Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T19:11:13.460Z Has data issue: false hasContentIssue false
  • English
  • Français

Digital Sequences with Best Possible Order of L2-Discrepancy

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

Published online by Cambridge University Press:  21 December 2009

Friedrich Pillichshammer
Friedrich Pillichshammer
Affiliation:
Institut für Finanzmathematik, Universität Linz, Altenbergstraße 69, A-4040 Linz, Austria. E-mail: [email protected]
Get access

Abstract

This paper treats the L2-discrepancy of digital (0, 1)-sequences over ℤ2, and gives conditions on the generator matrix of such a sequence which guarantee minimal possible order of L2-discrepancy of the generated sequence. The existence is proved for the first time of digital (0; 1)-sequences over ℤ2 with L2-discrepancy of order . This order is best possible by a result of K. Roth. The existence proof is constructive.

MSC classification

Secondary: 11K38: Irregularities of distribution, discrepancy 11K36: Well-distributed sequences and other variations
Type
Research Article
Information
Mathematika , Volume 53 , Issue 1 , June 2006 , pp. 149 - 160
Copyright
Copyright © University College London 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Chaix, H. and Faure, H., Discrépance et diaphonie en dimension un. Acta Arith. 63 (1993), 103–141.CrossRefGoogle Scholar
2Davenport, H., Note on irregularities of distribution. Mathematika 3 (1956), 131–135.CrossRefGoogle Scholar
3Drmota, M., Larcher, G. and Pillichshammer, F., Precise distribution properties of the van der Corput sequence and related sequences. Manuscripta Math. 118 (2005), 11–41.CrossRefGoogle Scholar
4Drmota, M. and Tichy, R. F., Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer-Verlag (Berlin, 1997).CrossRefGoogle Scholar
5Faure, H., Discrépance quadratique de la suite de van der Corput et de sa symétrique. Acta Arith. 55 (1990), 333–350.CrossRefGoogle Scholar
6Grozdanov, V.S., On the diaphony of one class of one-dimensional sequences. Int. J. Math. Math. Sci. 19 (1996), 115–124.CrossRefGoogle Scholar
7Haber, S., On a sequence of points of interest for numerical quadrature. J. Res. Nat. Bur. Standards Sect. B70 (1966), 127–136.CrossRefGoogle Scholar
8Hellekalek, P., General Discrepancy estimates: the Walsh function system. Acta Arith. 67 (1994), 313–322.CrossRefGoogle Scholar
9Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences. John Wiley (New York, 1974).Google Scholar
10Larcher, G. and Pillichshammer, F., Walsh series analysis of the L 2–discrepancy of symmetrisized point sets. Monatsh. Math. 132 (2001), 1–18.CrossRefGoogle Scholar
11Niederreiter, H., Application of diophantine approximations to numerical integration. In Diophantine Approximation and its Application (Osgood, C. F., ed.), Academic Press (New York, 1973), 129–199.Google Scholar
12Niederreiter, H., Point sets and sequences with small discrepancy. Monatsh. Math. 104 (1987), 273–337.CrossRefGoogle Scholar
13Niederreiter, H., Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Series in Applied Mathematics 63. S.I.A.M. (Philadelphia, 1992).CrossRefGoogle Scholar
14Pillichshammer, F., On the discrepancy of (0; 1)-sequences. J. Number Theory 104 (2004), 301–314.CrossRefGoogle Scholar
15Pirsic, G., Schnell konvergierende Walshreihen über Gruppen. Master's Thesis, University of Salzburg (1995). (Available at http://www.ricam.oeaw.ac.at/people/page/pirsic/)Google Scholar
16Proinov, P. D., On the L 2 discrepancy of some infinite sequences. Serdica 11 (1985), 3–12.Google Scholar
17Proinov, P. D. and Grozdanov, V.S., On the diaphony of the van der Corput-Halton sequence. J. Number Theory 30 (1988), 94–104.CrossRefGoogle Scholar
18Roth, K.F., On irregularities of distribution. Mathematika 1 (1959), 73–79.CrossRefGoogle Scholar
19Zinterhof, P., Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden. Sitzungsber. Österr. Akad. Wiss. Math.-Natur. Kl. II 185 (1976), 121–132.Google Scholar