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.

Abstract (VO)
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 .

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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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
Cité par

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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

Mordechay B. Levin ON THE UPPER BOUND OF THE L $_{p}$ DISCREPANCY OF HALTON’S SEQUENCE AND THE CENTRAL LIMIT THEOREM FOR HAMMERSLEY’S NET, Journal of Mathematical Sciences, Volume 280 (2024) no. 6, p. 1002 | DOI:10.1007/s10958-024-07311-w
Mordechay B. Levin ON THE UPPER BOUND OF THE $L_{p}$ DISCREPANCY OF HALTON’S SEQUENCE AND THE CENTRAL LIMIT THEOREM FOR HAMMERSLEY’S NET, II, Journal of Mathematical Sciences, Volume 280 (2024) no. 6, p. 1030 | DOI:10.1007/s10958-024-07312-9
Lai Zhang, Proceedings of the 2024 8th International Conference on Digital Signal Processing (2024), p. 146 | DOI:10.1145/3653876.3653889
Mordechay B. Levin On a bounded remainder set for a digital Kronecker sequence, Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, p. 163 | DOI:10.5802/jtnb.1197
Roswitha Hofer A lower bound on the star discrepancy of generalized Halton sequences in rational bases, Proceedings of the American Mathematical Society, Volume 147 (2019) no. 11, p. 4655 | DOI:10.1090/proc/14596
Lisa Kaltenböck; Wolfgang Stockinger On M. B. Levin’s Proofs for The Exact Lower Discrepancy Bounds of Special Sequences and Point Sets (A Survey), Uniform distribution theory, Volume 13 (2018) no. 2, p. 103 | DOI:10.2478/udt-2018-0014

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