Ergodic Properties of $k$-Free Integers in Number Fields

Abstract

Let $K/\mathbf{Q}$ be a degree $d$ extension. Inside the ring of integers ${\mathcal{O}}_K$ we define the set of $k$-free integers ${\mathcal{F}}_k$ and a natural ${\mathcal{O}}_K$-action on the space of binary ${\mathcal{O}}_K$-indexed sequences, equipped with an ${\mathcal{O}}_K$-invariant probability measure associated to ${\mathcal{F}}_k$. We prove that this action is ergodic, has pure point spectrum and is isomorphic to a ${\mathbf{Z}}^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the paper by the first author and Sinai arXiv:1112.4691 [math.DS] where $K=\mathbf{Q}$ and $k=2$.

arXiv comment31 pages, 3 figures. To appear in the Jounal of Modern Dynamics. Improvement of http://arxiv.org/abs/1304.0214v3