Tusnády's problem, the transference principle, and non-uniform QMC sampling
This provides a near-optimal solution to Tusnády's problem for general measures, improving upon previous results for quasi-Monte Carlo sampling.
The authors improve the discrepancy bound for arbitrary normalized measures on the unit cube from O((log N)^{(3d+1)/2} N^{-1}) to O((log N)^{d-1/2} N^{-1}), nearly matching the optimal bound for the Lebesgue measure.
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 $μ$ on $[0,1]^d$ there exist points $x_1, \dots, x_N$ whose discrepancy with respect to $μ$ 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 $μ$ there even exist points having discrepancy of order at most $(\log N)^{d-\frac12} N^{-1}$, which is almost as good as the discrepancy bound in the case of the Lebesgue measure.