{"paper":{"title":"A Discrete Fourier Kernel and Fraenkel's Tiling Conjecture","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Kevin O'Bryant, Ron Graham","submitted_at":"2004-07-17T01:00:39Z","abstract_excerpt":"The set B_{p,r}^q:=\\{\\floor{nq/p+r} \\colon n\\in Z \\} with integers p, q, r) is a Beatty set with density p/q. We derive a formula for the Fourier transform \\hat{B_{p,r}^q}(j):=\\sum_{n=1}^p e^{-2 \\pi i j \\floor{nq/p+r} / q}. A. S. Fraenkel conjectured that there is essentially one way to partition the integers into m>2 Beatty sets with distinct densities. We conjecture a generalization of this, and use Fourier methods to prove several special cases of our generalized conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0407306","kind":"arxiv","version":2},"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"}