Smooth fixed volume discrepancy, dispersion, and related problems
Let \(T\) be a set of \(n\) points in \([0,1)^d\). The dispersion of \(T\) is the volume of the largest empty axis-parallel box (i.e. the largest such set not containing a point of \(T\)) which can be inscribed in \([0,1)^d\). In the context of numerical integration, upper bounds for the minimal dispersion of any \(d\)-dimensional point set of fixed cardinality \(n\), denoted as \(\mathrm{disp}^{\ast}(n,d)\), are of particular interest. \par It is known that \(c/n \leq \mathrm{disp}^{\ast}(n,d) \leq C^{\ast}(d)/n\), for an absolute constant \(c\) with \(c<C^{\ast}(d)\) [\textit{C. Aistleitner} et al., Discrete Appl. Math. 230, 146--150 (2017; Zbl 1373.51002); \textit{A. Dumitrescu} and \textit{M. Jiang}, Algorithmica 66, No. 2, 225--248 (2013; Zbl 1262.68186); \textit{G. Rote} and \textit{R. F. Tichy}, Math. Comput. Modelling 23, No. 8--9, 9--23 (1996; Zbl 0855.11041)]. The author shows that Fibonacci and Frolov point sets have optimal rate of decay of dispersion with respect to the cardinality \(n\) of sets. The new concept of smooth fixed volume discrepancy is introduced which can be used to derive the upper bounds on the dispersion. Finally, it is also proven that the Fibonacci point sets provide the universal discretization of all integral norms.
scientific article; zbMATH DE number 6966596
scientific article