On normal lattice configurations and simultaneously normal numbers

Mordechay B. Levin

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 2, page 483-527
  • ISSN: 1246-7405

Abstract

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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 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 , ) . 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).

How to cite

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Levin, Mordechay B.. "On normal lattice configurations and simultaneously normal numbers." Journal de théorie des nombres de Bordeaux 13.2 (2001): 483-527. <http://eudml.org/doc/248700>.

@article{Levin2001,
abstract = {Let $q, q_1, \dots , q_s \ge 2$ be integers, and let $\alpha _ 1, \alpha _2, \dots $ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence\begin\{equation*\} (\left\lbrace \alpha \_mq^n\right\rbrace , \dots ,\left\lbrace \alpha \_\{m + s - 1\}q^n\right\rbrace )^\{MN\}\_\{m,n=1\} \end\{equation*\}coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences $(xn)^\{MN\}_\{n=1\}$ 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 $(\left\lbrace \alpha _1 q^n_1\right\rbrace , \dots , \left\lbrace \alpha _s q^n_s\right\rbrace )^N_\{n=1\}$ (Korobov’s problem).},
author = {Levin, Mordechay B.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {normal numbers; two-dimensional lattice configuration; -dimensional discrepancy},
language = {eng},
number = {2},
pages = {483-527},
publisher = {Université Bordeaux I},
title = {On normal lattice configurations and simultaneously normal numbers},
url = {http://eudml.org/doc/248700},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Levin, Mordechay B.
TI - On normal lattice configurations and simultaneously normal numbers
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 2
SP - 483
EP - 527
AB - Let $q, q_1, \dots , q_s \ge 2$ be integers, and let $\alpha _ 1, \alpha _2, \dots $ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence\begin{equation*} (\left\lbrace \alpha _mq^n\right\rbrace , \dots ,\left\lbrace \alpha _{m + s - 1}q^n\right\rbrace )^{MN}_{m,n=1} \end{equation*}coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences $(xn)^{MN}_{n=1}$ 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 $(\left\lbrace \alpha _1 q^n_1\right\rbrace , \dots , \left\lbrace \alpha _s q^n_s\right\rbrace )^N_{n=1}$ (Korobov’s problem).
LA - eng
KW - normal numbers; two-dimensional lattice configuration; -dimensional discrepancy
UR - http://eudml.org/doc/248700
ER -

References

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  7. [Ko2] N.M. Korobov, Distribution of fractional parts of exponential function. Vestnic Moskov. Univ. Ser. 1 Mat. Meh.21 (1966), no. 4, 42-46. Zbl0154.04801MR197435
  8. [Ko3] N.M. Korobov, Exponential sums and their applications, Kluwer Academic Publishers, Dordrecht, 1992. Zbl0754.11022MR1162539
  9. [KN] L. Kuipers, H. Niedrreiter, Uniform distribution of sequences. John Wiley, New York, 1974. Zbl0281.10001MR419394
  10. [Le1] M.B. Levin, On the uniform distribution of the sequence {αλx}. Math. USSR Sbornik27 (1975), 183-197. Zbl0357.10020
  11. [Le2] M.B. Levin, The distribution of fractional parts of the exponential function. Soviet. Math. (Iz. Vuz.)21 (1977), no. 11, 41-47. Zbl0389.10037MR506058
  12. [Le3] M.B. Levin, On the discrepancy estimate of normal numbers. Acta Arith.88 (1999), 99-111. Zbl0947.11023MR1700240
  13. [LS1] M.B. Levin, M. Smorodinsky, Explicit construction of normal lattice configuration, preprint. MR2150267
  14. [LS2] M.B. Levin, M. Smorodinsky, A Zd generalization of Davenport and Erdös theorem on normal numbers. Colloq. Math.84/85 (2000), 431-441. Zbl1014.11046MR1784206
  15. [Ni] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, 1992. Zbl0761.65002MR1172997
  16. [Ro] K. Roth, On irregularities of distributions. Mathematika1 (1954), 73-79. Zbl0057.28604MR66435

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