{"paper":{"title":"Univoque bases of real numbers: local dimension, Devil's staircase and isolated points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.DS"],"primary_cat":"math.NT","authors_text":"Derong Kong, Fan Lv, Jiayi Xu, Wenxia Li, Zhiqiang Wang","submitted_at":"2019-11-14T02:56:40Z","abstract_excerpt":"Given a positive integer $M$ and a real number $x>0$, let $\\mathcal U(x)$ be the set of all bases $q\\in(1, M+1]$ for which there exists a unique sequence $(d_i)=d_1d_2\\ldots$ with each digit $d_i\\in\\{0,1,\\ldots, M\\}$ satisfying $$ x=\\sum_{i=1}^\\infty\\frac{d_i}{q^i}. $$ The sequence $(d_i)$ is called a $q$-expansion of $x$. In this paper we investigate the local dimension of $\\mathcal U(x)$ and prove a `variation principle' for unique non-integer base expansions. We also determine the critical values of $\\mathcal U(x)$ such that when $x$ passes the first critical value the set $\\mathcal U(x)$ c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.05910","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/1911.05910/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"}