Tusnády's problem, the transference principle, and non-uniform QMC sampling

It is well-known that for every N ≥ 1 and d ≥ 1 there exist point sets x1, . . . , xN ∈ [0, 1] whose discrepancy with respect to the Lebesgue measure is of order at most (logN)d−1N−1. In a more general setting, the first author proved together with Josef Dick that for any normalized measure μ on [0, 1] there exist points x1, . . . , xN whose discrepancy with respect to μ is of order at most (logN)(3d+1)/2N−1. The proof used methods from combinatorial mathematics, and in particular a result of Banaszczyk on balancings of vectors. In the present note we use a version of the so-called transference principle together with recent results on the discrepancy of red-blue colorings to show that for any μ there even exist points having discrepancy of order at most (logN)d− 1 2N−1, which is almost as good as the discrepancy bound in the case of the Lebesgue measure.