{"paper":{"title":"Gap Sequence of Cutting Sequence with Slope $\\theta=[0;\\dot{d}]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Hanxiong Zhang, Yuke Huang","submitted_at":"2014-08-16T08:59:51Z","abstract_excerpt":"In this paper, we consider the factor properties and gap sequence of a special type of cutting sequence with slope $\\theta=[0;\\dot{d}]$, denoted by $F_{d,\\infty}$. Let $\\omega$ be a factor of $F_{d,\\infty}$, then it occurs in the sequence infinitely many times. Let $\\omega_p$ be the $p$-th occurrence of $\\omega$ and $G_p(\\omega)$ be the gap between $\\omega_p$ and $\\omega_{p+1}$. We define the $d$ types of kernel words and envelope words, give two versions of \"uniqueness of kernel decomposition property\". Using them, we prove the gap sequence $\\{G_p(\\omega)\\}_{p\\geq1}$ has exactly two distinct "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3724","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"}