynamics, including a lattice-point correspondence. Overall, the work advances the understanding of the interplay between arithmetic structure and topological/measure-theoretic dynamics in number fields.
Abstract
In 2010, Sarnak initiated the study of the dynamics of the system determined by the square of the Möbius 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 Følner 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.