L 2 discrepancy of generalized Zaremba point sets

Henri Faure[1]; Friedrich Pillichshammer[2]

  • [1] Institut de Mathématiques de Luminy, U.M.R. 6206 CNRS 163 avenue de Luminy, case 907 13288 Marseille Cedex 09, France
  • [2] Institut für Finanzmathematik, Universität Linz Altenbergerstraße 69 A-4040 Linz, Austria

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 1, page 121-136
  • ISSN: 1246-7405

Abstract

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We give an exact formula for the L 2 discrepancy of a class of generalized two-dimensional Hammersley point sets in base b , namely generalized Zaremba point sets. These point sets are digitally shifted Hammersley point sets with an arbitrary number of different digital shifts in base b . The Zaremba point set introduced by White in 1975 is the special case where the b shifts are taken repeatedly in sequential order, hence needing at least b b points to obtain the optimal order of L 2 discrepancy. On the contrary, our study shows that only one non-zero shift is enough for the same purpose, whatever the base b is.

How to cite

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Faure, Henri, and Pillichshammer, Friedrich. "$L_2$ discrepancy of generalized Zaremba point sets." Journal de Théorie des Nombres de Bordeaux 23.1 (2011): 121-136. <http://eudml.org/doc/219693>.

@article{Faure2011,
abstract = {We give an exact formula for the $L_2$ discrepancy of a class of generalized two-dimensional Hammersley point sets in base $b$, namely generalized Zaremba point sets. These point sets are digitally shifted Hammersley point sets with an arbitrary number of different digital shifts in base $b$. The Zaremba point set introduced by White in 1975 is the special case where the $b$ shifts are taken repeatedly in sequential order, hence needing at least $b^b$ points to obtain the optimal order of $L_2$ discrepancy. On the contrary, our study shows that only one non-zero shift is enough for the same purpose, whatever the base $b$ is.},
affiliation = {Institut de Mathématiques de Luminy, U.M.R. 6206 CNRS 163 avenue de Luminy, case 907 13288 Marseille Cedex 09, France; Institut für Finanzmathematik, Universität Linz Altenbergerstraße 69 A-4040 Linz, Austria},
author = {Faure, Henri, Pillichshammer, Friedrich},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Discrepancy; discrepancy; Hammersley point sets, Zaremba point sets; digitally shifted point sets},
language = {eng},
month = {3},
number = {1},
pages = {121-136},
publisher = {Société Arithmétique de Bordeaux},
title = {$L_2$ discrepancy of generalized Zaremba point sets},
url = {http://eudml.org/doc/219693},
volume = {23},
year = {2011},
}

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
DA - 2011/3//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 1
SP - 121
EP - 136
AB - We give an exact formula for the $L_2$ discrepancy of a class of generalized two-dimensional Hammersley point sets in base $b$, namely generalized Zaremba point sets. These point sets are digitally shifted Hammersley point sets with an arbitrary number of different digital shifts in base $b$. The Zaremba point set introduced by White in 1975 is the special case where the $b$ shifts are taken repeatedly in sequential order, hence needing at least $b^b$ points to obtain the optimal order of $L_2$ discrepancy. On the contrary, our study shows that only one non-zero shift is enough for the same purpose, whatever the base $b$ is.
LA - eng
KW - Discrepancy; discrepancy; Hammersley point sets, Zaremba point sets; digitally shifted point sets
UR - http://eudml.org/doc/219693
ER -

References

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