{"paper":{"title":"Tusn\\'ady's problem, the transference principle, and non-uniform QMC sampling","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.NA","math.PR"],"primary_cat":"math.CO","authors_text":"Aleksandar Nikolov, Christoph Aistleitner, Dmitriy Bilyk","submitted_at":"2017-03-17T17:43:59Z","abstract_excerpt":"It is well-known that for every $N \\geq 1$ and $d \\geq 1$ there exist point sets $x_1, \\dots, x_N \\in [0,1]^d$ whose discrepancy with respect to the Lebesgue measure is of order at most $(\\log N)^{d-1} N^{-1}$. In a more general setting, the first author proved together with Josef Dick that for any normalized measure $\\mu$ on $[0,1]^d$ there exist points $x_1, \\dots, x_N$ whose discrepancy with respect to $\\mu$ is of order at most $(\\log N)^{(3d+1)/2} N^{-1}$. The proof used methods from combinatorial mathematics, and in particular a result of Banaszczyk on balancings of vectors. In the presen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.06127","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"}