On linear normal lattices configurations

Levin, Mordechay B.; Smorodinsky, Meir

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 ${\mathbb{Z}}^d$, et obtenons une construction explicite du point générique de la transformation de l’ensemble ${\{0,1,...,b-1\}}^{{\mathbb{Z}}^d}$ par ${\mathbb{Z}}^d$ 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 ${\mathbb{Z}}^d$ case and obtain an explicit construction of the generic point of the ${\mathbb{Z}}^d$ shift transformation of the set ${\{0,1,...,b-1\}}^{{\mathbb{Z}}^d}$. We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base $b$ .

[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 ${\mathbb{Z}}^d$ 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.