{"paper":{"title":"Univoque bases and Hausdorff dimension","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Derong Kong, Fan L\\\"u, Martijn de Vries, Wenxia Li","submitted_at":"2016-06-13T02:00:54Z","abstract_excerpt":"Given a positive integer $M$ and a real number $q >1$, a \\emph{$q$-expansion} of a real number $x$ is a sequence $(c_i)=c_1c_2\\cdots$ with $(c_i) \\in \\{0,\\ldots,M\\}^\\infty$ such that \\[x=\\sum_{i=1}^{\\infty} c_iq^{-i}.\\]\n  It is well known that if $q \\in (1,M+1]$, then each $x \\in I_q:=\\left[0,M/(q-1)\\right]$ has a $q$-expansion. Let $\\mathcal{U}=\\mathcal{U}(M)$ be the set of \\emph{univoque bases} $q>1$ for which $1$ has a unique $q$-expansion.\n  The main object of this paper is to provide new characterizations of $\\mathcal{U}$ and to show that the Hausdorff dimension of the set of numbers $x \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03791","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"}