Numerical Verifications of Theoretical Results about the Weighted \(({\cal W}(b);\gamma)-\) Diaphony of the Generalized Van der Corput Sequence

  • Vesna Dimitrievska RistovskaEmail author
  • Vassil Grozdanov
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 207)


The weighted \(({\cal W}(b);\gamma)-\)diaphony is a new quantitative measure for the irregularity of distribution of sequences. In previous works of the authors it has been found the exact order \({\cal O}\left({1 \over N}\right)\) of the weighted \(({\cal W}(b);\gamma)-\)diaphony of the generalized Van der Corput sequence. Here, we give an upper bound of the weighted \(({\cal W}(b);\gamma)-\) diaphony, which is an analogue of the classical Erdös-Turán-Koksma inequality, with respect to this kind of the diaphony. This permits us to make a computational simu-lations of the weighted \(({\cal W}(b);\gamma)-\)diaphony of the generalized Van der Corput sequence. Different choices of sequences of permutations of the set {0,1, …, b − 1} are practically realized and the \(({\cal W}(b);\gamma)-\)diaphony of the corresponding generalized Van der Corput sequences is numerically calculated and discussed.


Uniform distribution of sequences Diaphony Generalized Van der Corput sequence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chrestenson, H.E.: A class of generalized Walsh functions. Pacific J. Math. 5, 17–31 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Dimitrievska Ristovska, V., Grozdanov, V., Kusakatov, V., Stoilova, S.: Computing complexity of a new type of the diaphony. In: The 9th International Conference for Informatics and Information Technology, CIIT, Bitola (2012) (inprint)Google Scholar
  3. 3.
    Dimitrievska Ristovska, V., Grozdanov, V., Mavrodieva, D., Stoilova, S.: On the weighted \(({\cal W}(b);\gamma)-\)diaphony of the generalized Van der Corput sequence and the Zaremba-Halton net (in submitting)Google Scholar
  4. 4.
    Faure, H.: Discrepances de suite associées à un systèm de numération (en dimension un). Bull. Soc. Math. France 109, 143–182 (1981)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Grozdanov, V., Stoilova, S.: The b −adic diaphony. Rendiconti di Matematica, Serie VII 22, 203–221 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2, 84–90 (1960)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kuipers, L., Niederreiter, H.: Uniform distribution of sequences. John Wiley & Sons, N. Y. (1974)zbMATHGoogle Scholar
  8. 8.
    Van der Corput, J.G.: Verteilungsfunktionen. Proc. Kon. Ned. Akad. Wetensch. 38, 813–821 (1935)Google Scholar
  9. 9.
    Zinterhof, P.: Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden. S. B. Akad. Wiss., Math.-Naturw. Klasse. Abt. II 185, 121–132 (1976)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Vesna Dimitrievska Ristovska
    • 1
    Email author
  • Vassil Grozdanov
    • 2
  1. 1.FINKIUniversity “Ss Cyril and Methodius”SkopjeMacedonia
  2. 2.Department of MathematicsSouth-West University “Neophit Rilsky”BlagoevgradBulgaria

Personalised recommendations