{"paper":{"title":"Almost Beatty Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"A.J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie","submitted_at":"2018-09-23T22:37:30Z","abstract_excerpt":"Given $0<\\alpha<1$, the Beatty sequence of density $\\alpha$ is the sequence $B_{\\alpha}=(\\lfloor n/\\alpha\\rfloor)_{n\\in\\mathbb{N}}$. Beatty's theorem states that if $\\alpha,\\beta$ are irrational numbers with $\\alpha+\\beta=1$, then the Beatty sequences $B_{\\alpha}$ and $B_{\\beta}$ partition the positive integers, that is, each positive integer belongs to exactly one of these two sequences. On the other hand, Uspensky showed that this result breaks down completely for partitions into three (or more) sequences: There does not exist a single triple $(\\alpha,\\beta,\\gamma)$ such that the Beatty sequ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.08690","kind":"arxiv","version":3},"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"}