Monatshefte für Mathematik

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

Dyadic Diaphony of Digital Nets Over ℤ2

  • Josef Dick
  • Friedrich Pillichshammer


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Chrestenson, HE 1955A class of generalized Walsh functions.Pacific J Math51731Google Scholar
  2. Dick J, Kuo F, Pillichshammer F, Sloan IH (2004) Construction algorithms for polynomial lattice rules for multivariate integration. Math Comp (to appear)Google Scholar
  3. Dick J, Pillichshammer F (2003) Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces. J Complexity (to appear)Google Scholar
  4. Dick J, Pillichshammer F (2003) On the mean square weighted L2 discrepancy of randomized digital (t,m,s)-nets over ℤ2. Acta Arithm (to appear)Google Scholar
  5. Dick J, Pillichshammer F (2004) Diaphony, discrepancy, spectral test and worst-case error (submitted)Google Scholar
  6. Dick, J, Sloan, IH, Wang, X, Woźniakowksi, H 2004Liberating the weights.J Complexity20593623CrossRefGoogle Scholar
  7. Drmota M, Tichy RF (1997) Sequences, discrepancies and applications. Lect Notes Math 1651. Berlin Heidelberg New York: SpringerGoogle Scholar
  8. Grozdanov, V, Stoilova, S 2001On the theory of b-adic diaphony.C R Acad Bulg Sci543134Google Scholar
  9. Grozdanov, V, Stoilova, S 2004The general diaphony.C R Acad Bulg Sci571318MathSciNetGoogle Scholar
  10. Grozdanov, V, Nikolova, E, Stoilova, S 2003Generalized b-adic diaphony.C R Acad Bulg Sci562330Google Scholar
  11. Hellekalek, P 2002Digital (t,m,s)-nets and the spectral test.Acta Arith105197204Google Scholar
  12. Hellekalek, P, Leeb, H 1997Dyadic diaphony.Acta Arith80187196Google Scholar
  13. Hickernell, FJ 1998A generalized discrepancy and quadrature error bound.Math Comp67299322CrossRefGoogle Scholar
  14. Kuipers L, Niederreiter H (1974) Uniform Distribution of Sequences. New York: WileyGoogle Scholar
  15. Larcher G (1998) Digital point sets: analysis and application. In: Hellekalek P, Larcher G (eds) Random and Quasi-Random Point Sets. Lect Notes Statistics 138: 167–222. Berlin Heidelberg New York: SpringerGoogle Scholar
  16. Larcher, G, Niederreiter, H, Schmid, WCh 1996Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration.Monatsh Math121231253CrossRefGoogle Scholar
  17. Niederdrenk K (1982) Die endliche Fourier- und Walshtransformation mit einer Einführung in die Bildverarbeitung. Braunschweig: ViewegGoogle Scholar
  18. Matoušek J (1999) Geometric Discrepancy. Berlin Heidelberg New York: SpringerGoogle Scholar
  19. Niederreiter H (1992) Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Series in Appl Math 63 Philadelphia: SIAMGoogle Scholar
  20. Niederreiter H, Xing CP (2001) Rational Points on Curves over Finite Fields. London Math Soc Lect Notes Series 285. Cambridge: Univ PressGoogle Scholar
  21. Pirsic G (1995) Schnell konvergierende Walshreihen über Gruppen. Master’s Thesis, University of Salzburg (available at Scholar
  22. Rivlin TJ, Saff EB (2000) Joseph L. Walsh Selected Papers. Berlin Heidelberg New York: SpringerGoogle Scholar
  23. Roth, KF 1959On irregularities of distribution.Mathematika17379Google Scholar
  24. Sloan, IH, Woźniakowski, H 1998When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?J Complexity14133CrossRefGoogle Scholar
  25. Walsh, JL 1923A closed set of normal orthogonal functions.Amer J Math55524Google Scholar

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

Personalised recommendations