On the analytic capacity and curvature of some Cantor sets with non-\(\sigma\)-finite length
type: Item
instance of: scholarly article
title: On the analytic capacity and curvature of some Cantor sets with non-\(\sigma\)-finite length
zbMATH Open document ID: 0888.30026
DOI: 10.5565/PUBLMAT_40196_12
published in: Publicacions Matemàtiques
publication date: 1996-07-31T00:00:00Z
full work available at URL: https://eudml.org/doc/41245
review text: Let \(\gamma(E)\) and \(c(E)\) be the analytic and Newtonian, respectively, capacities of a compact set \(E\) in \(\mathbb{C}\), and let \(\Lambda_h\) be the Hausdorff measure generated by an increasing function \(h:[0,\infty)\to [0,\infty)\), and set \(h_1(r)=r\). The standard classical results in this subject are: \(c(E)>0\) implies \(\gamma(E)>0\), and \(\Lambda_{h_1}(E)=0\) implies \(\gamma(E)=0\). In this lecture notes [Analytic capacity and measure (1972; Zbl 0253.30014)] \textit{J. Garnett} introduced certain Cantor sets \(E(\lambda)\) in \(\mathbb{C}\) which are associated with decreasing sequences \(\lambda=(\lambda_n)\), \(0<\lambda_n<\frac12\) \((n\in\mathbb{N})\), such that \(0<\Lambda_h(E(\lambda))< \infty\), where \(h=h_{(\lambda)}\), and \(h_{(\lambda)}: [0,\infty)\to [0,\infty)\) is an increasing function with \[ h_{(\lambda)} (\lambda_1\cdots \lambda_n)= 4^{-n} \qquad (n\in\mathbb{N}). \] It was claimed there that \(\gamma(E(\lambda))>0\) implies \(c(E(\lambda))>0\), i.e. that \(\gamma(E(\lambda))>0\), is equivalent to \(c(E(\lambda))>0\). However, it was pointed out by Eiderman [see the commentary in the book of \textit{L. D. Ivanov}, ``Variations of sets and functions'', Nauka, 1975 (Russian)] that Garnett's proof contains a mistake. The main result of the present paper shows further that Garnett's claim is also false. In fact, in the present paper, the author shows that \(\gamma(E(\lambda))>0\) provided \(\int_0^1 r^{-3}h(r)^2 dr<\infty\), where \(h=h_{(\lambda)}\) as above. From this follows that many of the sets \(E=E(\lambda)\) have \(\gamma(E)>0\) but \(c(E)=0\). The author also conjectures the truthfulness of the converse of his main result. The method of proof is based on the so-called Menger three-points curvature, which is the reciprocal of the radius of the circle that passes through the three points in \(\mathbb{C}\), and \textit{M. S. Melnikov}'s identity [Sb. Math. 186, No. 6, 827-846 (1995; Zbl 0840.30008)] which relates this curvatu…
Mathematics Subject Classification ID: 30E20
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 911933
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: Cantor sets
zbMATH Keywords: Menger three-points curvature
author: Pertti Mattila
reviewed by: Jacob Burbea
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 911933