On the (VilB2;α;γ)-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/(N^(1−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/N^(1+ε)) for some ε > 0. If α1 > α2, then the following holds: if 1 < α2 < 2 the exact order is O(1/N^(1−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/N^(1+ε)) for some ε > 0. Here N = Bν , where Bν denotes the number of the points of the nets Zκ,μ,B2,ν.