Functiones et Approximatio Commentarii Mathematici

The $p$-adic diaphony of the Halton sequence

Friedrich Pillichshammer

Full-text: Open access


The $p$-adic diaphony as introduced by Hellekalek is a quantitative measure for the irregularity of distribution of a sequence in the unit cube. In this paper we show how this notion of diaphony can be interpreted as worst-case integration error in a certain reproducing kernel Hilbert space. Our main result is an upper bound on the $p$-adic diaphony of the Halton sequence.

Article information

Funct. Approx. Comment. Math., Volume 49, Number 1 (2013), 91-102.

First available in Project Euclid: 20 September 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11K06: General theory of distribution modulo 1 [See also 11J71]
Secondary: 11K38: Irregularities of distribution, discrepancy [See also 11Nxx] 11K41: Continuous, $p$-adic and abstract analogues

irregularity of distribution diaphony Halton sequence quasi-Monte Carlo


Pillichshammer, Friedrich. The $p$-adic diaphony of the Halton sequence. Funct. Approx. Comment. Math. 49 (2013), no. 1, 91--102. doi:10.7169/facm/2013.49.1.6.

Export citation


  • N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.
  • J. Dick and F. Pillichshammer, Diaphony, discrepancy, spectral test and worst-case error, Math. Comput. Simulation 70 (2005), 159–171.
  • J. Dick and F. Pillichshammer, Digital nets and sequences – Discrepancy theory and quasi-Monte Carlo integration, Cambridge University Press, Cambridge, 2010.
  • M. Drmota and R.F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Mathematics 1651, Springer-Verlag, Berlin, 1997.
  • V. Grozdanov and S. Stoilova, The general diaphony, C.R. Acad. Bulgare Sci. 57 (2004), 13–18.
  • P. Hellekalek, A general discrepancy estimate based on $p$-adic arithmetics, Acta Arith. 139 (2009), 117–129.
  • P. Hellekalek, A notion of diaphony based on $p$-adic arithmetic, Acta Arith. 145 (2010), 273–284.
  • P. Hellekalek and H. Leeb, Dyadic diaphony, Acta Arith. 80 (1997), 187–196.
  • F.J. Hickernell, A generalized discrepancy and quadrature error bound, Math. Comp. 67 (1998), 299–322.
  • L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. John Wiley, New York, 1974; reprint, Dover Publications, Mineola, NY, 2006.
  • J. Matoušek, Geometric discrepancy. Springer, Berlin Heidelberg New York, 1999.
  • H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, 1992.
  • I.H. Sloan and H. Woźniakowski, When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?, J. Complexity 14 (1998), 1–33.
  • P. Zinterhof, Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden, Sitzungsber. Österr. Akad. Wiss. Math.-Natur. Kl. II 185 (1976), 121–132.