Remarks on the descriptive metric characterization of singular sets of analytic functions
type: Item
instance of: scholarly article
title: Remarks on the descriptive metric characterization of singular sets of analytic functions
zbMATH Open document ID: 0917.30019
DOI: 10.1007/BF02316142
author: S. V. Kolesnikov
published in: Mathematical Notes
publication date: 1999-03-09T00:00:00Z
review text: A classical theorem due to A. Plessner states that for a function \(f\) meromorphic in the unit disk \(D\), the unit circle \(\Gamma\) can be expressed as the union \(\Gamma=I(f)\cup F(f)\cup E(f)\), where \(I(f)\) is the collection of points \(\zeta\) at which the cluster set of \(f\) in each Stolz angle at \(\zeta\) is the full Riemannian sphere, \(F(f)\) is the set of points in \(\Gamma\) at which \(f\) has an angular limit, and \(E(f)\) is a set of linear measure zero in \(\Gamma\). \textit{P. Lappan} [Bull. Lond. Math. Soc. 2, 60-62 (1970; Zbl 0197.35505)] has shown that the set \(I(f)\) is a \(G_\delta\) set and that each \(G_\delta\) subset of \(\Gamma\) is precisely the set \(I(f)\) for an appropriate meromorphic function \(f\). Here, the author proves that for three disjoint subsets \(E_1\), \(E_2\), and \(E_3\) of \(\Gamma\) such that \(\Gamma=E_1\cup E_2\cup E_3\), in order for there to exist a function \(f\) meromorphic in \(D\) such that \(E_1=I(f)\), \(E_2=F(f)\), and \(E_3=E(f)\), it is necessary and sufficient that both \(E_1\) is a \(G_\delta\) set and \(E_3\) is a \(G_{\delta\sigma}\) set of linear measure zero. In addition, the author proves a more specialized result: For any set \(E\subset\Gamma\), where \(E\) is a \(G_\delta\) set of logarithmic capacity zero, there exists a bounded function \(f\) analytic and one sheeted on the disk \(D\), such that \(E=E(f)\).
reviewed by: Peter Lappan
Mathematics Subject Classification ID: 30D40
zbMATH DE Number: 1226022
zbMATH Keywords: Stolz angle
MaRDI profile type: Publication
cites work: A Characterization of Plessner Points
cites work: Q3244558
cites work: ON SETS OF NONEXISTENCE OF RADIAL LIMITS OF BOUNDED ANALYTIC FUNCTIONS
cites work: Ensembles exceptionnels
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description: scientific article; zbMATH DE number 1226022