Ergodic Properties of \(k\)-Free Integers in Number Fields

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 probabi...

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Bibliographic Details
Published in:arXiv.org 2013-10
Main Authors: Cellarosi, Francesco, Vinogradov, Ilya
Format: Article
Language:English
Subjects:
Ergodic processes
Group theory
Integers
Number theory
Rings (mathematics)
Online Access:Get full text
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Summary: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 \(k=2\).
ISSN:2331-8422