June, 2020

27-50

Vesna Dimitrievska Ristovska, Vassil Grozdanov and Tsvetelina Petrova

In the present paper the so-called (Vil_Bs; α; γ)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of two-dimensional nets ZB2,νκ,μ of type of Zaremba-Halton constructed in a generalized B₂-adic system or Cantor system is introduced and the (Vil_B2; 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 (Vil_B2; α; γ)-diaphony of the nets ZB2,νκ,μ is shown. If α₁ = α₂, then the following holds: if 1 < α₂ < 2 the exact order is 𝒪 (logNN1-ε) for some ε > 0, if α₂ = 2 the exact order is 𝒪 (logNN) and if α₂ > 2 the exact order is 𝒪 (logNN1+ε) for some ε > 0. If α₁ > α₂, then the following holds: if 1 < α₂ < 2 the exact order is 𝒪 (logNN1-ε) for some ε > 0, if α₂ = 2 the exact order is 𝒪 (1N) and if α₂ > 2 the exact order is 𝒪 (logNN1+ε) for some ε > 0. Here N = B_ν, where B_ν denotes the number of the points of the nets ZB2,νκ,μ.

Ristovska, V.D., Grozdanov, V. & Petrova, T. (2020). On the (VilB2; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System. Uniform distribution theory, 15(1), 27-50. https://doi.org/10.2478/udt-2020-0002