{"paper":{"title":"On small bases for which $1$ has countably many expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jian Lu, Lijin Wang, Simon Baker, Yuru Zou","submitted_at":"2015-02-25T15:45:24Z","abstract_excerpt":"Let $q\\in(1,2)$. A $q$-expansion of a number $x$ in $[0,\\frac{1}{q-1}]$ is a sequence $(\\delta_i)_{i=1}^\\infty\\in\\{0,1\\}^{\\mathbb{N}}$ satisfying $$ x=\\sum_{i=1}^\\infty\\frac{\\delta_i}{q^i}.$$ Let $\\mathcal{B}_{\\aleph_0}$ denote the set of $q$ for which there exists $x$ with a countable number of $q$-expansions, and let $\\mathcal{B}_{1, \\aleph_0}$ denote the set of $q$ for which $1$ has a countable number of $q$-expansions. In \\cite{Sidorov6} it was shown that $\\min\\mathcal{B}_{\\aleph_0}=\\min\\mathcal{B}_{1,\\aleph_0}=\\frac{1+\\sqrt{5}}{2},$ and in \\cite{Baker} it was shown that $\\mathcal{B}_{\\ale"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.07212","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"}