Let $\gamma_q(n)$ be the van der Corput sequence in the base $q$and $g(x,y,z,u)$ be an asymptotic distribution function of the$4$-dimensional sequence\begin{equation*}(\gamma_q(n),\gamma_q(n+1),\gamma_q(n+2),\gamma_q(n+3)),\quad n=1,2,\dots.\end{equation*}Weyl's limit relation is the limit\begin{align*}\lim_{N\to\infty}&\frac{1}{N}\sum_{n=0}^{N-1}F(\gamma_q(n),\gamma_q(n+1),\gamma_q(n+2),\gamma_q(n+3))\nonumber\\&=\int_0^1\int_0^1\int_0^1\int_0^1F(x,y,z,u)\dd_x\dd_y\dd_z\dd_u g(x,y,z,u).\end{align*}In this paper we find an explicit formula for $g(x,x,x,x)$ and thenas an example we find the limit\begin{equation*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\max(\gamma_q(n),\gamma_q(n+1),\gamma_q(n+2),\gamma_q(n+3))=\frac{1}{2}+\frac{3}{q}-\frac{6}{q^2}\end{equation*}for the base $q=4,5,6,\dots$.Also we find an explicit form of $s$-th iteration $T^{(s)}(x)$ of thevon Neumann-Kakutani transformation defined by $T(\gamma_q(n))=\gamma_q(n+1)$.
@article{389,
title = {An asymptotic distribution function of the 4-dimensional shifted van der Corput sequence},
journal = {Tatra Mountains Mathematical Publications},
volume = {62},
year = {2015},
doi = {10.2478/tatra.v64i0.389},
language = {EN},
url = {http://dml.mathdoc.fr/item/389}
}
Baláž, Vladimír; Fialová, Jana; Hofer, Markus; Iacò, M. R.; Strauch, Oto. An asymptotic distribution function of the 4-dimensional shifted van der Corput sequence. Tatra Mountains Mathematical Publications, Tome 62 (2015) . doi : 10.2478/tatra.v64i0.389. http://gdmltest.u-ga.fr/item/389/