Journal of Modern Dynamics (JMD)

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

Pages: 461 - 488, Issue 3, September 2013      doi:10.3934/jmd.2013.7.461

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Francesco Cellarosi - Department of Mathematics, Altgeld Hall, 1409 W Green Street, Urbana, IL 61801, United States (email)
Ilya Vinogradov - School of Mathematics, University Walk, Bristol, BS8 1TW, United Kingdom (email)

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 a natural $\mathscr O_K$-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 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 work of Cellarosi and Sinai [J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1343--1374] that considered the case $K=\mathbf{Q}$ and $k=2$.

Keywords:  Square-free and $k$-free integers in number fields, correlation functions, group actions with pure point spectrum, ergodicity, isomorphism of group actions.
Mathematics Subject Classification:  Primary: 37A35, 37A45; Secondary: 11R04, 11N25, 37C85, 28D15.

Received: March 2013;      Revised: September 2013;      Available Online: December 2013.