An asymptotic distribution function of the 4-dimensional shifted van der Corput sequence

Abstract

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 then
as 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 the
von Neumann-Kakutani transformation defined by $T(\gamma_q(n))=\gamma_q(n+1)$.