Monatshefte für Mathematik

, Volume 145, Issue 4, pp 285–299 | Cite as

Dyadic Diaphony of Digital Nets Over ℤ2

  • Josef Dick
  • Friedrich Pillichshammer
Article

Abstract.

The dyadic diaphony, introduced by Hellekalek and Leeb, is a quantitative measure for the irregularity of distribution of point sets in the unit-cube. In this paper we study the dyadic diaphony of digital nets over ℤ2. We prove an upper bound for the dyadic diaphony of nets and show that the convergence order is best possible. This follows from a relation between the dyadic diaphony and the \({\cal L}_2\) discrepancy. In order to investigate the case where the number of points is small compared to the dimension we introduce the limiting dyadic diaphony, which is defined as the limiting case where the dimension tends to infinity. We obtain a tight upper and lower bound and we compare this result with the limiting dyadic diaphony of a random sample.

2000 Mathematics Subject Classifications: 11K06, 11K38 
Key words: Dyadic diaphony, digital nets, \({\cal L}_2\) discrepancy 

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Copyright information

© Springer-Verlag/Wien 2005

Authors and Affiliations

  • Josef Dick
    • 1
  • Friedrich Pillichshammer
    • 2
  1. 1.University of New South WalesSydneyAustralia
  2. 2.Universität LinzAustria

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