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We extend the Davenport and Erdős construction of normal numbers to the ${\mathbb{Z}}^d$ case.

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).

We prove the central limit theorem for the multisequence ${\sum }_{1\le n₁\le N₁}\cdots {\sum }_{1\le n_d\le N_d}{a}_{n₁,...,n_d}cos(⟨2\pi m,A{₁}^{n₁}...A_d^{n_d}x⟩)$ where $m\in {\mathbb{Z}}^s$, $a_{n₁,...,n_d}$ are reals, $A₁,...,A_d$ 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.

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

We extend Champernowne’s construction of normal numbers to base b to the ${\mathbb{Z}}^d$ case and obtain an explicit construction of a generic point of the ${\mathbb{Z}}^d$ shift transformation of the set ${0,1,...,b-1}^{{\mathbb{Z}}^d}$.