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The fractal dimensions of the spectrum have been determined by Fan, Liu and Wen (Erg. Th. Dyn. Sys.,2011) when $\\{a_n\\}_{n\\ge1}$ is bounded. The present paper will treat the most difficult case, i.e, $\\{a_n\\}_{n\\ge1}$ is unbounded. We prove that for $V\\ge24$, $$ \\dim_H\\ \\Sigma_{\\alpha,V}=s_*(V)\\ \\ \\ \\text{and}\\ \\ \\ \\bar{\\dim}_B\\ \\Sigma_{\\alpha,V}=s"},"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":"1310.1473","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-10-05T13:14:56Z","cross_cats_sorted":["math.DS","math.MP"],"title_canon_sha256":"65e0f553c11465b79379f5312c1962aa0d2edf4d452d60032d4c0a615c253b0f","abstract_canon_sha256":"b4b5b3fcff84d7e485e6a391b668e285db9be74966aa3a3c1daa9b4c3f3fc693"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:43.868564Z","signature_b64":"3CMH3ND2kn+JM5/TP4JIYZojUZqPBY8jMKFcDo1WIpYy0bED2wjkDMJNA2GcJBDZJO41gw+hQEICJ32Tro3+DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e06c408a1880de48428cc5eef7e54197446b11d50fd81366d9b39ccb46eb99e","last_reissued_at":"2026-05-18T01:22:43.868005Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:43.868005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The fractal dimensions of the spectrum of Sturm Hamiltonian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.MP"],"primary_cat":"math-ph","authors_text":"Qinghui Liu, Yanhui Qu, Zhiying Wen","submitted_at":"2013-10-05T13:14:56Z","abstract_excerpt":"Let $\\alpha\\in(0,1)$ be irrational and $[0;a_1,a_2,\\cdots]$ be the continued fraction expansion of $\\alpha$. Let $H_{\\alpha,V}$ be the Sturm Hamiltonian with frequency $\\alpha$ and coupling $V$, $\\Sigma_{\\alpha,V}$ be the spectrum of $H_{\\alpha,V}$. The fractal dimensions of the spectrum have been determined by Fan, Liu and Wen (Erg. Th. Dyn. Sys.,2011) when $\\{a_n\\}_{n\\ge1}$ is bounded. The present paper will treat the most difficult case, i.e, $\\{a_n\\}_{n\\ge1}$ is unbounded. We prove that for $V\\ge24$, $$ \\dim_H\\ \\Sigma_{\\alpha,V}=s_*(V)\\ \\ \\ \\text{and}\\ \\ \\ \\bar{\\dim}_B\\ \\Sigma_{\\alpha,V}=s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.1473","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":"1310.1473","created_at":"2026-05-18T01:22:43.868085+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.1473v1","created_at":"2026-05-18T01:22:43.868085+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.1473","created_at":"2026-05-18T01:22:43.868085+00:00"},{"alias_kind":"pith_short_12","alias_value":"HYDMICFBRAG6","created_at":"2026-05-18T12:27:46.883200+00:00"},{"alias_kind":"pith_short_16","alias_value":"HYDMICFBRAG6JBBI","created_at":"2026-05-18T12:27:46.883200+00:00"},{"alias_kind":"pith_short_8","alias_value":"HYDMICFB","created_at":"2026-05-18T12:27:46.883200+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/HYDMICFBRAG6JBBIZRPO67SUDF","json":"https://pith.science/pith/HYDMICFBRAG6JBBIZRPO67SUDF.json","graph_json":"https://pith.science/api/pith-number/HYDMICFBRAG6JBBIZRPO67SUDF/graph.json","events_json":"https://pith.science/api/pith-number/HYDMICFBRAG6JBBIZRPO67SUDF/events.json","paper":"https://pith.science/paper/HYDMICFB"},"agent_actions":{"view_html":"https://pith.science/pith/HYDMICFBRAG6JBBIZRPO67SUDF","download_json":"https://pith.science/pith/HYDMICFBRAG6JBBIZRPO67SUDF.json","view_paper":"https://pith.science/paper/HYDMICFB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.1473&json=true","fetch_graph":"https://pith.science/api/pith-number/HYDMICFBRAG6JBBIZRPO67SUDF/graph.json","fetch_events":"https://pith.science/api/pith-number/HYDMICFBRAG6JBBIZRPO67SUDF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HYDMICFBRAG6JBBIZRPO67SUDF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HYDMICFBRAG6JBBIZRPO67SUDF/action/storage_attestation","attest_author":"https://pith.science/pith/HYDMICFBRAG6JBBIZRPO67SUDF/action/author_attestation","sign_citation":"https://pith.science/pith/HYDMICFBRAG6JBBIZRPO67SUDF/action/citation_signature","submit_replication":"https://pith.science/pith/HYDMICFBRAG6JBBIZRPO67SUDF/action/replication_record"}},"created_at":"2026-05-18T01:22:43.868085+00:00","updated_at":"2026-05-18T01:22:43.868085+00:00"}