{"paper":{"title":"A characterization of eventually periodicity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"cs.CC","authors_text":"Dong Han Kim, Teturo Kamae","submitted_at":"2014-04-17T02:35:59Z","abstract_excerpt":"In this article, we show that the Kamae-Xue complexity function for an infinite sequence classifies eventual periodicity completely. We prove that an infinite binary word $x_1x_2 \\cdots $ is eventually periodic if and only if $\\Sigma(x_1x_2\\cdots x_n)/n^3$ has a positive limit, where $\\Sigma(x_1x_2\\cdots x_n)$ is the sum of the squares of all the numbers of appearance of finite words in $x_1 x_2 \\cdots x_n$, which was introduced by Kamae-Xue as a criterion of randomness in the sense that $x_1x_2\\cdots x_n$ is more random if $\\Sigma(x_1x_2\\cdots x_n)$ is smaller. In fact, it is known that the l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.4416","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"}