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

TL;DR

This paper studies the ergodic properties of the set of $k$-free integers within the ring of integers of a number field, revealing that the associated dynamical system has pure point spectrum, is ergodic, and is isomorphic to a $ extbf{Z}^d$-action on a compact abelian group.

Contribution

It generalizes previous work by establishing ergodic properties of $k$-free integers in arbitrary number fields, extending beyond the rational case.

Findings

The action is ergodic and has pure point spectrum.

It is isomorphic to a $ extbf{Z}^d$-action on a compact abelian group.

The system has zero measure-theoretical entropy and is not weakly mixing.

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 $k=2$.