{"paper":{"title":"Range-Renewal Structure in Continued Fractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Jian-Sheng Xie, Jun Wu","submitted_at":"2013-05-09T13:39:20Z","abstract_excerpt":"Let $\\omega=[a_1, a_2, \\cdots]$ be the infinite expansion of continued fraction for an irrational number $\\omega \\in (0,1)$; let $R_n (\\omega)$ (resp. $R_{n, \\, k} (\\omega)$, $R_{n, \\, k+} (\\omega)$) be the number of distinct partial quotients each of which appears at least once (resp. exactly $k$ times, at least $k$ times) in the sequence $a_1, \\cdots, a_n$. In this paper it is proved that for Lebesgue almost all $\\omega \\in (0,1)$ and all $k \\geq 1$, $$ \\displaystyle \\lim_{n \\to \\infty} \\frac{R_n (\\omega)}{\\sqrt{n}}=\\sqrt{\\frac{\\pi}{\\log 2}}, \\quad \\lim_{n \\to \\infty} \\frac{R_{n, \\, k} (\\ome"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2088","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"}