On the (VilB;α)-diaphony of the Van Der Corput sequence constructed in Cantor Systems

Abstract

In the present paper the authors consider the so-called $({\mathrm{Vil}}_{{\mathcal{B}}_s};\alpha ;\gamma )$-diaphony as a suitable tool to investigate sequences constructed in arbitrary Cantor systems. The definition of this kind of the diaphony is based on using Vilenkin function system and depends on two arguments -- a vector $\alpha$ of exponential parameters and a vector $\gamma$ of coordinate weights. This diaphony is used to investigate the distribution of the points of the Van der Corput sequence ${\omega }_B$ constructed in the same $B$-adic Cantor system. In this way a process of synchronization between the technique of a construction of the sequence ${\omega }_B$ and the tool of its studying is realized. Upper and low bounds of the $({\mathrm{Vil}}_B;\alpha )$-diaphony of the sequence ${\omega }_B$ are presented. This permit us to show the influence of the exponential parameter $\alpha$ to the exact order of the $({\mathrm{Vil}}_B;\alpha )$-diaphony of this sequence. When $\alpha =2$ the exact order is $\mathcal{O}\left({\displaystyle \frac{\sqrt{\log N}}{N}}\right)$ and when $\alpha >2$ the exact order is $\mathcal{O}\left({\displaystyle \frac{1}{N}}\right)$.