On the mean square weighted L₂discrepancy of randomized digital (t,m,s)-nets over Z₂

Abstract

On the mean square weighted L 2 discrepancy of randomized digital (t, m, s)-nets over Z 2 by Josef Dick (Sydney) and Friedrich Pillichshammer (Linz)1. Introduction. We study distribution properties of point sets in the s-dimensional unit cube [0, 1) s . There are various measures for the equidistribution of such point sets (see for example [7,10,11,16,19]). The one we consider here is based on the following function. For a set P N = {x 0 , . . . , x N −1 } of points in the s-dimensional unit cube [0, 1) s the discrepancy function is defined asThe 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 L 2 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 L 2 discrepancy of a point set, which will be the focus of this paper.Let D = {1, . . . , s}. For u ⊆ D let γ u be a non-negative real number, |u| the cardinality of u and for a vector x ∈ [0, 1) s let x u denote the vector from 2000 Mathematics Subject Classification: 11K38, 11K06.

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