{"paper":{"title":"Besicovitch covering numbers for $\\mathcal B$-free and other shifts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Gerhard Keller, Stanis{\\l}aw Kasjan","submitted_at":"2025-05-14T09:59:26Z","abstract_excerpt":"For a finite alphabet $A$ define by $d_1(x,y):=\\limsup_{n\\to\\infty}\\frac{1}{2n+1}\\#\\{|i|\\le n: x_i\\neq y_i\\}$ the Besicovitch pseudo-metric on $A^{\\mathbb Z}$. It is well known that a closed subshift of $A^{\\mathbb Z}$ has finite covering numbers w.r.t. $d_1$ if and only if it is mean-equicontinuous. Here we study, more generally, the scaling behavior of these covering numbers for individual orbits which are generic for an ergodic measure $\\mu$ on $A^{\\mathbb Z}$ with discrete spectrum, and we explore their usefulness as invariants for block code equivalence. We illustrate this by developing t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.09253","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.09253/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}