{"paper":{"title":"Gaps in N-expansions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"C. Kraaikamp, H. Nakada, J. de Jonge","submitted_at":"2021-07-14T14:13:12Z","abstract_excerpt":"For a natural number $N\\geq 2$ and a real $\\alpha$ such that $0 < \\alpha \\leq \\sqrt{N}-1$, we define $I_\\alpha:=[\\alpha,\\alpha+1]$ and $I_\\alpha^-:=[\\alpha,\\alpha+1)$ and investigate the continued fraction map $T_\\alpha:I_\\alpha \\to I_\\alpha^-$, which is defined as $T_\\alpha(x):= N/x-d(x),$ where $d(x):=\\left \\lfloor N/x -\\alpha\\right \\rfloor$. For all natural $N \\geq 7$, for certain values of $\\alpha$, open intervals $(a,b) \\subset I_\\alpha$ exist such that for almost every $x \\in I_{\\alpha}$ there is an natural number $n_0$ for which $T_\\alpha^n(x) \\notin (a,b)$ for all $n\\geq n_0$. These \\e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.06722","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2107.06722/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}