On linear normal lattices configurations
Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 825-858.

Dans cet article nous prolongeons la construction de Champernowne de nombres normaux dans la base b pour le cas d , et obtenons une construction explicite du point générique de la transformation de l’ensemble {0,1,...,b-1} d par d déplacement. Nous prouvons que l’intersection de la configuration de réseau considérée avec une droite arbitraire est une suite normale dans la base b .

In this paper we extend Champernowne’s construction of normal numbers in base b to the d case and obtain an explicit construction of the generic point of the d shift transformation of the set {0,1,...,b-1} d . We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base b .

DOI : 10.5802/jtnb.523
Levin, Mordechay B. 1 ; Smorodinsky, Meir 2

1 Department of Mathematics Bar-Ilan University 52900, Ramat-Gan, Israel
2 School of Mathematical Sciences Tel Aviv University 69978, Tel-Aviv, Israel
@article{JTNB_2005__17_3_825_0,
     author = {Levin, Mordechay B. and Smorodinsky, Meir},
     title = {On linear normal lattices configurations},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {825--858},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {3},
     year = {2005},
     doi = {10.5802/jtnb.523},
     zbl = {05016590},
     mrnumber = {2212128},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.523/}
}
TY  - JOUR
AU  - Levin, Mordechay B.
AU  - Smorodinsky, Meir
TI  - On linear normal lattices configurations
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2005
SP  - 825
EP  - 858
VL  - 17
IS  - 3
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.523/
DO  - 10.5802/jtnb.523
LA  - en
ID  - JTNB_2005__17_3_825_0
ER  - 
%0 Journal Article
%A Levin, Mordechay B.
%A Smorodinsky, Meir
%T On linear normal lattices configurations
%J Journal de théorie des nombres de Bordeaux
%D 2005
%P 825-858
%V 17
%N 3
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.523/
%R 10.5802/jtnb.523
%G en
%F JTNB_2005__17_3_825_0
Levin, Mordechay B.; Smorodinsky, Meir. On linear normal lattices configurations. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 825-858. doi : 10.5802/jtnb.523. http://archive.numdam.org/articles/10.5802/jtnb.523/

[1] R. Adler, M. Keane, M. Smorodinsky, A construction of a normal number for the continued fraction transformation. Journal of Number Theory 13 (1981), 95–105. | MR | Zbl

[2] D. J. Champernowne, The construction of decimals normal in the scale of ten. J. London Math. Soc. 8 (1933), 254–260. | Zbl

[3] J. Cigler, Asymptotische Verteilung reeller Zahlen mod 1. Monatsh. Math. 64 (1960), 201–225. | MR | Zbl

[4] M. Drmota, R. F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Math, vol. 1651, Springer, 1997. | MR | Zbl

[5] T. Kamae, Subsequences of normal sequences. Israel J. Math. 16 (1973), 121–149. | MR | Zbl

[6] N. M. Korobov, Exponential Sums and their Applications. Kluwer Academic Publishers, Dordrecht, 1992. | MR | Zbl

[7] L. Kuipers , H. Niederreiter, Uniform Distribution of Sequences. Pure and Applied Mathematics, Wiley–Interscience, New York, 1974. | MR | Zbl

[8] P. Kirschenhofer, R.F. Tichy, On uniform distribution of double sequences. Manuscripta Math. 35 (1981), 195–207. | MR | Zbl

[9] M. B. Levin, On normal lattice configurations and simultaneously normal numbers. J. Théor. Nombres Bordeaux 13 (2001), 483–527. | Numdam | MR | Zbl

[10] M. B. Levin, Discrepancy estimate of completely uniform distributed double sequences. In preparation.

[11] M. B. Levin, M. Smorodinsky, A d generalisation of the Davenport–Erdös construction of normal numbers. Colloq. Math. 84/85 (2000), 431–441. | MR | Zbl

[12] M. B. Levin, M. Smorodinsky, Explicit construction of normal lattice configurations. Colloq. Math. 102 (2005), 33–47. | MR | Zbl

[13] M. B. Levin, M. Smorodinsky, On polynomial normal lattice configurations. Monatsh. Math. (2005) | MR | Zbl

[14] A. G. Postnikov, Arithmetic modeling of random processes. Proc. Steklov. Inst. Math. 57 (1960), 84 pp. | MR | Zbl

[15] M. Smorodinsky, B. Weiss, Normal sequences for Markov shifts and intrinsically ergodic subshifts. Israel J. Math. 59 (1987), 225–233. | MR | Zbl

[16] B. Weiss, Normal sequences as collectives. Proc. Symp. on Topological Dynamics and Ergodic Theory, Univ. of Kentucky, 1971.

Cité par Sources :