A Brief Discussion of the Discrepancy Bounds

Abstract

In many applications the dimension s can be rather large. In this case, the asymptotically almost optimal bounds on the discrepancy which we obtained, e.g., for the Hammersley point set or for (t, m, s)-nets soon become useless for a modest number N of points. For example, assume that for every $s,N \in \mathbb{N}$ we have a point set $\mathcal{P}_{s,N}$ in the s-dimensional unit cube of cardinality N with star discrepancy of at most

$$\displaystyle\begin{array}{rcl} D_{N}^{{\ast}}(\mathcal{P}_{ s,N}) \leq c_{s}\frac{(\log N)^{s-1}} {N},& &{}\end{array}$$

(6.1)

with some c _s  > 0 that is independent of N.