L 2 discrepancy of generalized Zaremba point sets
Journal de Théorie des Nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 121-136.

Nous donnons une formule exacte pour la discrépance L 2 des ensembles généralisés de Zaremba, une sous-classe des ensembles plans généralisés de Hammersley en base b. Ces ensembles de points sont des décalés digitaux des ensembles de Hammersley obtenus avec un nombre arbitraire des différents décalages en base b. L’ensemble de Zaremba introduit par White en 1975 est le cas particulier où les b décalages possibles sont pris et répétés dans l’ordre, ce qui exige au moins b b points pour atteindre la discrépance L 2 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 b.

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.

@article{JTNB_2011__23_1_121_0,
     author = {Henri Faure and Friedrich Pillichshammer},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.753/}
}
Henri Faure; Friedrich Pillichshammer. $L_2$ discrepancy of generalized Zaremba point sets. Journal de Théorie des Nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 121-136. doi : 10.5802/jtnb.753. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.753/

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