A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets
type: Item
instance of: scholarly article
title: A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets
zbMATH Open document ID: 1150.11026
author: Peter Kritzer
author: Gerhard Larcher
author: Friedrich Pillichshammer
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2007-08-07T00:00:00Z
review text: If \(\omega=\{{\mathbf x}_n=(x_n,y_n)\), \(0\leq n\leq 2^m-1\}\) is a Hammersley net (\(x_n=n/2^m\), \(y_n=\sum_{i=0}^{m-1} a_i/2^{i+1}\), \(n=\sum_{i=0}^{m-1}a_i2^i\)), then a \(\sigma\)-shifted Hammersley net with \(\sigma=(\sigma_1,\dots,\sigma_m)^{m-1}\in{\mathbb Z}_2^m\) is the net \(\omega_\sigma=\{{\mathbf x}_n=(x_n,y_n)\), \(0\leq n\leq 2^m-1\}\) with \(y_n=\sum_{i=1}^m (a_i\oplus \sigma_{i+1})/2^i\), where \(\oplus\) is addition modulo 2. The shifted Hammersley nets are introduced because the digital shifts can improve the distribution properties of the classical Hammersley nets. The authors focus their investigation on the leading term in the estimations of the star discrepancy \(D^*\). For instance, while the leading term of the upper estimate of \(2^mD^*_{2^m}(\omega)\) is \(m/3\), there is a shifted Hammersley net \(\omega_\sigma\) with \(2^mD^*_{2^m}(\omega_\sigma)\leq m/6+O(1)\). A detailed investigation of the star discrepancy of shifted Hammersley nets contained in the paper shows that for the star discrepancy of any \(\sigma\)-shifted Hammersley net \(\omega_\sigma\) we have \(2^mD^*_{2^m}(\omega)\geq m/6-4/9\) and that the value of \(2^mD^*_{2^m}(\omega)\) is influenced by the number of ones in \(\sigma\) and by the number of changes from ones to zeros in \(\sigma\). The estimates of the star discrepancy of shifted Hammerley nets depend on certain sums of distances to the nearest integer, which are also studied in the paper. Finally, some interesting results concerning the star discrepancy of shifted van der Corput sequences are proved in the last section of the paper.
reviewed by: Štefan Porubský
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11K31
Mathematics Subject Classification ID: 11K06
zbMATH DE Number: 5177471
zbMATH Keywords: low discrepancy sequence
zbMATH Keywords: Hammersley point set
zbMATH Keywords: van der Corput point set
zbMATH Keywords: distance to the nearest integer
zbMATH Keywords: Hammersley sequence (net)
zbMATH Keywords: star discrepancy
MaRDI profile type: Publication
full work available at URL: s10231-006-0002-5
OpenAlex ID: W2078669974
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DOI: 10.1007/S10231-006-0002-5
description: scientific article; zbMATH DE number 5177471