On the $(\mathrm{Vil}_B;\alpha)$-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 $\mathcal{O}\left(\dfrac{\sqrt{\log N}}{N}\right)$ and when $\alpha>2$ the exact order is $\mathcal{O}\left(\dfrac1N\right)$.