Converse of Meier's theorem for tangential paths with arbitrary approach function (Q1063730)

This paper is a continuation of work of the first author [Dokl. Akad. Nauk SSSR 260, 777-780 (1981; Zbl 0487.30021)], and a familiarity with the previous paper, including a number of non-standard definitions and notations, is assumed in the present paper. A collection of curves and regions bounded by these curves are determined in the unit disk D by an ''approach function'' h. For a function f meromorphic in D, these regions determine subsets of the unit circle \(\Gamma\), including the sets \(M_ h(f)\) and \(I_ h(f)\), which are analogues of the classical sets of Meier points and Plessner points (where the regions determined by the approach function h replace the usual Stolz angles). In the earlier paper, an exact analogue to a theorem of Meier was proved, showing that for an arbitrary approach function h and an arbitrary function f meromorphic in D, the set \(\Gamma -(I_ h(f)\cup M_ h(f))\) is an \(F_{\sigma}\)-set of first category. In the current paper, it is proved that if \(\Gamma =E_ 1\cup E_ 2\cup E_ 3\), where \(E_ 1\cap \bar E_ 2=\emptyset\), \(E_ 2\) is a \(G_{\delta}\)-set, and \(E_ 3\) is an \(F_{\sigma}\)-set of first category which is disjoint from \(E_ 1\) and \(E_ 2\), and if h is an arbitrary approach function, then there exists a function f analytic in D with \(M_ h(f)=E_ 1\) and \(I_ h(f)=E_ 2\). The proof is a construction using the Mergelyan theorem on polynomial approximation.