Well dispersed sequences in \([0,1]^{d}\)
type: Item
instance of: scholarly article
title: Well dispersed sequences in \([0,1]^{d}\)
zbMATH Open document ID: 1404.11099
published in: Journal of Number Theory
publication date: 2018-05-02T00:00:00Z
review text: For a sequence \(\overline{z}=(z_1,z_2,\ldots)\) of points in \([0,1]^d\), define the dispersion measure \(\mu_d(\overline{z})=\inf_{n\geq 1}\inf_{m\geq 1} n^{1/d}\|z_m-z_{m+n}\|\) and \(\alpha_d=\sup_{\overline{z}} \mu_d(\overline{z})\), where \(\|\cdot\|\) denotes the \(L_1\) norm and \(\overline{z}\) ranges over all sequences. In the present paper, the authors show that \(\alpha_d\geq ((2^{d-1}(2^d-1))^{1/d}(1+\sum_{k\geq 1}\frac 1{F_{2k}}))^{-1}\) for \(d\geq 1\), where \(F_n\) denotes the \(n\)-th Fibonacci number defined by \(F_0=0\) and \(F_1=1\). They also obtain upper bounds on \(\alpha_d\), for example \(\alpha_2\leq \frac{2}{1/\sqrt{2}+1/\sqrt{3}}\).
reviewed by: Yukio Ohkubo
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11K36
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 6865875
zbMATH Keywords: distribution of sequences
zbMATH Keywords: dispersion
author: Fan Chung
author: Ron Graham
MaRDI profile type: Publication
full work available at URL: j.jnt.2018.02.013
OpenAlex ID: W2794259272
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DOI: 10.1016/J.JNT.2018.02.013
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description: scientific article; zbMATH DE number 6865875