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The limit superior (respectively limit inferior) of $\\frac{r_n(x,\\beta)}{n}$ is linked to the classical Diophantine approximation (respectively uniform Diophantine approximation). We obtain the Hausdorff dimension of the level set $$E_{a,b}=\\left\\{x \\in [0,1]: \\liminf_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{n}=a,\\ \\limsup_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{n}=b\\right\\}\\ (0\\leq a\\leq b\\leq1).$$ Furthermore, we show that the extremely"},"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":"1805.04744","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-05-12T15:58:49Z","cross_cats_sorted":[],"title_canon_sha256":"ed56cdbc0e2d2d4d0ae8f7076295d73f156f35eef5b55e7601c8fe73d444369c","abstract_canon_sha256":"543bc5ec154509046e7ba34935cf6209914e365abf18a932a96702c9b407c8b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:44.570660Z","signature_b64":"nu1HgBb0aRb9eIgXq8e0I3VSD8kxlSW8m/pw60Xzjavh8HU8Pc/ismLzDoaPu59p5vnMXKNmaiJk21IhgGwnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88133eea87735592b560f520ba901b044763ced0206f067fc5f552a48b85e28a","last_reissued_at":"2026-05-18T00:10:44.570269Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:44.570269Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Diophantine approximation and run-length function on \\beta-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Lixuan Zheng","submitted_at":"2018-05-12T15:58:49Z","abstract_excerpt":"For any $\\beta > 1$, denoted by $r_n(x,\\beta)$ the maximal length of consecutive zeros amongst the first $n$ digits of the $\\beta$-expansion of $x\\in[0,1]$. The limit superior (respectively limit inferior) of $\\frac{r_n(x,\\beta)}{n}$ is linked to the classical Diophantine approximation (respectively uniform Diophantine approximation). We obtain the Hausdorff dimension of the level set $$E_{a,b}=\\left\\{x \\in [0,1]: \\liminf_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{n}=a,\\ \\limsup_{n\\rightarrow \\infty}\\frac{r_n(x,\\beta)}{n}=b\\right\\}\\ (0\\leq a\\leq b\\leq1).$$ Furthermore, we show that the extremely"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.04744","kind":"arxiv","version":3},"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":"1805.04744","created_at":"2026-05-18T00:10:44.570325+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.04744v3","created_at":"2026-05-18T00:10:44.570325+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.04744","created_at":"2026-05-18T00:10:44.570325+00:00"},{"alias_kind":"pith_short_12","alias_value":"RAJT52UHONKZ","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_16","alias_value":"RAJT52UHONKZFNLA","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_8","alias_value":"RAJT52UH","created_at":"2026-05-18T12:32:50.500415+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/RAJT52UHONKZFNLA6UQLVEA3AR","json":"https://pith.science/pith/RAJT52UHONKZFNLA6UQLVEA3AR.json","graph_json":"https://pith.science/api/pith-number/RAJT52UHONKZFNLA6UQLVEA3AR/graph.json","events_json":"https://pith.science/api/pith-number/RAJT52UHONKZFNLA6UQLVEA3AR/events.json","paper":"https://pith.science/paper/RAJT52UH"},"agent_actions":{"view_html":"https://pith.science/pith/RAJT52UHONKZFNLA6UQLVEA3AR","download_json":"https://pith.science/pith/RAJT52UHONKZFNLA6UQLVEA3AR.json","view_paper":"https://pith.science/paper/RAJT52UH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.04744&json=true","fetch_graph":"https://pith.science/api/pith-number/RAJT52UHONKZFNLA6UQLVEA3AR/graph.json","fetch_events":"https://pith.science/api/pith-number/RAJT52UHONKZFNLA6UQLVEA3AR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RAJT52UHONKZFNLA6UQLVEA3AR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RAJT52UHONKZFNLA6UQLVEA3AR/action/storage_attestation","attest_author":"https://pith.science/pith/RAJT52UHONKZFNLA6UQLVEA3AR/action/author_attestation","sign_citation":"https://pith.science/pith/RAJT52UHONKZFNLA6UQLVEA3AR/action/citation_signature","submit_replication":"https://pith.science/pith/RAJT52UHONKZFNLA6UQLVEA3AR/action/replication_record"}},"created_at":"2026-05-18T00:10:44.570325+00:00","updated_at":"2026-05-18T00:10:44.570325+00:00"}