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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"},"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":"0902.0488","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-02-03T12:25:46Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"d129defa2b0dac00bd7ce74512c7312cf1efc41be871d90b8ebbc5d7d8c9b479","abstract_canon_sha256":"01ed4ff9fe78e895a837d8633fb76ad85bf750bde4ede5f8a815c0f95f8a0fd6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:52.221948Z","signature_b64":"Jw2zZ73Udrd3lZgCFebgvMjS41ocUZa5GgwSakUIqSmKJ7HFPX+Vozn/FMxwkVI+fda/Oeer0fj/HzJeRZ9yDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70cd817a58292fb21c5b62f91ef45820fb8e69ce86f3929007538879c68981dc","last_reissued_at":"2026-05-18T04:19:52.221333Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:52.221333Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0902.0488","created_at":"2026-05-18T04:19:52.221428+00:00"},{"alias_kind":"arxiv_version","alias_value":"0902.0488v4","created_at":"2026-05-18T04:19:52.221428+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0902.0488","created_at":"2026-05-18T04:19:52.221428+00:00"},{"alias_kind":"pith_short_12","alias_value":"ODGYC6SYFEX3","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_16","alias_value":"ODGYC6SYFEX3EHC3","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_8","alias_value":"ODGYC6SY","created_at":"2026-05-18T12:26:01.383474+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ODGYC6SYFEX3EHC3ML4R55CYED","json":"https://pith.science/pith/ODGYC6SYFEX3EHC3ML4R55CYED.json","graph_json":"https://pith.science/api/pith-number/ODGYC6SYFEX3EHC3ML4R55CYED/graph.json","events_json":"https://pith.science/api/pith-number/ODGYC6SYFEX3EHC3ML4R55CYED/events.json","paper":"https://pith.science/paper/ODGYC6SY"},"agent_actions":{"view_html":"https://pith.science/pith/ODGYC6SYFEX3EHC3ML4R55CYED","download_json":"https://pith.science/pith/ODGYC6SYFEX3EHC3ML4R55CYED.json","view_paper":"https://pith.science/paper/ODGYC6SY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0902.0488&json=true","fetch_graph":"https://pith.science/api/pith-number/ODGYC6SYFEX3EHC3ML4R55CYED/graph.json","fetch_events":"https://pith.science/api/pith-number/ODGYC6SYFEX3EHC3ML4R55CYED/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ODGYC6SYFEX3EHC3ML4R55CYED/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ODGYC6SYFEX3EHC3ML4R55CYED/action/storage_attestation","attest_author":"https://pith.science/pith/ODGYC6SYFEX3EHC3ML4R55CYED/action/author_attestation","sign_citation":"https://pith.science/pith/ODGYC6SYFEX3EHC3ML4R55CYED/action/citation_signature","submit_replication":"https://pith.science/pith/ODGYC6SYFEX3EHC3ML4R55CYED/action/replication_record"}},"created_at":"2026-05-18T04:19:52.221428+00:00","updated_at":"2026-05-18T04:19:52.221428+00:00"}