On the (Vil;;)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

Abstract

In the present paper the so-called (VilBs ; α; γ)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of two-dimensional nets Zκ,μ B2,ν of type of Zaremba-Halton constructed in a generalized B2-adic system or Cantor system is introduced and the (VilB2 ; α; γ)-diaphony of these nets is studied. The influence of the vector α = (α1, α2) of exponential parameters to the exact order of the (VilB2 ; α; γ)-diaphony of the nets Zκ,μ B2,ν is shown. If α1 = α2, then the following holds: if 1 < α2 < 2 the exact order is O √log N N1−ε for some ε > 0, if α2 = 2 the exact order is O √log N N and if α2 > 2 the exact order is O √log N N1+ε

for some ε > 0. If α1 > α2, then the following holds: if 1 < α2 < 2 the exact order is O 1 N1−ε for some ε > 0, if α2 = 2 the exact order is O 1 N and if α2 > 2 the exact order is O 1 N1+ε for some ε > 0. Here N = Bν , where Bν denotes the number of the points of the nets Zκ,μ B2,ν.