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On normal lattice configurations and simultaneously normal numbers

Mordechay B. Levin — 2001

Journal de théorie des nombres de Bordeaux

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).

A multiparameter variant of the Salem-Zygmund central limit theorem on lacunary trigonometric series

Mordechay B. Levin — 2013

Colloquium Mathematicae

We prove the central limit theorem for the multisequence 1 n N 1 n d N d a n , . . . , n d c o s ( 2 π m , A n . . . A d n d x ) where m s , a n , . . . , n d are reals, A , . . . , A d are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in [ 0 , 1 ] s . The main tool is the S-unit theorem.

On linear normal lattices configurations

Mordechay B. LevinMeir Smorodinsky — 2005

Journal de Théorie des Nombres de Bordeaux

In this paper we extend Champernowne’s construction of normal numbers in base b to the d case and obtain an explicit construction of the generic point of the d shift transformation of the set { 0 , 1 , . . . , b - 1 } d . We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base b .

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