{"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"}