On normal lattice configurations and simultaneously normal numbers
Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 483-527.

Let q,q 1 ,,q s 2 be integers, and let α 1 ,α 2 , 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 ,,α m+s-1 q n ) m,n=1 MN coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences (xn) n=1 MN in s-dimensional unit cube (s,M,N=1,2,). We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence (α 1 q 1 n ,,α s q s n ) n=1 N (Korobov’s problem).

Soient q,q 1 ,,q s 2 des entiers et α 1 ,α 2 , des nombres réels. Dans cet article, on montre que la borne inférieure de la discrépance de la suite double (α m q n ,,α m+s-1 q n ) m,n=1 MN coïncide (à un facteur logarithmique près) avec la borne inférieure de la discrépance des suites ordinaires (xn) n=1 MN dans un cube de dimension (s,M,N=1,2,). Nous calculons aussi une borne inférieure de la discrépance (à un facteur logarithmique près) de la suite (α 1 q 1 n ,,α s q s n ) n=1 N (problème de Korobov).

@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},
     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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