{"paper":{"title":"Beta-expansions of rational numbers in quadratic Pisot bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO","math.NT"],"primary_cat":"math.DS","authors_text":"Tom\\'a\\v{s} Hejda, Wolfgang Steiner","submitted_at":"2014-11-10T13:30:20Z","abstract_excerpt":"We study rational numbers with purely periodic R\\'enyi $\\beta$-expansions. For bases $\\beta$ satisfying $\\beta^2=a\\beta+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for $\\gamma(\\beta)=1$, i.e., that all rational numbers $p/q\\in[0,1)$ with $\\gcd(q,b)=1$ have a purely periodic $\\beta$-expansion. A simple algorithm for determining the value of $\\gamma(\\beta)$ for all quadratic Pisot numbers $\\beta$ is described."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2419","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"}