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

Rok, strany: 2015, 75 - 92

Let $γ_q(n)$ 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*} \bl(γ_q(n),γ_q(n+1),γ_q(n+2),γ_q(n+3)\br),   n=1,2,… \end{equation*} Weyl's limit relation is the equality \begin{multline*} \lim_N\to∞((1) / (N))∑_n=0^N-1F\bl(γ_q(n),γ_q(n+1),γ_q(n+2),γ_q(n+3)\br) \nonumber\\[-5pt] = \int\limits₀¹ \int\limits₀¹ \int\limits₀¹ \int\limits₀¹ F(x,y,z,u)\dd_x\dd_y\dd_z\dd_u g(x,y,z,u). \end{multline*} 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∞((1) / (N))∑_n=0^N-1\max\bl(γ_q(n),γ_q(n+1),γ_q(n+2),γ_q(n+3)\br) =((1) / (2))+((3) / (q))-((6) / (q²)) \end{equation*} for the base $q=4,5,6,…$ Also we find an explicit form of $s$th iteration $T^(s)(x)$ of the von Neumann-Kakutani transformation defined by $T\bl(γ_q(n)\br)=γ_q(n+1)$.

ISO 690:

Baláž, V., Hofer, M., Fialová, J., Iacò, M., Strauch, O. 2015. The asymptotic distribution function of the 4-dimensional shifted van der Corput sequence. In Tatra Mountains Mathematical Publications, vol. 64, no.3, pp. 75-92. 1210-3195.

APA:

Baláž, V., Hofer, M., Fialová, J., Iacò, M., Strauch, O. (2015). The asymptotic distribution function of the 4-dimensional shifted van der Corput sequence. Tatra Mountains Mathematical Publications, 64(3), 75-92. 1210-3195.