Generalized discrepancies on the sphere

Abstract

Generalized discrepancies are a class of discrepancies introduced in the seminal paper [1] to measure uniformity of points over the unit sphere in R 3 . However, convergence to 0 of this quantity has been shown only in the case of spherical t−designs. In the following, we completely characterize sequences for which convergence to 0 of D (PN ; A) holds. The interest of this result is that, when evaluating uniformity on the sphere, generalized discrepancies are much simpler to compute than the well-known spherical cap discrepancy.