Ergodic Properties of k-Free Integers in Number Fields

2013/03/31 by Cellarosi, Francesco, Vinogradov, Ilya · 1 citation
#11N25 #11R04 #28D15 #37A35 #37A45 #37C85 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1304.0214

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.

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