Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-20T16:19:50.575Z Has data issue: false hasContentIssue false
  • English
  • Français

ON HYBRID SEQUENCES BUILT FROM NIEDERREITER–HALTON SEQUENCES AND KRONECKER SEQUENCES

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  29 July 2011

ROSWITHA HOFER  and
PETER KRITZER
ROSWITHA HOFER
Affiliation:
Institut für Finanzmathematik, Universität Linz, Altenbergerstr. 69, 4040 Linz, Austria (email: [email protected])
PETER KRITZER*
Affiliation:
Institut für Finanzmathematik, Universität Linz, Altenbergerstr. 69, 4040 Linz, Austria (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We discuss the distribution properties of hybrid sequences whose components stem from Niederreiter–Halton sequences on the one hand, and Kronecker sequences on the other. In this paper, we give necessary and sufficient conditions on the uniform distribution of such sequences, and derive a result regarding their discrepancy. We conclude with a short summary and a discussion of topics for future research.

Keywords

uniform distribution modulo onediscrepancyKronecker sequenceNiederreiter–Halton sequence

MSC classification

Secondary: 11K06: General theory of distribution modulo $1$ 11K38: Irregularities of distribution, discrepancy
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 2 , October 2011 , pp. 238 - 254
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The authors gratefully acknowledge the support of the Austrian Science Fund (Project P21943).

References

[1]Dick, J. and Pillichshammer, F., Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration (Cambridge University Press, Cambridge, 2010).Google Scholar
[2]Drmota, M. and Larcher, G., ‘The sum-of-digits function and uniform distribution modulo 1’, J. Number Theory 89 (2001), 65–96.CrossRefGoogle Scholar
[3]Drmota, M. and Tichy, R. F., Sequences, Discrepancies and Applications, Lecture Notes in Mathematics, 1651 (Springer, Berlin, 1997).Google Scholar
[4]Hofer, R., ‘Note on the joint distribution of the weighted sum-of-digits function modulo one in case of pairwise coprime bases’, Unif. Distrib. Theory 2 (2007), 35–47.Google Scholar
[5]Hofer, R., ‘On subsequences of Niederreiter–Halton sequences’, in: Monte Carlo and Quasi-Monte Carlo Methods 2008 (eds. L’Ecuyer, P. and Owen, B.) (Springer, Berlin, 2009), pp. 423–438.CrossRefGoogle Scholar
[6]Hofer, R., ‘On the distribution properties of Niederreiter–Halton sequences’, J. Number Theory 129 (2009), 451–463.Google Scholar
[7]Hofer, R., Kritzer, P., Larcher, G. and Pillichshammer, F., ‘Distribution properties of generalized van der Corput–Halton sequences and their subsequences’, Int. J. Number Theory 5 (2009), 719–746.CrossRefGoogle Scholar
[8]Hofer, R. and Larcher, G., ‘Metric results on the discrepancy of Halton–Kronecker sequences’, Math. Z. (2011), to appear.CrossRefGoogle Scholar
[9]Hofer, R. and Larcher, G., ‘On existence and discrepancy of certain Niederreiter–Halton sequences’, Acta Arith. 141 (2010), 369–394.Google Scholar
[10]Hofer, R., Pillichshammer, F. and Pirsic, G., ‘Distribution properties of sequences generated by q-additive functions with respect to Cantor representation of integers’, Acta Arith. 138 (2009), 179–200.Google Scholar
[11]Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences (Wiley, New York, 1974).Google Scholar
[12]Larcher, G. and Niederreiter, H., ‘Generalized (t,s)-sequences, Kronecker-type sequences, and diophantine approximations of formal Laurent series’, Trans. Amer. Math. Soc. 347 (1995), 2051–2073.Google Scholar
[13]Lemieux, C., Monte Carlo and Quasi-Monte Carlo Sampling, Springer Series in Statistics (Springer, New York, 2009).Google Scholar
[14]Niederreiter, H., ‘Methods for estimating discrepancy’, in: Applications of Number Theory to Numerical Analysis (ed. Zaremba, S. K.) (Academic Press, New York, 1972), pp. 203–236.CrossRefGoogle Scholar
[15]Niederreiter, H., ‘Point sets and sequences with small discrepancy’, Monatsh. Math. 104 (1987), 273–337.Google Scholar
[16]Niederreiter, H., Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Series in Applied Mathematics, 63 (SIAM, Philadelphia, 1992).CrossRefGoogle Scholar
[17]Niederreiter, H., ‘On the discrepancy of some hybrid sequences’, Acta Arith. 138 (2009), 373–398.Google Scholar
[18]Niederreiter, H., ‘Further discrepancy bounds and an Erdős–Turán–Koksma inequality for hybrid sequences’, Monatsh. Math. 161 (2010), 193–222.Google Scholar
[19]Pillichshammer, F., ‘Uniform distribution of sequences connected with the weighted sum-of-digits function’, Unif. Distrib. Theory 2 (2007), 1–10.Google Scholar
[20]Spanier, J., ‘Quasi-Monte Carlo methods for particle transport problems’, in: Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Lecture Notes in Statistics, 106 (eds. Niederreiter, H. and Shiue, P. J.-S.) (Springer, New York, 1995), pp. 121–408.Google Scholar