Tusnády’s Problem, the Transference Principle, and Non-uniform QMC Sampling

Abstract

It is well-known that for every $N \ge 1$ and $d \ge 1$ 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 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.