Tusn\'ady's problem, the transference principle, and non-uniform QMC sampling

TL;DR

This paper improves bounds on the discrepancy of point sets in high-dimensional spaces for arbitrary measures, using the transference principle and recent coloring discrepancy results, approaching the optimal bounds known for Lebesgue measure.

Contribution

It introduces a new approach combining the transference principle with coloring discrepancy results to nearly match Lebesgue measure discrepancy bounds for arbitrary measures.

Findings

Discrepancy bounds of order (log N)^{d-1/2} N^{-1} for any measure.

Extension of discrepancy results from Lebesgue to arbitrary measures.

Use of transference principle and coloring discrepancy techniques.

Abstract

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 $\mu$ 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 $(\log N)^{d-\frac12} N^{-1}$, which is almost as good as the discrepancy bound in the case of the Lebesgue measure.