Irregularities of distribution. I
type: Item
instance of: scholarly article
title: Irregularities of distribution. I
zbMATH Open document ID: 0631.10034
DOI: 10.1007/BF02392553
author: József Beck
published in: Acta Mathematica
publication date: 1987-01-01T00:00:00Z
review text: Several important new results in the theory of irregularities of point distributions are proved. Let \(\mathcal P=\{z_1,\dots,z_n\}\) be a distribution of \(n\) points in the \(K\)-dimensional unit cube \(U^K\), let \(A\) be a measurable set in \(U^K\) and define the discrepancy by \((\mu\) is the Lebesgue measure) \[ D[\mathcal P;A]=\sum_{x\in\mathcal P\cap A}1-n\mu(A). \] The main theorem says that for every \(\varepsilon >0\) there exists a ball \(A\) in \(U^K\) such that \[ | D[\mathcal P;A] | > c\cdot n^{1/2-1/2k-\varepsilon} \] with a constant \(c=c(K,\varepsilon)>0\). This result improves a well-known result of W. M. Schmidt (where the exponent \(K-1/2K(K+2)-\varepsilon\) was obtained instead of \(1/2-1/2k-\varepsilon\). For a detailed survey on irregularities of distributions we refer to the monographs of \textit{W. M. Schmidt} [Lectures on irregularities of distribution. Bombay: Tata Inst. Fundamental Research (1977; Zbl 0434.10031)] and of the author and \textit{W. W. L. Chen} [Irregularities of distributions. Cambridge etc.: Cambridge University Press (1987; Zbl 0617.10039)]. Furthermore the author proves lower bounds for the discrepancy with respect to convex bodies where rotations are allowed. This rotation discrepancy is always large and behaves like the square root of the surface area. Probabilistic arguments show that this lower bound is nearly best possible. The proofs of the theorems are based on an ingenious application of the Fourier transformation method. \{Part II, cf. Proc. Lond. Math. Soc., III. Ser. 56, No. 1, 1--50 (1988; Zbl 0644.10038)\}.
reviewed by: Robert F. Tichy
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11K06
zbMATH DE Number: 4025531
zbMATH Keywords: irregularities of point distributions
zbMATH Keywords: discrepancy
zbMATH Keywords: lower bounds
zbMATH Keywords: rotation discrepancy
zbMATH Keywords: Fourier transformation method
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02392553
OpenAlex ID: W2342092426
description: scientific article; zbMATH DE number 4025531