{"paper":{"title":"Periodic representations in algebraic bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tom\\'a\\v{s} V\\'avra, V\\'it\\v{e}zslav Kala","submitted_at":"2017-09-13T05:36:29Z","abstract_excerpt":"We study periodic representations in number systems with an algebraic base $\\beta$ (not a rational integer). We show that if $\\beta$ has no Galois conjugate on the unit circle, then there exists a finite integer alphabet $\\mathcal A$ such that every element of $\\mathbb Q(\\beta)$ admits an eventually periodic representation with base $\\beta$ and digits in $\\mathcal A$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04143","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"}