{"paper":{"title":"The dimension of irregular set in parameter space","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":"2016-11-30T12:10:18Z","abstract_excerpt":"For any real number $\\beta>1$. The $n$th cylinder of $\\beta$ in the parameter space $\\{\\beta\\in \\mathbb{R}: \\beta>1\\}$ is a set of real numbers in $(1,\\infty)$ having the same first $n$ digits in their $\\beta$-expansion of $1$, denote by $I^P_n(\\beta)$. We study the quantities which describe the growth of the length of $I^P_n(\\beta)$. The Huasdorff dimension of the set of given growth rate of the length of $I^P_n(\\beta)$ will be determined in this paper."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.10111","kind":"arxiv","version":1},"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"}