{"paper":{"title":"Growth rate for beta-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"De-Jun Feng, Nikita Sidorov","submitted_at":"2009-02-03T12:25:46Z","abstract_excerpt":"Let $\\beta>1$ and let $m>\\be$ be an integer. Each $x\\in I_\\be:=[0,\\frac{m-1}{\\beta-1}]$ can be represented in the form \\[ x=\\sum_{k=1}^\\infty \\epsilon_k\\beta^{-k}, \\] where $\\epsilon_k\\in\\{0,1,...,m-1\\}$ for all $k$ (a $\\beta$-expansion of $x$). It is known that a.e. $x\\in I_\\beta$ has a continuum of distinct $\\beta$-expansions. In this paper we prove that if $\\beta$ is a Pisot number, then for a.e. $x$ this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by $\\beta$.\n  When $\\beta<\\frac{1+\\sqr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.0488","kind":"arxiv","version":4},"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"}