{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:ZVWSBPIJ5DQSGW3IQB2VIK5R7T","short_pith_number":"pith:ZVWSBPIJ","schema_version":"1.0","canonical_sha256":"cd6d20bd09e8e1235b688075542bb1fcee7dfdd1b83a7b14ff26cffab5dc05c0","source":{"kind":"arxiv","id":"1604.01470","version":5},"attestation_state":"computed","paper":{"title":"The topological property of the irregular sets on the lengths of basic intervals in beta-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bing Li, Lixuan Zheng, Min Wu","submitted_at":"2016-04-06T03:24:39Z","abstract_excerpt":"Let $\\beta > 1$ be a real number and $(\\epsilon_1(x, \\beta), \\epsilon_2(x, \\beta), \\ldots)$ be the $\\beta$-expansion of a point $x \\in (0, 1]$. For all $x \\in (0,1]$, let $A(D(x))$ be the set of accumulation points of $\\frac{-\\log_\\beta |I_n(x)|}{n}$ as $n \\rightarrow \\infty$, where $|I_n(x)|$ is the length of the basic interval of order $n$ containing $x \\in (0, 1]$. In this paper, we prove that $A(D(x))$ is always a closed interval for any $x \\in (0,1]$. Furthermore, if $\\lambda(\\beta)>0$, the extremely irregular set containing points $x \\in [0, 1]$ whose upper limit of $\\frac{-\\log_\\beta |I"},"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":"1604.01470","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-04-06T03:24:39Z","cross_cats_sorted":[],"title_canon_sha256":"6d1cb341bdfaa27f708ee7b4db40ba26d063bab9c2664419138ba08326af6072","abstract_canon_sha256":"b8341fbd6b5c8fb2d2ca7b505c11fd7c0945d4f492087e623e2016a86eae631d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:58.069131Z","signature_b64":"yut/0WIjtemP6kBLp5OCr7HtMk6+5AEBtbKPvqkkJRVQL1+FbkWinO0PRDaR93ssAhrbL4/Rfg5+otduJq3UAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd6d20bd09e8e1235b688075542bb1fcee7dfdd1b83a7b14ff26cffab5dc05c0","last_reissued_at":"2026-05-18T00:54:58.068681Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:58.068681Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The topological property of the irregular sets on the lengths of basic intervals in beta-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bing Li, Lixuan Zheng, Min Wu","submitted_at":"2016-04-06T03:24:39Z","abstract_excerpt":"Let $\\beta > 1$ be a real number and $(\\epsilon_1(x, \\beta), \\epsilon_2(x, \\beta), \\ldots)$ be the $\\beta$-expansion of a point $x \\in (0, 1]$. For all $x \\in (0,1]$, let $A(D(x))$ be the set of accumulation points of $\\frac{-\\log_\\beta |I_n(x)|}{n}$ as $n \\rightarrow \\infty$, where $|I_n(x)|$ is the length of the basic interval of order $n$ containing $x \\in (0, 1]$. In this paper, we prove that $A(D(x))$ is always a closed interval for any $x \\in (0,1]$. Furthermore, if $\\lambda(\\beta)>0$, the extremely irregular set containing points $x \\in [0, 1]$ whose upper limit of $\\frac{-\\log_\\beta |I"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.01470","kind":"arxiv","version":5},"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":"1604.01470","created_at":"2026-05-18T00:54:58.068751+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.01470v5","created_at":"2026-05-18T00:54:58.068751+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.01470","created_at":"2026-05-18T00:54:58.068751+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZVWSBPIJ5DQS","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZVWSBPIJ5DQSGW3I","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZVWSBPIJ","created_at":"2026-05-18T12:30:55.937587+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/ZVWSBPIJ5DQSGW3IQB2VIK5R7T","json":"https://pith.science/pith/ZVWSBPIJ5DQSGW3IQB2VIK5R7T.json","graph_json":"https://pith.science/api/pith-number/ZVWSBPIJ5DQSGW3IQB2VIK5R7T/graph.json","events_json":"https://pith.science/api/pith-number/ZVWSBPIJ5DQSGW3IQB2VIK5R7T/events.json","paper":"https://pith.science/paper/ZVWSBPIJ"},"agent_actions":{"view_html":"https://pith.science/pith/ZVWSBPIJ5DQSGW3IQB2VIK5R7T","download_json":"https://pith.science/pith/ZVWSBPIJ5DQSGW3IQB2VIK5R7T.json","view_paper":"https://pith.science/paper/ZVWSBPIJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.01470&json=true","fetch_graph":"https://pith.science/api/pith-number/ZVWSBPIJ5DQSGW3IQB2VIK5R7T/graph.json","fetch_events":"https://pith.science/api/pith-number/ZVWSBPIJ5DQSGW3IQB2VIK5R7T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZVWSBPIJ5DQSGW3IQB2VIK5R7T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZVWSBPIJ5DQSGW3IQB2VIK5R7T/action/storage_attestation","attest_author":"https://pith.science/pith/ZVWSBPIJ5DQSGW3IQB2VIK5R7T/action/author_attestation","sign_citation":"https://pith.science/pith/ZVWSBPIJ5DQSGW3IQB2VIK5R7T/action/citation_signature","submit_replication":"https://pith.science/pith/ZVWSBPIJ5DQSGW3IQB2VIK5R7T/action/replication_record"}},"created_at":"2026-05-18T00:54:58.068751+00:00","updated_at":"2026-05-18T00:54:58.068751+00:00"}