Van der Corput sequences towards general (0,1)–sequences in base b

Henri Faure[1]

  • [1] Institut de Mathématiques de Luminy, U.M.R. 6206 CNRS 163 avenue de Luminy, case 907 13288 Marseille Cedex 09, France

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 125-140
  • ISSN: 1246-7405

Abstract

top
As a result of recent studies on unidimensional low discrepancy sequences, we can assert that the original van der Corput sequences are the worst distributed with respect to various measures of irregularities of distribution among two large families of ( 0 , 1 ) –sequences, and even among all ( 0 , 1 ) –sequences for the star discrepancy D * . We show in the present paper that it is not the case for the extreme discrepancy D by producing two kinds of sequences which are the worst distributed among all ( 0 , 1 ) –sequences, with a discrepancy D essentially twice greater. In addition, we give an unified presentation for the two generalizations presently known of van der Corput sequences.

How to cite

top

Faure, Henri. "Van der Corput sequences towards general (0,1)–sequences in base b." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 125-140. <http://eudml.org/doc/249975>.

@article{Faure2007,
abstract = {As a result of recent studies on unidimensional low discrepancy sequences, we can assert that the original van der Corput sequences are the worst distributed with respect to various measures of irregularities of distribution among two large families of $(0,1)$–sequences, and even among all $(0,1)$–sequences for the star discrepancy $D^*$. We show in the present paper that it is not the case for the extreme discrepancy $D$ by producing two kinds of sequences which are the worst distributed among all $(0,1)$–sequences, with a discrepancy $D$ essentially twice greater. In addition, we give an unified presentation for the two generalizations presently known of van der Corput sequences.},
affiliation = {Institut de Mathématiques de Luminy, U.M.R. 6206 CNRS 163 avenue de Luminy, case 907 13288 Marseille Cedex 09, France},
author = {Faure, Henri},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {1},
pages = {125-140},
publisher = {Université Bordeaux 1},
title = {Van der Corput sequences towards general (0,1)–sequences in base b},
url = {http://eudml.org/doc/249975},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Faure, Henri
TI - Van der Corput sequences towards general (0,1)–sequences in base b
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 125
EP - 140
AB - As a result of recent studies on unidimensional low discrepancy sequences, we can assert that the original van der Corput sequences are the worst distributed with respect to various measures of irregularities of distribution among two large families of $(0,1)$–sequences, and even among all $(0,1)$–sequences for the star discrepancy $D^*$. We show in the present paper that it is not the case for the extreme discrepancy $D$ by producing two kinds of sequences which are the worst distributed among all $(0,1)$–sequences, with a discrepancy $D$ essentially twice greater. In addition, we give an unified presentation for the two generalizations presently known of van der Corput sequences.
LA - eng
UR - http://eudml.org/doc/249975
ER -

References

top
  1. H. Chaix, H. Faure, Discrépance et diaphonie en dimension un. Acta Arith. 63.2 (1993), 103–141. Zbl0772.11022MR1206080
  2. J. Dick, P. Kritzer, A best possible upper bound on the star discrepancy of ( t , m , 2 ) –nets. Monte Carlo Meth. Appl. 12.1 (2006), 1–17. Zbl1103.11022MR2229922
  3. M. Drmota, G. Larcher, F. Pillichshammer, Precise distribution properties of the van der Corput sequence and related sequences. Manuscripta Math. 118 (2005), 11–41. Zbl1088.11060MR2171290
  4. H. Faure, Discrépance de suites associées à un système de numération (en dimension u n ). Bull. Soc. math. France 109 (1981), 143–182. Zbl0488.10052MR623787
  5. H. Faure, Good permutations for extreme discrepancy. J. Number Theory. 42 (1992), 47–56. Zbl0768.11026MR1176419
  6. H. Faure, Discrepancy and diaphony of digital ( 0 , 1 ) –sequences in prime base. Acta Arith. 117.2 (2005), 125–148. Zbl1080.11054MR2139596
  7. H. Faure, Irregularities of distribution of digital ( 0 , 1 ) –sequences in prime base. Integers, Electronic Journal of Combinatorial Number Theory 5(3) A07 (2005), 1–12. Zbl1084.11041MR2191753
  8. P. Kritzer, A new upper bound on the star discrepancy of ( 0 , 1 ) –sequences. Integers, Electronic Journal of Combinatorial Number Theory 5(3) A11 (2005), 1–9. Zbl1092.11034MR2191757
  9. G. Larcher, F. Pillichshammer, Walsh series analysis of the L 2 –discrepancy of symmetrisized points sets. Monatsh. Math. 132 (2001), 1–18. Zbl1108.11309MR1825715
  10. G. Larcher, F. Pillichshammer, Sums of distances to the nearest integer and the discrepancy of digital nets. Acta Arith. 106.4 (2003), 379–408. Zbl1054.11039MR1957912
  11. W. J. Morokoff, R. E. Caflisch, Quasi random sequences and their discrepancies. SIAM J. Sci. Comput. 15.6 (1994), 1251–1279. Zbl0815.65002MR1298614
  12. H. Niederreiter, Random number generation and quasi-Monte Carlo methods. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992. Zbl0761.65002MR1172997
  13. H. Niederreiter, C. P. Xing, Quasirandom points and global functions fields. Finite Fields and Applications (S. Cohen and H. Niederreiter, eds), London Math. Soc. Lectures Notes Series Vol. 233 (1996), 269–296. Zbl0932.11050MR1433154
  14. H. Niederreiter, C. P.Xing, Nets, ( t , s ) –sequences and algebraic geometry. Random and Quasi-random Point Sets (P. Hellekalek and G. Larcher, eds), Lectures Notes in Statistics, Springer Vol. 138 (1998), 267–302. Zbl0923.11113MR1662844
  15. F. Pillichshammer, On the discrepancy of (0,1)–sequences. J. Number Theory 104 (2004), 301–314. Zbl1048.11061MR2029508
  16. S. Tezuka, Polynomial arithmetic analogue of Halton sequences. ACM Trans. Modeling and Computer Simulation 3 (1993), 99–107. Zbl0846.11045

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.