On the (Vil B 2; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

Abstract

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