On the (VilB2; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

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,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } 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,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } is shown. If α₁ = α₂, then the following holds: if 1 < α₂ < 2 the exact order is 𝒪 (logNN1-ε{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}) for some ε > 0, if α₂ = 2 the exact order is 𝒪 (logNN{{\sqrt {\log N} } \over N}) and if α₂ > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. If α₁ > α₂, then the following holds: if 1 < α₂ < 2 the exact order is 𝒪 (logNN1-ε{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}) for some ε > 0, if α₂ = 2 the exact order is 𝒪 (1N{1 \over N}) and if α₂ > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. Here N = B_ν, where B_ν denotes the number of the points of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu }.