On normal lattice configurations and simultaneously normal numbers

Mordechay B. Levin

doi:10.5802/jtnb.335

Mordechay B. Levin

Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 483-527.

Abstract
Résumé

Let $q, q_{1}, \dots, q_{s} \geq 2$ be integers, and let $α_{1}, α_{2}, \dots$ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence ${(\{α_{m} q^{n}\}, \dots, \{α_{m + s - 1} q^{n}\})}_{m, n = 1}^{M N}$ coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ${(x n)}_{n = 1}^{M N}$ in $s$ -dimensional unit cube $(s, M, N = 1, 2, \dots)$ . We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ${(\{α_{1} q_{1}^{n}\}, \dots, \{α_{s} q_{s}^{n}\})}_{n = 1}^{N}$ (Korobov’s problem).

Soient $q, q_{1}, \dots, q_{s} \geq 2$ des entiers et $α_{1}, α_{2}, \dots$ des nombres réels. Dans cet article, on montre que la borne inférieure de la discrépance de la suite double ${(\{α_{m} q^{n}\}, \dots, \{α_{m + s - 1} q^{n}\})}_{m, n = 1}^{M N}$ coïncide (à un facteur logarithmique près) avec la borne inférieure de la discrépance des suites ordinaires ${(x n)}_{n = 1}^{M N}$ dans un cube de dimension $(s, M, N = 1, 2, \dots)$ . Nous calculons aussi une borne inférieure de la discrépance (à un facteur logarithmique près) de la suite ${(\{α_{1} q_{1}^{n}\}, \dots, \{α_{s} q_{s}^{n}\})}_{n = 1}^{N}$ (problème de Korobov).

MR: 1879670 | Zbl: 0999.11039

DOI: 10.5802/jtnb.335

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     author = {Mordechay B. Levin},
     title = {On normal lattice configurations and simultaneously normal numbers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {483--527},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {2},
     year = {2001},
     doi = {10.5802/jtnb.335},
     zbl = {0999.11039},
     mrnumber = {1879670},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.335/}
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Mordechay B. Levin. On normal lattice configurations and simultaneously normal numbers. Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 483-527. doi : 10.5802/jtnb.335. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.335/

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