Power-free points in quadratic number fields: Stabiliser, dynamics and entropy
Michael Baake, Alvaro Bustos, Andreas Nickel
TL;DR
The paper studies $k$-free integers in quadratic fields through number-theoretic shift spaces obtained via the Minkowski embedding, linking dynamical invariants to algebraic data. It establishes that, for any quadratic field, the stabiliser of the $k$-free set is generated by the unit group and the Galois action and is independent of $k$, with explicit finite dihedral structures in imaginary cases and infinite dihedral structures in real cases. By constructing the associated $\mathcal{B}$-free subshift and showing it is a hereditary Erdős weak model set, the authors determine the centraliser (trivial) and normaliser (a nontrivial semidirect product involving $\mathcal{O}^{\times}$ and $\mathrm{Gal}(K/\mathbb{Q})$), enabling a topological distinction between shifts from real versus imaginary quadratics. They compute the topological entropy of these shifts as $s=\log(2)/\zeta_K(k)$, tying a dynamical invariant directly to Dedekind zeta values and extracting number-theoretic information (e.g., when entropy uniquely identifies $(K,k)$ in certain families). The results illuminate how extended symmetries and entropy classify number-theoretic shift spaces and reveal deep connections between algebraic invariants and dynamical behavior.
Abstract
The sets of $k$-free integers in general quadratic number fields are studied, with special emphasis on (extended) symmetries and their impact on the topological dynamical systems induced by such integers. We establish correspondences between number-theoretic and dynamical quantities, and use symmetries and entropy to distinguish the systems.