{"paper":{"title":"Numbers with simply normal $\\beta$-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Derong Kong, Simon Baker","submitted_at":"2017-07-04T14:34:25Z","abstract_excerpt":"In [Bak] the first author proved that for any $\\beta\\in (1,\\beta_{KL})$ every $x\\in(0,\\frac{1}{\\beta-1})$ has a simply normal $\\beta$-expansion, where $\\beta_{KL}\\approx 1.78723$ is the Komornik-Loreti constant. This result is complemented by an observation made in [JSS], where it was shown that whenever $\\beta\\in (\\beta_T, 2]$ there exists an $x\\in(0,\\frac{1}{\\beta-1})$ with a unique $\\beta$-expansion, and this expansion is not simply normal. Here $\\beta_T\\approx 1.80194$ is the unique zero in $(1,2]$ of the polynomial $x^3-x^2-2x+1$. This leaves a gap in our understanding within the interval"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01013","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"}