{"paper":{"title":"Multiple expansions of real numbers with digits set $\\{0,1,q\\}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Derong Kong, Kan Jiang, Karma Dajani, Wenxia Li","submitted_at":"2015-08-25T13:13:58Z","abstract_excerpt":"For $q>1$ we consider expansions in base $q$ over the alphabet $\\{0,1,q\\}$. Let $\\mathcal{U}_q$ be the set of $x$ which have a unique $q$-expansions. For $k=2, 3,\\cdots,\\aleph_0$ let $\\mathcal{B}_k$ be the set of bases $q$ for which there exists $x$ having $k$ different $q$-expansions, and for $q\\in \\mathcal{B}_k$ let $\\mathcal{U}_q^{(k)}$ be the set of all such $x$'s which have $k$ different $q$-expansions. In this paper we show that \\[ \\mathcal{B}_{\\aleph_0}=[2,\\infty),\\quad \\mathcal{B}_k=(q_c,\\infty)\\quad \\textrm{for any}\\quad k\\ge 2, \\] where $q_c\\approx 2.32472$ is the appropriate root of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06138","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"}