Nous examinons une classe spéciale de -réseaux en base 2. Particulièrement, nous nous occupons du réseau de Hammersley en deux dimensions qui joue un rôle spécial parmi ce type de réseaux, puisque nous démontrons que c’est le plus mal distribué quant à la discrépance à l’origine. En le montrant, nous améliorons un majorant connu pour la discrépance à l’origine de -réseaux en base 2. De plus, nous démontrons qu’on peut obtenir des réseaux avec une discrépance à l’origine très basse en transformant le réseau de Hammersley d’une manière appropriée.
We study a special class of -nets in base 2. In particular, we are concerned with the two-dimensional Hammersley net that plays a special role among these since we prove that it is the worst distributed with respect to the star discrepancy. By showing this, we also improve an existing upper bound for the star discrepancy of digital -nets over . Moreover, we show that nets with very low star discrepancy can be obtained by transforming the Hammersley point set in a suitable way.
@article{JTNB_2006__18_1_203_0,
author = {Kritzer, Peter},
title = {On some remarkable properties of the two-dimensional {Hammersley} point set in base 2},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {203--221},
publisher = {Universit\'e Bordeaux 1},
volume = {18},
number = {1},
year = {2006},
doi = {10.5802/jtnb.540},
zbl = {1103.11024},
mrnumber = {2245882},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.540/}
}
TY - JOUR AU - Kritzer, Peter TI - On some remarkable properties of the two-dimensional Hammersley point set in base 2 JO - Journal de théorie des nombres de Bordeaux PY - 2006 SP - 203 EP - 221 VL - 18 IS - 1 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.540/ DO - 10.5802/jtnb.540 LA - en ID - JTNB_2006__18_1_203_0 ER -
%0 Journal Article %A Kritzer, Peter %T On some remarkable properties of the two-dimensional Hammersley point set in base 2 %J Journal de théorie des nombres de Bordeaux %D 2006 %P 203-221 %V 18 %N 1 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.540/ %R 10.5802/jtnb.540 %G en %F JTNB_2006__18_1_203_0
Kritzer, Peter. On some remarkable properties of the two-dimensional Hammersley point set in base 2. Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 203-221. doi : 10.5802/jtnb.540. http://www.numdam.org/articles/10.5802/jtnb.540/
[1] L. De Clerck, A method for exact calculation of the stardiscrepancy of plane sets applied to the sequences of Hammersley. Monatsh. Math. 101 (1986), 261–278. | MR | Zbl
[2] J. Dick, P. Kritzer, Star-discrepancy estimates for digital -nets and -sequences over . Acta Math. Hungar. 109 (3) (2005), 239–254. | MR | Zbl
[3] M. Drmota, R. F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer, Berlin, 1997. | MR | Zbl
[4] H. Faure, On the star-discrepancy of generalized Hammersley sequences in two dimensions. Monatsh. Math. 101 (1986), 291–300. | MR | Zbl
[5] J. H. Halton, S. K. Zaremba, The extreme and the discrepancies of some plane sets. Monatsh. Math. 73 (1969), 316–328. | MR | Zbl
[6] L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences. John Wiley, New York, 1974. | MR | Zbl
[7] G. Larcher, F. Pillichshammer, Sums of distances to the nearest integer and the discrepancy of digital nets. Acta Arith. 106 (2003), 379–408. | MR | Zbl
[8] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. CBMS–NSF Series in Applied Mathematics 63, SIAM, Philadelphia, 1992. | MR | Zbl
[9] F. Zhang, Matrix Theory. Springer, New York, 1999. | MR | Zbl
Cité par Sources :
