{"paper":{"title":"Ergodic Properties of $k$-Free Integers in Number Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Francesco Cellarosi, Ilya Vinogradov","submitted_at":"2013-03-31T15:02:24Z","abstract_excerpt":"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 Si"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0214","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}