On the analytic capacity and curvature of some Cantor sets with non-\(\sigma\)-finite length
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 curvature to Cauchy's kernel. Other related, and more general, results are also given.
scientific article; zbMATH DE number 911933