{"paper":{"title":"An Upper Bound of the Minimal Dispersion via Delta Covers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"cs.CG","authors_text":"Daniel Rudolf","submitted_at":"2017-01-05T12:00:37Z","abstract_excerpt":"For a point set of $n$ elements in the $d$-dimensional unit cube and a class of test sets we are interested in the largest volume of a test set which does not contain any point. For all natural numbers $n$, $d$ and under the assumption of a $delta$-cover with cardinality $\\vert \\Gamma_\\delta \\vert$ we prove that there is a point set, such that the largest volume of such a test set without any point is bounded by $\\frac{\\log \\vert \\Gamma_\\delta \\vert}{n} + \\delta$. For axis-parallel boxes on the unit cube this leads to a volume of at most $\\frac{4d}{n}\\log(\\frac{9n}{d})$ and on the torus to $\\f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.06430","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"}