A Brief Discussion of the Discrepancy Bounds

  • Gunther Leobacher
  • Friedrich Pillichshammer
Part of the Compact Textbooks in Mathematics book series (CTM)


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
$$\displaystyle\begin{array}{rcl} D_{N}^{{\ast}}(\mathcal{P}_{ s,N}) \leq c_{s}\frac{(\log N)^{s-1}} {N},& &{}\end{array}$$
with some c s  > 0 that is independent of N.


Variable Boundary Variable Stars Cardinality Polynomial Tractability Aistleitner 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Gunther Leobacher
    • 1
  • Friedrich Pillichshammer
    • 2
  1. 1.Institute of Financial MathematicsUniversity of LinzLinzAustria
  2. 2.University of LinzLinzAustria

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