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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1101.0036","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2010-12-30T05:06:11Z","cross_cats_sorted":["cs.DM","math.NT"],"title_canon_sha256":"bb010c955166c2fd049bd92190e1838e5104a2be7f157963bf21710fbda901cf","abstract_canon_sha256":"38e82e5e2109ba31da6ceb533ecf15fead79ae0086bb6c65fcc9401b9dbc9f83"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:11.178348Z","signature_b64":"KWfNxADglg2qGCeHu4GVwk2DzXkOp4DvGM3QGtihwBv0tajF1IYk//5W7F2U150W8QGFHd5pnYCSjjFEjtGtBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3913bc841ac97d209ab621b7a0e3699d3189b16b0aaa69b6748f7cc7e6d0ec41","last_reissued_at":"2026-05-18T04:32:11.177844Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:11.177844Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $r\\ge 1$. 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