On the (Vil;;)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System
Journal
Uniform distribution theory
Date Issued
2020-06-01
Author(s)
Grozdanov, Vassil
Petrova, Tsvetelina
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.
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.
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