On the mean square weighted L 2 discrepancy of randomized digital ( t , m , s )-nets over Z

∆(t1, . . . , ts) = AN ([0, t1)× · · · × [0, ts)) N − t1 · · · ts, where 0 ≤ tj ≤ 1 and AN ([0, t1)×· · ·× [0, ts)) denotes the number of indices n with xn ∈ [0, t1)× · · · × [0, ts). The discrepancy function measures the difference of the portion of points in an axis parallel box containing the origin and the volume of this box. Hence it is a measure of the irregularity of distribution of a point set in [0, 1)s. There are of course other functions which serve a comparable purpose, though this function has drawn a great deal of attention as various connections with applications have been pointed out, notably in numerical integration of functions (see for example [19, 29]). Further, we can use different norms of the discrepancy function, again yielding different quality measures. Amongst those norms especially the L2 norm and the L∞ norm are of considerable interest and have been studied extensively (see for example [19, 29]). In the following we introduce some notation and define the weighted L2 discrepancy of a point set, which will be the focus of this paper. Let D = {1, . . . , s}. For u ⊆ D let γu be a non-negative real number, |u| the cardinality of u and for a vector x ∈ [0, 1)s let xu denote the vector from