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The exceptional set $$E_{\\max}^{\\varphi}=\\left\\{x \\in [0,1]:\\liminf_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{\\varphi(n)}=0, \\limsup_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{\\varphi(n)}=+\\infty\\right\\}$$ is investigated, where $\\varphi: \\mathbb{N} \\rightarrow \\mathbb{R}^+$ is a monotonically increasing function with $\\lim\\limits_{n\\rightarrow \\infty }\\varphi(n)=+\\infty$. We prove that the set $E_{\\max}^{\\varphi}$ is either empt"},"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":"1704.01317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-04-05T09:19:38Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"046abdf3fad1c9ce7e4b2defad44e2e6da92e8b30349591d9b798458c4b6ac0e","abstract_canon_sha256":"7f418346681bfabf03688fefc6e32e11a7db9911ea64b4d48f8d03b48780233d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:56.085231Z","signature_b64":"wo2y9A9iUO72cVU7w/2AqwSHO//sIpzl6gTGnModZVKi1Fq/NwuP981f+f3Oqky45cuxs1qNNg0bdNpRLs3xDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"afde5864b65327d433071c9e1cc3f28f09189043870ef93c91225235ff92ef6d","last_reissued_at":"2026-05-18T00:28:56.084784Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:56.084784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The exceptional sets on the run-length function of beta-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Bing Li, Lixuan Zheng, Min Wu","submitted_at":"2017-04-05T09:19:38Z","abstract_excerpt":"Let $\\beta > 1$ and the run-length function $r_n(x,\\beta)$ be the maximal length of consecutive zeros amongst the first n digits in the $\\beta$-expansion of $x\\in[0,1]$. The exceptional set $$E_{\\max}^{\\varphi}=\\left\\{x \\in [0,1]:\\liminf_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{\\varphi(n)}=0, \\limsup_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{\\varphi(n)}=+\\infty\\right\\}$$ is investigated, where $\\varphi: \\mathbb{N} \\rightarrow \\mathbb{R}^+$ is a monotonically increasing function with $\\lim\\limits_{n\\rightarrow \\infty }\\varphi(n)=+\\infty$. We prove that the set $E_{\\max}^{\\varphi}$ is either empt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.01317","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1704.01317","created_at":"2026-05-18T00:28:56.084857+00:00"},{"alias_kind":"arxiv_version","alias_value":"1704.01317v1","created_at":"2026-05-18T00:28:56.084857+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.01317","created_at":"2026-05-18T00:28:56.084857+00:00"},{"alias_kind":"pith_short_12","alias_value":"V7PFQZFWKMT5","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_16","alias_value":"V7PFQZFWKMT5IMYH","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_8","alias_value":"V7PFQZFW","created_at":"2026-05-18T12:31:49.984773+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/V7PFQZFWKMT5IMYHDSPBZQ7SR4","json":"https://pith.science/pith/V7PFQZFWKMT5IMYHDSPBZQ7SR4.json","graph_json":"https://pith.science/api/pith-number/V7PFQZFWKMT5IMYHDSPBZQ7SR4/graph.json","events_json":"https://pith.science/api/pith-number/V7PFQZFWKMT5IMYHDSPBZQ7SR4/events.json","paper":"https://pith.science/paper/V7PFQZFW"},"agent_actions":{"view_html":"https://pith.science/pith/V7PFQZFWKMT5IMYHDSPBZQ7SR4","download_json":"https://pith.science/pith/V7PFQZFWKMT5IMYHDSPBZQ7SR4.json","view_paper":"https://pith.science/paper/V7PFQZFW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1704.01317&json=true","fetch_graph":"https://pith.science/api/pith-number/V7PFQZFWKMT5IMYHDSPBZQ7SR4/graph.json","fetch_events":"https://pith.science/api/pith-number/V7PFQZFWKMT5IMYHDSPBZQ7SR4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V7PFQZFWKMT5IMYHDSPBZQ7SR4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V7PFQZFWKMT5IMYHDSPBZQ7SR4/action/storage_attestation","attest_author":"https://pith.science/pith/V7PFQZFWKMT5IMYHDSPBZQ7SR4/action/author_attestation","sign_citation":"https://pith.science/pith/V7PFQZFWKMT5IMYHDSPBZQ7SR4/action/citation_signature","submit_replication":"https://pith.science/pith/V7PFQZFWKMT5IMYHDSPBZQ7SR4/action/replication_record"}},"created_at":"2026-05-18T00:28:56.084857+00:00","updated_at":"2026-05-18T00:28:56.084857+00:00"}