Algebroid functions, Wirsing's theorem and their relations
The author deals with the relation between Nevanlinna theory and Diophantine approximation. The author first points out a relationship between the Second Main Theorem for algebroid functions in Nevanlinna theory and Wirsing's theorem [\textit{E. Wirsing}, Proc. Sympos. Pure Math. 20, 213-247 (1971; Zbl 0223.10017)] in Diophantine approximation. The second part of this paper deals with moving target problem for holomorphic mappings into compact Riemann surfaces. Its counterpart in Diophantine approximation follows from a recent work of J. Tucker (preprint pp. 30). Indeed, Tucker actually proved a more general result. The author also deals with the higher dimensional case. In this part the author follows Stoll's method of algebroid reduction to obtain a weak version of the Second Main Theorem on a finite analytic covering space over \(\pi:M\to \mathbb{C}^m\). Related to this, the author mentions that the counting function \(N(r,\beta)\) of the ramification divisor of \(\pi:M\to \mathbb{C}^m\) is estimated by the characteristic function of a meromorphic mapping \(M\to\mathbb{P}(V)\) if \(f\) separates the fibers of \(\pi:M\to \mathbb{C}^m\). However, this assumption on the meromorphic mapping \(f\), that is, \(f\) separates the fibers of \(\pi:M\to \mathbb{C}^m\), does not need to obtain the estimate on \(N(r,\beta)\). Indeed, this estimate always holds for nonconstant meromorphic mappings \(f:M\to \mathbb{P}(V)\) by a theorem due to \textit{J. Noguchi} [Hiroshima Math. J. 6, 265-280 (1976; Zbl 0338.32016)]. The author finally gives a conjecture of Wirsing's version of Schmidt's subspace theorem.
scientific article; zbMATH DE number 1406452