{"paper":{"title":"Arithmetics in number systems with negative base","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"E. Pelantov\\'a, T. V\\'avra, Z. Mas\\'akov\\'a","submitted_at":"2010-02-04T15:02:28Z","abstract_excerpt":"We study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set ${\\rm Fin}(-\\beta)$ and $\\Z_{-\\beta}$ of numbers having finite resp. integer $(-\\beta)$-expansions. We show that ${\\rm Fin}(-\\beta)$ is trivial if $\\beta$ is smaller than the golden ratio $\\frac12(1+\\sqrt5)$. For $\\beta\\geq\\frac12(1+\\sqrt5)$ we prove that ${\\rm Fin}(-\\beta)$ is a ring, only if $\\beta$ is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that ${\\rm Fin}(-\\beta)$ is a ring if $\\beta$ is a quadratic Pisot n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.1009","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"}