Low-discrepancy point sets for non-uniform measures
We prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure μ on the d-dimensional unit cube. We improve a theorem of Beck, by showing that for any d ≥ 1, N ≥ 1, and any non-negative, normalized Borel measure μ on there exists a point set whose star-discrepancy with respect to μ is of order . For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy...