Fibonacci sets and symmetrization in discrepancy theory
type: Item
instance of: scholarly article
title: Fibonacci sets and symmetrization in discrepancy theory
zbMATH Open document ID: 1257.11074
author: Dmitriy Bilyk
author: V. N. Temlyakov
author: Rui Yu
published in: Journal of Complexity
publication date: 2012-01-10T00:00:00Z
review text: From the authors' abstract: Let \(\{b_n\}\) be the sequence of Fibonacci numbers. The \(b_n\)-point Fibonacci set \(\mathcal F_n\subset[0,1]^2\) is defined as \(\mathcal F_n:=\{(\mu\slash b_n,\{\mu b_{n-1}\slash b_n\}\}_{\mu=1}^{b_n},\) where \(\{x\}\) is the fractional part of a number \(x\in\mathbb R.\) We give a Fourier analytic proof of the fact that the symmetrized Fibonacci set \(\mathcal F_n^\prime=\mathcal F_n\cup\{(p_1,1-p_2):(p_1,p_2)\in\mathcal F_n\}\) has asymptotically minimal \(L_2\) discrepancy. We also introduce \textit{quartered} \(L_p\) discrepancy, which is a modification of the \(L_p\) discrepancy symmetrized with respect to the center of the unit square.
reviewed by: Roman Urban
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 65D30
zbMATH DE Number: 5996042
zbMATH Keywords: discrepancy
zbMATH Keywords: Fibonacci numbers
zbMATH Keywords: cubature formulas
zbMATH Keywords: numerical integration
zbMATH Keywords: Fourier coefficients
MaRDI profile type: Publication
full work available at URL: j.jco.2011.07.001
OpenAlex ID: W2072179573
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DOI: 10.1016/J.JCO.2011.07.001
description: scientific article; zbMATH DE number 5996042