On normal lattice configurations and simultaneously normal numbers

Levin, Mordechay B.

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

Soient $q,q_1,\cdots ,q_s\ge 2$ des entiers et ${\alpha }_1,{\alpha }_2,\cdots$ des nombres réels. Dans cet article, on montre que la borne inférieure de la discrépance de la suite double

$${(\left\{{\alpha }_m{q}^n\right\},\cdots ,\left\{{\alpha }_{m+s-1}{q}^n\right\})}_{m,n=1}^{MN}$$

coïncide (à un facteur logarithmique près) avec la borne inférieure de la discrépance des suites ordinaires ${\left(xn\right)}_{n=1}^{MN}$ dans un cube de dimension $(s,M,N=1,2,\cdots )$. Nous calculons aussi une borne inférieure de la discrépance (à un facteur logarithmique près) de la suite ${(\left\{{\alpha }_1{q}_1^n\right\},\cdots ,\left\{{\alpha }_s{q}_s^n\right\})}_{n=1}^N$ (problème de Korobov).

Let $q,q_1,\cdots ,q_s\ge 2$ be integers, and let ${\alpha }_1,{\alpha }_2,\cdots$ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence

$${(\left\{{\alpha }_m{q}^n\right\},\cdots ,\left\{{\alpha }_{m+s-1}{q}^n\right\})}_{m,n=1}^{MN}$$

coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ${\left(xn\right)}_{n=1}^{MN}$ in $s$-dimensional unit cube $(s,M,N=1,2,\cdots )$. We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ${(\left\{{\alpha }_1{q}_1^n\right\},\cdots ,\left\{{\alpha }_s{q}_s^n\right\})}_{n=1}^N$ (Korobov’s problem).

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