On the lower bound of the discrepancy of Halton's sequence I

Mordechay B. Levin

doi:10.1016/j.crma.2016.02.003

Number theory

On the lower bound of the discrepancy of Halton's sequence I
[Sur la limite inférieure de la discrépance de la suite de Halton I]

Présenté par : the Editorial Board

Mordechay B. Levin ¹

¹ Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel

Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 445-448.

Résumé
Abstract

Soit ${(H_{s} (n))}_{n \geq 1}$ une suite de Halton á s dimensions. Soit $D_{N}$ la discrépance de la suite ${(H_{s} (n))}_{n = 1}^{N}$ . Il est connu que $N D_{N} = O (\ln^{s} N)$ lorsque $N \to \infty$ . Dans cet article, nous montrons que cette estimation est exacte :

{\lim^{‾}}_{N \to \infty} N \ln^{- s} (N) D_{N} > 0 .

Let ${(H_{s} (n))}_{n \geq 1}$ be an s-dimensional Halton's sequence. Let $D_{N}$ be the discrepancy of the sequence ${(H_{s} (n))}_{n = 1}^{N}$ . It is known that $N D_{N} = O (\ln^{s} N)$ as $N \to \infty$ . In this paper, we prove that this estimate is exact:

{\lim^{‾}}_{N \to \infty} N \ln^{- s} (N) D_{N} > 0 .

Reçu le : 2015-05-25
Accepté le : 2016-02-10
Publié le : 2016-03-21

DOI : 10.1016/j.crma.2016.02.003

Affiliations des auteurs :

Mordechay B. Levin ¹

¹ Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel

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     title = {On the lower bound of the discrepancy of {Halton's} sequence {I}},
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Mordechay B. Levin. On the lower bound of the discrepancy of Halton's sequence I. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 445-448. doi : 10.1016/j.crma.2016.02.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.02.003/

Bibliographie
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[5] M.B. Levin On the lower bound in the lattice point remainder problem for a parallelepiped, Discrete Comput. Geom., Volume 54 (2015), pp. 826-870

[6] M.B. Levin On the lower bound of the discrepancy of $(t, s)$ sequence II http://arXiv.org/abs/1505.04975

[7] M.B. Levin On the lower bound of the discrepancy of Halton's sequence II, Eur. J. Math. (2016) (in press) | DOI

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