On normal lattice configurations and simultaneously normal numbers

Mordechay B. Levin

Mordechay B. Levin

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 483-527.

Résumé
Abstract

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).

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).

MR Zbl

@article{JTNB_2001__13_2_483_0,
     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},
     zbl = {0999.11039},
     mrnumber = {1879670},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2001__13_2_483_0/}
}

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Mordechay B. Levin. On normal lattice configurations and simultaneously normal numbers. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 483-527. https://jtnb.centre-mersenne.org/item/JTNB_2001__13_2_483_0/

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