On the (VilB2;α;γ)-diaphony of the nets of type of Zaremba-Halton constructed in generalized number system
Journal
Uniform distribution theory,The Journal of Slovak Academy of Sciences
Date Issued
2020
Author(s)
Vasil Grozdanov
Tsvetelina Petrova
DOI
10.2478
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.
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.
Subjects
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