We give an exact formula for the discrepancy of a class of generalized two-dimensional Hammersley point sets in base , namely generalized Zaremba point sets. These point sets are digitally shifted Hammersley point sets with an arbitrary number of different digital shifts in base . The Zaremba point set introduced by White in 1975 is the special case where the shifts are taken repeatedly in sequential order, hence needing at least points to obtain the optimal order of discrepancy. On the contrary, our study shows that only one non-zero shift is enough for the same purpose, whatever the base is.
Nous donnons une formule exacte pour la discrépance des ensembles généralisés de Zaremba, une sous-classe des ensembles plans généralisés de Hammersley en base . Ces ensembles de points sont des décalés digitaux des ensembles de Hammersley obtenus avec un nombre arbitraire des différents décalages en base . L’ensemble de Zaremba introduit par White en 1975 est le cas particulier où les décalages possibles sont pris et répétés dans l’ordre, ce qui exige au moins points pour atteindre la discrépance optimale. Au contraire, notre étude montre qu’il suffit d’un seul décalage non nul pour obtenir le même résultat, quelle que soit la base .
@article{JTNB_2011__23_1_121_0,
author = {Faure, Henri and Pillichshammer, Friedrich},
title = {$L_2$ discrepancy of generalized {Zaremba} point sets},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {121--136},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {23},
number = {1},
year = {2011},
doi = {10.5802/jtnb.753},
zbl = {1277.11081},
mrnumber = {2780622},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.753/}
}
TY - JOUR AU - Faure, Henri AU - Pillichshammer, Friedrich TI - $L_2$ discrepancy of generalized Zaremba point sets JO - Journal de théorie des nombres de Bordeaux PY - 2011 SP - 121 EP - 136 VL - 23 IS - 1 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.753/ DO - 10.5802/jtnb.753 LA - en ID - JTNB_2011__23_1_121_0 ER -
%0 Journal Article %A Faure, Henri %A Pillichshammer, Friedrich %T $L_2$ discrepancy of generalized Zaremba point sets %J Journal de théorie des nombres de Bordeaux %D 2011 %P 121-136 %V 23 %N 1 %I Société Arithmétique de Bordeaux %U http://www.numdam.org/articles/10.5802/jtnb.753/ %R 10.5802/jtnb.753 %G en %F JTNB_2011__23_1_121_0
Faure, Henri; Pillichshammer, Friedrich. $L_2$ discrepancy of generalized Zaremba point sets. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 121-136. doi : 10.5802/jtnb.753. http://www.numdam.org/articles/10.5802/jtnb.753/
[1] H. Chaix and H. Faure, Discrépance et diaphonie en dimension un. Acta Arith. 63 (1993), 103–141. | MR | Zbl
[2] W .W .L .Chen and M. M. Skriganov, Explicit constructions in the classical mean squares problem in irregularities of point distribution. J. Reine Angew. Math. 545 (2002), 67–95. | MR | Zbl
[3] W .W .L .Chen and M. M. Skriganov, Orthogonality and digit shifts in the classical mean squares problem in irregularities of point distribution. In Diophantine approximation, Dev. Math. 16, 141–159, Springer, New York Vienna, 2008. | MR
[4] H. Davenport, Note on irregularities of distribution. Mathematika 3 (1956), 131–135. | MR | Zbl
[5] J. Dick and F. Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010. | MR
[6] H. Faure, Discrépance de suites associées à un système de numération (en dimension un). Bull. Soc. Math. France 109 (1981), 143–182. | Numdam | MR | Zbl
[7] H. Faure and F. Pillichshammer, discrepancy of generalized two-dimensional Hammersley point sets. Monatsh. Math. 158 ( 2009), 31–61. | MR | Zbl
[8] H. Faure and F. Pillichshammer, discrepancy of two-dimensional digitally shifted Hammersley point sets in base . In Monte Carlo and Quasi-Monte Carlo Methods 2008, P. L’Ecuyer and A. Owen (eds.), 355–368, Springer-Verlag, Berlin Heidelberg, 2009.
[9] H. Faure, F. Pillichshammer, G. Pirsic and W. Ch. Schmid, discrepancy of generalized two-dimensional Hammersley point sets scrambled with arbitrary permutations. Acta. Arith. 141 (2010), 395–418. | MR | Zbl
[10] J.H. Halton and S.K. Zaremba, The extreme and the discrepancies of some plane sets. Monatsh. Math. 73 (1969), 316–328. | MR | Zbl
[11] P. Kritzer and F. Pillichshammer, An exact formula for the discrepancy of the shifted Hammersley point set. Uniform Distribution Theory 1 (2006), 1–13. | MR | Zbl
[12] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. John Wiley, New York, 1974; reprint, Dover Publications, Mineola, NY, 2006. | MR | Zbl
[13] G. Larcher and F. Pillichshammer, Walsh series analysis of the -discrepancy of symmetrisized point sets. Monatsh. Math. 132 (2001), 1–18. | MR | Zbl
[14] F. Pillichshammer, On the -discrepancy of the Hammersley point set. Monatsh. Math. 136 (2002), 67–79. | MR | Zbl
[15] P. D. Proinov, Symmetrization of the van der Corput generalized sequences. Proc. Japan Acad. 64 (1988), Ser. A, 159–162. | MR | Zbl
[16] K.F. Roth, On irregularities of distribution. Mathematika 1 (1954), 73–79. | MR | Zbl
[17] I.V. Vilenkin, Plane nets of Integration. Ž. Vyčisl. Mat. i Mat. Fiz. 7 (1967), 189–196. (English translation in: U.S.S.R. Computational Math. and Math. Phys. 7 (1) (1967), 258–267.) | MR | Zbl
[18] B.E. White, Mean-square discrepancies of the Hammersley and Zaremba sequences for arbitrary radix. Monatsh. Math. 80 (1975), 219–229. | MR | Zbl
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