Abstract
We study the star discrepancy of Hammersley nets and van der Corput sequences which are important examples of low-dimensional quasi-Monte Carlo point sets. By a so-called digital shift, the quality of distribution of these point sets can be improved. In this paper, we advance and extend existing bounds on digitally shifted Hammersley and van der Corput point sets and establish criteria for the choice of digital shifts leading to optimal results. Our investigations are partly based on a close analysis of certain sums of distances to the nearest integer.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Béjian, R.: Minoration de la discrépance d'une suite quelconque sur T. Acta Arith. 41, 185–202 (1982)
Béjian, R., Faure, H.: Discrépance de la suite de van der Corput. C. R. Acad. Sci., Paris, Sér. A 285, 313–316 (1977)
De Clerck, L.: A method for exact calculation of the stardiscrepancy of plane sets applied to the sequences of Hammersley. Monatsh. Math. 101, 261–278 (1986)
Dick, J., Kritzer, P.: A best possible upper bound on the star discrepancy of (t,m,2). To Appear in: Monte Carlo Methods Appl. (2006)
Drmota, M., Larcher, G., Pillichshammer, F.: Precise distripution properties of the van der Corput sequence and related sequences. In: Manuscripta Math. 118, 11–41 (2005)
Drmota, M., Tichy, R.F.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, vol. 1651. Springer, Berlin Heidelberg New York (1997)
Faure, H.: On the star-discrepancy of generalized Hammersley sequences in two dimensions. Monatsh. Math. 101, 291–300 (1986)
Halton, J.H., Zaremba, S.K.: The extreme and the L 2 discrepancies of some plane sets. Monatsh. Math. 73, 316–328 (1969)
Kritzer, P.: On some remarkable properties of the two-dimensional Hammersley net in base 2. J. Théor. Nombres Bordeaux (in press)
Kritzer, P., Pillichshammer, F.: Point sets with low L p -discrepancy. To Appear in: Math Scovaca (2006)
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (1974)
Larcher, G., Pillichshammer, F.: Sums of distances to the nearest integer and the discrepancy of digital nets. Acta Arith. 106, 379–408 (2003)
Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. No. 63 in CBMS-NSF Series in Applied Mathematics. SIAM, Philadelphia (1992)
Pillichshammer, F.: On the discrepancy of (0,1)-sequences. J. Number Theory 104, 301–314 (2004)
Schmidt, W.M.: Irregularities of distribution. VII. Acta Arith. 21, 45–50 (1972)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classi cation (2000) 11K38; 11K09
Rights and permissions
About this article
Cite this article
Kritzer, P., Larcher, G. & Pillichshammer, F. A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets. Annali di Matematica 186, 229–250 (2007). https://doi.org/10.1007/s10231-006-0002-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-006-0002-5