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Denote $$E^\\varphi_{\\alpha,\\beta}=\\left\\{x\\in\\Sigma: \\liminf_{n\\to\\infty}\\frac{\\log R_n(x)}{\\varphi(n)}=\\alpha,\\ \\limsup_{n\\to\\infty}\\frac{\\log R_n(x)}{\\varphi(n)}=\\beta\\right\\},$$ where $\\varphi: \\mathbb{N}\\to \\mathbb{R}^+$ is a monotonically increasing function and $0\\leq\\alpha\\leq\\beta\\leq +\\infty$. We show that the Hausdorff dimension of the set $E^\\varphi_{\\alpha,\\beta}$ admits a dichotomy: it is either zero or one depending on $"},"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":"1510.00495","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-10-02T05:18:46Z","cross_cats_sorted":[],"title_canon_sha256":"027e282991d7456359bd69e0a0343ab3f8cdb4d1605efd493b69ae041bd62ec1","abstract_canon_sha256":"1d90de867cc4840888f28605cc59d03499811ff635757f3be5a6d5a90a15a518"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:52.444217Z","signature_b64":"+LyAsbTvrRyuCuRd4WX4S1fTuz+oa59REUS5ncUvSU9pokCBtJd3C+Koc5aeDH0I96THWSroeJiCsGsfN/Q8Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2a627946bbb013f2549bb624df850e3e590f1efbbffedf46082dbf5442a1337","last_reissued_at":"2026-05-18T01:17:52.443749Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:52.443749Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zero-one law of Hausdorff dimensions of the recurrent sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bing Li, Dong Han Kim","submitted_at":"2015-10-02T05:18:46Z","abstract_excerpt":"Let $(\\Sigma, \\sigma)$ be the one-sided shift space with $m$ symbols and $R_n(x)$ be the first return time of $x\\in\\Sigma$ to the $n$-th cylinder containing $x$. Denote $$E^\\varphi_{\\alpha,\\beta}=\\left\\{x\\in\\Sigma: \\liminf_{n\\to\\infty}\\frac{\\log R_n(x)}{\\varphi(n)}=\\alpha,\\ \\limsup_{n\\to\\infty}\\frac{\\log R_n(x)}{\\varphi(n)}=\\beta\\right\\},$$ where $\\varphi: \\mathbb{N}\\to \\mathbb{R}^+$ is a monotonically increasing function and $0\\leq\\alpha\\leq\\beta\\leq +\\infty$. We show that the Hausdorff dimension of the set $E^\\varphi_{\\alpha,\\beta}$ admits a dichotomy: it is either zero or one depending on $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00495","kind":"arxiv","version":2},"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":"1510.00495","created_at":"2026-05-18T01:17:52.443818+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.00495v2","created_at":"2026-05-18T01:17:52.443818+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.00495","created_at":"2026-05-18T01:17:52.443818+00:00"},{"alias_kind":"pith_short_12","alias_value":"2KTCPFDLXMAT","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"2KTCPFDLXMAT6JKJ","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"2KTCPFDL","created_at":"2026-05-18T12:29:02.477457+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/2KTCPFDLXMAT6JKJXNRE36CQ4P","json":"https://pith.science/pith/2KTCPFDLXMAT6JKJXNRE36CQ4P.json","graph_json":"https://pith.science/api/pith-number/2KTCPFDLXMAT6JKJXNRE36CQ4P/graph.json","events_json":"https://pith.science/api/pith-number/2KTCPFDLXMAT6JKJXNRE36CQ4P/events.json","paper":"https://pith.science/paper/2KTCPFDL"},"agent_actions":{"view_html":"https://pith.science/pith/2KTCPFDLXMAT6JKJXNRE36CQ4P","download_json":"https://pith.science/pith/2KTCPFDLXMAT6JKJXNRE36CQ4P.json","view_paper":"https://pith.science/paper/2KTCPFDL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.00495&json=true","fetch_graph":"https://pith.science/api/pith-number/2KTCPFDLXMAT6JKJXNRE36CQ4P/graph.json","fetch_events":"https://pith.science/api/pith-number/2KTCPFDLXMAT6JKJXNRE36CQ4P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2KTCPFDLXMAT6JKJXNRE36CQ4P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2KTCPFDLXMAT6JKJXNRE36CQ4P/action/storage_attestation","attest_author":"https://pith.science/pith/2KTCPFDLXMAT6JKJXNRE36CQ4P/action/author_attestation","sign_citation":"https://pith.science/pith/2KTCPFDLXMAT6JKJXNRE36CQ4P/action/citation_signature","submit_replication":"https://pith.science/pith/2KTCPFDLXMAT6JKJXNRE36CQ4P/action/replication_record"}},"created_at":"2026-05-18T01:17:52.443818+00:00","updated_at":"2026-05-18T01:17:52.443818+00:00"}