{"paper":{"title":"On the Discrepancy of Jittered Sampling","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Florian Pausinger, Stefan Steinerberger","submitted_at":"2015-10-01T14:24:41Z","abstract_excerpt":"We study the discrepancy of jittered sampling sets: such a set $\\mathcal{P} \\subset [0,1]^d$ is generated for fixed $m \\in \\mathbb{N}$ by partitioning $[0,1]^d$ into $m^d$ axis aligned cubes of equal measure and placing a random point inside each of the $N = m^d$ cubes. We prove that, for $N$ sufficiently large, $$ \\frac{1}{10}\\frac{d}{N^{\\frac{1}{2} + \\frac{1}{2d}}} \\leq \\mathbb{E} D_N^*(\\mathcal{P}) \\leq \\frac{\\sqrt{d} (\\log{N})^{\\frac{1}{2}}}{N^{\\frac{1}{2} + \\frac{1}{2d}}},$$ where the upper bound with an unspecified constant $C_d$ was proven earlier by Beck. Our proof makes crucial use of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00251","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"}