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 .

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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://www.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 602450 | Zbl 0448.10050

[2] D. J. Champernowne, The construction of decimals normal in the scale of ten. J. London Math. Soc. 8 (1933), 254–260. | Zbl 0007.33701

[3] J. Cigler, Asymptotische Verteilung reeller Zahlen mod 1. Monatsh. Math. 64 (1960), 201–225. | MR 121358 | Zbl 0111.25301

[4] M. Drmota, R. F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Math, vol. 1651, Springer, 1997. | MR 1470456 | Zbl 0877.11043

[5] T. Kamae, Subsequences of normal sequences. Israel J. Math. 16 (1973), 121–149. | MR 338321 | Zbl 0272.28012

[6] N. M. Korobov, Exponential Sums and their Applications. Kluwer Academic Publishers, Dordrecht, 1992. | MR 1162539 | Zbl 0754.11022

[7] L. Kuipers , H. Niederreiter, Uniform Distribution of Sequences. Pure and Applied Mathematics, Wiley–Interscience, New York, 1974. | MR 419394 | Zbl 0281.10001

[8] P. Kirschenhofer, R.F. Tichy, On uniform distribution of double sequences. Manuscripta Math. 35 (1981), 195–207. | MR 627933 | Zbl 0478.10036

[9] M. B. Levin, On normal lattice configurations and simultaneously normal numbers. J. Théor. Nombres Bordeaux 13 (2001), 483–527. | Numdam | MR 1879670 | Zbl 0999.11039

[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 1784206 | Zbl 1014.11046

[12] M. B. Levin, M. Smorodinsky, Explicit construction of normal lattice configurations. Colloq. Math. 102 (2005), 33–47. | MR 2150267 | Zbl 1080.11057

[13] M. B. Levin, M. Smorodinsky, On polynomial normal lattice configurations. Monatsh. Math. (2005) | MR 2216557 | Zbl 05036002

[14] A. G. Postnikov, Arithmetic modeling of random processes. Proc. Steklov. Inst. Math. 57 (1960), 84 pp. | MR 148639 | Zbl 0106.12101

[15] M. Smorodinsky, B. Weiss, Normal sequences for Markov shifts and intrinsically ergodic subshifts. Israel J. Math. 59 (1987), 225–233. | MR 920084 | Zbl 0643.10041

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

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