Numerical Verifications of Theoretical Results about the Weighted $({\cal W}(b);\gamma)-$ Diaphony of the Generalized Van der Corput Sequence

Abstract

The weighted $({\cal W}(b);\gamma)-$diaphony is a new quantitative measure for the irregularity of distribution of sequences. In previous works of the authors it has been found the exact order ${\cal O}\left({1 \over N}\right)$ of the weighted $({\cal W}(b);\gamma)-$diaphony of the generalized Van der Corput sequence. Here, we give an upper bound of the weighted $({\cal W}(b);\gamma)-$ diaphony, which is an analogue of the classical Erdös-Turán-Koksma inequality, with respect to this kind of the diaphony. This permits us to make a computational simu-lations of the weighted $({\cal W}(b);\gamma)-$diaphony of the generalized Van der Corput sequence. Different choices of sequences of permutations of the set {0,1, …, b − 1} are practically realized and the $({\cal W}(b);\gamma)-$diaphony of the corresponding generalized Van der Corput sequences is numerically calculated and discussed.