Ergodic properties of $k$-free integers in number fields

Abstract

Let $K/\mathbf{Q}$ be a degree-$d$ extension. Inside the ring of integers$\mathscr O_K$ we define the set of $k$-free integers $\mathscr F_k$ and anatural $\mathscr O_K$-action on the space of binary $\mathscr O_K$-indexed sequences,equipped with an $\mathscr O_K$-invariant probability measure associated to$\mathscr F_k$. We prove that this action is ergodic, has pure pointspectrum, and is isomorphic to a $\mathbf Z^d$-action on a compact abeliangroup. In particular, it is not weakly mixing and has zeromeasure-theoretical entropy. This work generalizes the work ofCellarosi and Sinai [J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1343--1374] that considered the case $K=\mathbf{Q}$ and $k=2$.