Quasi-Monte-Carlo methods and the dispersion of point sequences

Quasi-Monte-Carlo methods are well known for solving different problems of numerical analysis such as integration, optimization, etc. The error estimates for global optimization depend on the dispersion of the point sequence with respect to balls. In general, the dispersion of a point set with respect to various classes of range spaces, like balls, squares, triangles, axis-parallel and arbitrary rectangles, spherical caps and slices, is the area of the largest empty range, and it is a measure for the distribution of the points. The main purpose of our paper is to give a survey about this topic, including some folklore results. Furthermore, we prove several properties of the dispersion, generalizing investigations of Niederreiter and others concerning balls. For several well-known uniformly distributed point sets, we estimate the dispersion with respect to triangles, and we also compare them computationally. For the dispersion with respect to spherical slices, we mention an application to the polygonal approximation of curves in space.