A method for exact calculation of the discrepancy of low-dimensional finite point sets. I
type: Item
instance of: scholarly article
title: A method for exact calculation of the discrepancy of low-dimensional finite point sets. I
zbMATH Open document ID: 0789.11041
DOI: 10.1007/BF02941337
author: Peter Bundschuh
author: Yaochen Zhu
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1994-03-17T00:00:00Z
review text: The discrepancy of a point sequence \((u_ k)\), \(1 \leq k \leq n\) in \([0,1)^ d\) is defined by \[ D_ n=\sup_ J \bigl| {1\over n} \text{ card} \{k:u_ k \in J\}-\text{vol} J \bigr|, \] where the supremum is extended over all axis parallel intervals \(J\). For \(d=1\) an explicit formula is due to \textit{H. Niederreiter} [Appl. Number Theory Numer. Anal., Proc. Symp. Univ. Montreal 1971, 203-236 (1972; Zbl 0248.10025)]: \[ D_ n={1 \over 2n}+\max_{1 \leq i \leq n} \left( u_ i- \left| {2i-1 \over 2n} \right| \right). \] In the present paper the authors obtain an interesting generalization for \(d=2,3\). The expression for \(D_ n\) is a three-fold maximum only containing terms of the sequence. In a subsequent paper a generalization to arbitrary dimensions will be obtained.
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 11J71
zbMATH DE Number: 492588
zbMATH Keywords: low-dimensional finite point sets
zbMATH Keywords: discrepancy
zbMATH Keywords: explicit formula
reviewed by: Robert F. Tichy
MaRDI profile type: Publication
cites work: A method for exact calculation of the stardiscrepancy of plane sets applied to the sequences of Hammersley
cites work: Applications of Number Theory to Numerical Analysis
cites work: Q4767350
cites work: Discrepancy and convex programming
cites work: Quasi-Monte Carlo methods and pseudo-random numbers
cites work: A quasirandom approach to integration in Bayesian statistics
cites work: Number theoretic method in applied statistics
full work available at URL: https://doi.org/10.1007/bf02941337
OpenAlex ID: W1973676116
description: scientific article; zbMATH DE number 492588