{"paper":{"title":"The topological property of the irregular sets on the lengths of basic intervals in beta-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bing Li, Lixuan Zheng, Min Wu","submitted_at":"2016-04-06T03:24:39Z","abstract_excerpt":"Let $\\beta > 1$ be a real number and $(\\epsilon_1(x, \\beta), \\epsilon_2(x, \\beta), \\ldots)$ be the $\\beta$-expansion of a point $x \\in (0, 1]$. For all $x \\in (0,1]$, let $A(D(x))$ be the set of accumulation points of $\\frac{-\\log_\\beta |I_n(x)|}{n}$ as $n \\rightarrow \\infty$, where $|I_n(x)|$ is the length of the basic interval of order $n$ containing $x \\in (0, 1]$. In this paper, we prove that $A(D(x))$ is always a closed interval for any $x \\in (0,1]$. Furthermore, if $\\lambda(\\beta)>0$, the extremely irregular set containing points $x \\in [0, 1]$ whose upper limit of $\\frac{-\\log_\\beta |I"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.01470","kind":"arxiv","version":5},"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"}