{"paper":{"title":"On the regularity of $\\{\\lfloor\\log_b(\\alpha n+\\beta)\\rfloor\\}_{n\\geq0}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"Jiemeng Zhang, Yingjun Guo, Zhixiong Wen","submitted_at":"2015-08-26T12:32:00Z","abstract_excerpt":"Let $\\alpha,\\beta$ be real numbers and $b\\geq2$ be an integer. Allouche and Shallit showed that the sequence $\\{\\lfloor\\alpha n+\\beta\\rfloor\\}_{n\\geq0}$ is $b$-regular if and only if $\\alpha$ is rational. In this paper, using a base-independent regular language, we prove a similar result that the sequence $\\{\\lfloor\\log_b(\\alpha n+\\beta)\\rfloor\\}_{n\\geq0}$ is $b$-regular if and only if $\\alpha$ is rational. In particular, when $\\alpha=\\sqrt{2},\\beta=0$ and $b=2$, we answer the question of Allouche and Shallit that the sequence $\\{\\lfloor\\frac{1}{2}+\\log_2n\\rfloor\\}_{n\\geq0}$ is not $2$-regular"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06466","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"}