ERGODIC PROPERTIES OF k-FREE INTEGERS IN NUMBER FIELDS

作者：Cellarosi Francesco^*; Vinogradov Ilya

来源：Journal of Modern Dynamics, 2013, 7(3): 461-488.

DOI：10.3934/jmd.2013.7.461

• Square-free and k-free integers in number fields
• correlation functions
• group actions with pure point spectrum
• ergodicity
• isomorphism of group actions

Let K/Q be a degree-d extension. Inside the ring of integers O-K we define the set of k-free integers F-k and a natural O-K-action on the space of binary O-K-indexed sequences, equipped with an O-K-invariant probability measure associated to F-k. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a 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 = Q and k = 2.

• 出版日期2013-9