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

Presented by: 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.

Abstract
Résumé

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 .

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 .

Received: 2015-05-25
Accepted: 2016-02-10
Published online: 2016-03-21

DOI: 10.1016/j.crma.2016.02.003

Author's affiliations:

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/

References
Cited by

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[3] M. Drmota; R. Tichy Sequences, Discrepancies and Applications, Lecture Notes in Mathematics, vol. 1651, 1997

[4] H. Faure; H. Chaix Minoration de discrépance en dimension deux, Acta Arith., Volume 76 (1996) no. 2, pp. 149-164

[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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