{"paper":{"title":"Arithmetics in numeration systems with negative quadratic base","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.NT","authors_text":"T. V\\'avra, Z. Mas\\'akov\\'a","submitted_at":"2010-11-05T14:15:02Z","abstract_excerpt":"We consider positional numeration system with negative base $-\\beta$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $\\beta$ is a quadratic Pisot number. We study a class of roots $\\beta>1$ of polynomials $x^2-mx-n$, $m\\geq n\\geq 1$, and show that in this case the set ${\\rm Fin}(-\\beta)$ of finite $(-\\beta)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $\\beta=\\tau=\\frac12(1+\\sqrt5)$, the golden ratio. For such $\\beta$, we determine the exact bound on the number of fractional"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1403","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"}