Dyadic diaphony of digital nets over ℤ₂

Abstract

The dyadic diaphony, introduced by Hellekalek and Leeb, is a quantitative measure for the irregularity of distribution of point sets in the unit-cube. In this paper we study the dyadic diaphony of digital nets over ℤ 2. We prove an upper bound for the dyadic diaphony of nets and show that the convergence order is best possible. This follows from a relation between the dyadic diaphony and the ℒ discrepancy. In order to investigate the case where the number of points is small compared to the dimension we introduce the limiting dyadic diaphony, which is defined as the limiting case where the dimension tends to infinity. We obtain a tight upper and lower bound and we compare this result with the limiting dyadic diaphony of a random sample. © Springer-Verlag 2005.