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, 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 1879670 | Zbl 0999.11039

DOI : https://doi.org/10.5802/jtnb.335

@article{JTNB_2001__13_2_483_0,
     author = {Levin, Mordechay B.},
     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/}
}

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. doi : 10.5802/jtnb.335. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.335/

Bibliographie

[C] D.J. Champernowne, The construction of decimal normal in the scale ten. J. London Math. Soc. 8 (1935), 254-260. | JFM 59.0214.01 | Zbl 0007.33701

[Ci] J. Cigler, Asymptotische Verteilung reeller Zahlen mod 1. Monatsh. Math. 64 (1960), 201-225. | EuDML 177097 | MR 121358 | Zbl 0111.25301

[DrTi] M. Drmota, R.F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer-Verlag, Berlin, 1997. | MR 1470456 | Zbl 0877.11043

[Ei] J. Eichenauer-Herrmann, A unified approach to the analysis of compound pseudorandom numbers. Finite Fields Appl. 1 (1995), 102-114. | MR 1334628 | Zbl 0822.11053

[KT] P. Kirschenhofer, R.F. Tichy, On uniform distribution of double sequences, Manuscripta Math. 35 (1981), 195-207. | EuDML 154792 | MR 627933 | Zbl 0478.10036

[Ko1] N.M. Korobov, Numbers with bounded quotient and their applications to questions of Diophantine approximation. Izv. Akad. Nauk SSSR Ser. Mat. 19 (1955), 361-380. | MR 74464 | Zbl 0065.03202

[Ko2] N.M. Korobov, Distribution of fractional parts of exponential function. Vestnic Moskov. Univ. Ser. 1 Mat. Meh. 21 (1966), no. 4, 42-46. | MR 197435 | Zbl 0154.04801

[Ko3] N.M. Korobov, Exponential sums and their applications, Kluwer Academic Publishers, Dordrecht, 1992. | MR 1162539 | Zbl 0754.11022

[KN] L. Kuipers, H. Niedrreiter, Uniform distribution of sequences. John Wiley, New York, 1974. | MR 419394 | Zbl 0281.10001

[Le1] M.B. Levin, On the uniform distribution of the sequence {αλx}. Math. USSR Sbornik 27 (1975), 183-197. | Zbl 0357.10020

[Le2] M.B. Levin, The distribution of fractional parts of the exponential function. Soviet. Math. (Iz. Vuz.) 21 (1977), no. 11, 41-47. | MR 506058 | Zbl 0389.10037

[Le3] M.B. Levin, On the discrepancy estimate of normal numbers. Acta Arith. 88 (1999), 99-111. | MR 1700240 | Zbl 0947.11023

[LS1] M.B. Levin, M. Smorodinsky, Explicit construction of normal lattice configuration, preprint. | MR 2150267

[LS2] M.B. Levin, M. Smorodinsky, A Zd generalization of Davenport and Erdös theorem on normal numbers. Colloq. Math. 84/85 (2000), 431-441. | MR 1784206 | Zbl 1014.11046

[Ni] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, 1992. | MR 1172997 | Zbl 0761.65002

[Ro] K. Roth, On irregularities of distributions. Mathematika 1 (1954), 73-79. | MR 66435 | Zbl 0057.28604