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

It is well-known that for every (N ge 1) and (d ge 1) there exist point sets (x_1, dots , x_N [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 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 (mu ) there even exist points having discrepancy of order at most ((log N)^{d-frac{1}{2}} N^{-1}), which is almost as good as the discrepancy bound in the case of the Lebesgue measure.