Skip to main content
SpringerLink
Log in

Improvements on Low Discrepancy One-Dimensional Sequences and Two-Dimensional Point Sets

  • Conference paper
Monte Carlo and Quasi-Monte Carlo Methods 2006

Summary

We obtain significant improvements for the star discrepancy D* of generalized van der Corput sequences by means of linear digit scramblings (see Section 5.2 for the definition). We also find a new lower bound for the extreme discrepancy D of these sequences which permits to show that linearly-scrambled sequences are set in a good place among generalized van der Corput sequences. Finally, we derive the corresponding properties for generalized Hammersley point sets in arbitrary bases and link recent developments in base 2 by Kritzer, Larcher and Pillichshammer to former studies of Béjian and the author.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Béjian. Sur certaines suites présentant une faible discrépance à l'origine. C. R. Acad. Sc.,Paris, 286A, 135–138 (1978).

    Google Scholar 

  2. R. Béjian. Minoration de la discrépance d'une suite quelconque sur T. Acta Arith., 41, 185–202 (1982).

    MATH  MathSciNet  Google Scholar 

  3. H. Chaix and H. Faure. Discrépance et diaphonie en dimension un. Acta Arith., 63, 103–141 (1993).

    MATH  MathSciNet  Google Scholar 

  4. L. De Clerck. A method for the exact calculation of the star-discrepancy of plane sets applied to the sequences of Hammersley. Mh. Math., 101, 261–278 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Drmota, G. Larcher, and F. Pillichshammer. Precise distribution properties of the van der Corput sequence and related sequences. Manuscripta Math. 118, 11–41 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Faure. Discrépance de suites associées à un système de numération (en dimension un). Bull. Soc. Math. France, 109, 143–182 (1981).

    MATH  MathSciNet  Google Scholar 

  7. H. Faure. On the star-discrepancy of generalized Hammersley sequences in two dimensions. Mh. Math., 101, 291–300 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Faure. Good permutations for extreme discrepancy. J. Number Theory, 41, 47–56 (1992).

    Article  MathSciNet  Google Scholar 

  9. H. Faure. Discrepancy and diaphony of digital (0,1)-sequences in prime base. Acta Arith. 117.2, 125–148 (2005).

    Article  MathSciNet  Google Scholar 

  10. H. Faure. Irregularities of distribution of digital (0, 1)-sequences in prime base. Integers, Electronic Journal of Combinatorial Number Theory 5(3) A07, 1–12 (2005).

    MathSciNet  Google Scholar 

  11. H. Faure. Selection criteria for (random) generation of digital (0, s)–sequences. In: Niederreiter, H., Talay, D. (Eds.) Monte Carlo and Quasi-Monte Carlo Methods 2004. Springer, Berlin Heidelberg New York, 113–126 (2006).

    Chapter  Google Scholar 

  12. H. Faure. Van der Corput sequences towards general (0, 1)–sequences in base b. J. Théor. Nombres Bordeaux, to appear (2007).

    Google Scholar 

  13. J.H. Halton and S.K. Zaremba. The extreme and L 2-discrepancies of some plane sets. Mh. Math., 73, 316–328 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  14. P. Kritzer. A new upper bound on the star discrepancy of (0, 1)-sequences. Integers, Electronic Journal of Combinatorial Number Theory 5(3) A11, 1–9 (2005).

    MathSciNet  Google Scholar 

  15. P. Kritzer. On some remarkable properties of the two-dimensional Hammersley point set in base 2. J. Théor. Nombres Bordeaux 18, 203–221 (2006).

    MATH  MathSciNet  Google Scholar 

  16. P. Kritzer, G. Larcher, and F. Pillichshammer. A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets. Ann. Math. Pura Appl., to appear (2006).

    Google Scholar 

  17. G. Larcher and F. Pillichshammer. Walsh series analysis of the L 2 -discrepancy of symmetrisized points sets. Mh. Math., 132, 1–18 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  18. G. Larcher and F. Pillichshammer. Sums of distances to the nearest integer and the discrepancy of digital nets. Acta Arith., 106.4, 379–408 (2003).

    Article  MathSciNet  Google Scholar 

  19. J. Matoušek. On the L 2–discrepancy for anchored boxes. J. Complexity, 14, 527–556 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  20. H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1992).

    Google Scholar 

  21. H. Niederreiter and C.P. Xing. Quasirandom points and global functions fields. In: Cohen, S. and Niederreiter, H. (eds) Finite Fields and Applications. London Math. Soc. Lectures Notes Series, 233, 269–296 (1996).

    Google Scholar 

  22. A.B. Owen. Randomly permuted (t, m, s)-nets and (t, s)-sequences. In: Niederreiter, H., Shiue, P. (eds.) Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Springer, Lectures Notes in Statistics, 126, 299–317 (1995).

    Google Scholar 

  23. F. Pillichshammer. On the discrepancy of (0,1)-sequences. J. Number Theory, 104, 301–314 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  24. S. Tezuka. A generalization of Faure sequences and its efficient implementation. Research Report IBM RT0105 (1994).

    Google Scholar 

  25. S. Tezuka. Uniform Random Numbers: Theory and Practice. Kluwer Academic Publishers, Boston (1995).

    MATH  Google Scholar 

  26. B.E. White. Mean-square discrepancies of the Hammersley and Zaremba sequences for arbitrary radix. Mh. Math., 80, 219–229 (1975).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Luminy, UMR 6206 CNRS, 163 Av. de Luminy, case 907, 13288, Marseille cedex 9, France

    Henri Faure

Authors
  1. Henri Faure
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Ulm University, Albert-Einstein-Allee 11, 89069, Ulm, Germany

    Alexander Keller

  2. Department of Computer Science, University of Kaiserslautern, 67653, Kaiserslautern, Germany

    Stefan Heinrich

  3. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Harald Niederreiter

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Faure, H. (2008). Improvements on Low Discrepancy One-Dimensional Sequences and Two-Dimensional Point Sets. In: Keller, A., Heinrich, S., Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74496-2_19

Download citation

Publish with us

Policies and ethics

Search

Navigation