{"paper":{"title":"Smallest bases of expansions with multiple digits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Derong Kong, Wenxia Li, Yuru Zou","submitted_at":"2015-07-29T13:21:29Z","abstract_excerpt":"Given two positive integers $M$ and $k$, let $\\B_k$ be the set of bases $q>1$ such that there exists a real number $x$ having exactly $k$ different $q$-expansions over the alphabet $\\{0,1,\\cdots,M\\}$. In this paper we investigate the smallest base $q_2$ of $\\B_2$, and show that if $M=2m$ the smallest base $$q_2 =\\frac{m+1+\\sqrt{m^2+2m+5}}{2},$$ and if $M=2m-1$ the smallest base $q_2$ is the appropriate root of $$ x^4=(m-1)\\,x^3+2 m\\, x^2+m \\,x+1. $$ Moreover, for $M=2$ we show that $q_2$ is also the smallest base of $\\B_k$ for all $k\\ge 3$. This turns out to be different from that for $M=1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08135","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"}