{"paper":{"title":"The growth function of S-recognizable sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.NT"],"primary_cat":"cs.FL","authors_text":"Emilie Charlier, Narad Rampersad","submitted_at":"2010-12-30T05:06:11Z","abstract_excerpt":"A set $X\\subseteq\\mathbb N$ is S-recognizable for an abstract numeration system S if the set $\\rep_S(X)$ of its representations is accepted by a finite automaton. We show that the growth function of an S-recognizable set is always either $\\Theta((\\log(n))^{c-df}n^f)$ where $c,d\\in\\mathbb N$ and $f\\ge 1$, or $\\Theta(n^r \\theta^{\\Theta(n^q)})$, where $r,q\\in\\mathbb Q$ with $q\\le 1$. If the number of words of length n in the numeration language is bounded by a polynomial, then the growth function of an S-recognizable set is $\\Theta(n^r)$, where $r\\in \\mathbb Q$ with $r\\ge 1$. Furthermore, for eve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.0036","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"}