On normal lattice configurations and simultaneously normal numbers

@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 -