Remarks on the descriptive metric characterization of singular sets of analytic functions (Q1272030)

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 \(E=E(f)\).