A metric result for special sequences related to the Halton sequences
type: Item
instance of: scholarly article
title: A metric result for special sequences related to the Halton sequences
zbMATH Open document ID: 1446.11142
author: Roswitha Hofer
published in: Journal of Complexity
publication date: 2017-11-21T00:00:00Z
review text: The Halton sequence \((\mathbf{y}_n)_{n\geq 0}\) is a well-known and important example of a low discrepancy sequence. It is defined via the so-called radical inverse function \(\phi_b\) which mirrors the \(b\)-adic expansion of an integer \(n\) at the decimal point and gives a number in the unit interval \([0,1)\). The Halton sequence in pairwise coprime integer bases \(b_1, \ldots, b_d \geq 2\) is a \(d\)-dimensional sequence of points in \([0,1)^d\) defined by \[ \mathbf{y}_n = \left( \phi_{b_1}(n), \ldots, \phi_{b_d}(n)\right). \] The author of the paper under review studies related sequences of the form \[ \mathbf{x}_n = \left( \phi_{b_1}(\lfloor n \beta \rfloor ), \ldots, \phi_{b_d}(\lfloor n \beta \rfloor)\right), \] for nonzero \(\beta \in \mathbb{R}\) and pairwise coprime integers \(b_1, \ldots, b_d \geq 2\). The main result of the paper (Theorem 1) provides an upper bound on the discrepancy of such sequences for almost all \(\beta \in \mathbb{R}\) (in the sense of Lebesgue measure) which is close to the upper bound satisfied by low discrepancy sequences.
reviewed by: Florian Pausinger
Mathematics Subject Classification ID: 11K31
Mathematics Subject Classification ID: 11K36
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 65C10
zbMATH DE Number: 6810591
zbMATH Keywords: Halton sequences
zbMATH Keywords: subsequences
zbMATH Keywords: discrepancy
MaRDI profile type: Publication
full work available at URL: j.jco.2017.09.001
OpenAlex ID: W2754418962
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DOI: 10.1016/J.JCO.2017.09.001
description: scientific article; zbMATH DE number 6810591