{"paper":{"title":"Zero-one law of Hausdorff dimensions of the recurrent sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bing Li, Dong Han Kim","submitted_at":"2015-10-02T05:18:46Z","abstract_excerpt":"Let $(\\Sigma, \\sigma)$ be the one-sided shift space with $m$ symbols and $R_n(x)$ be the first return time of $x\\in\\Sigma$ to the $n$-th cylinder containing $x$. Denote $$E^\\varphi_{\\alpha,\\beta}=\\left\\{x\\in\\Sigma: \\liminf_{n\\to\\infty}\\frac{\\log R_n(x)}{\\varphi(n)}=\\alpha,\\ \\limsup_{n\\to\\infty}\\frac{\\log R_n(x)}{\\varphi(n)}=\\beta\\right\\},$$ where $\\varphi: \\mathbb{N}\\to \\mathbb{R}^+$ is a monotonically increasing function and $0\\leq\\alpha\\leq\\beta\\leq +\\infty$. We show that the Hausdorff dimension of the set $E^\\varphi_{\\alpha,\\beta}$ admits a dichotomy: it is either zero or one depending on $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00495","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":""},"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"}