An analogue of continued fractions in number theory for Nevanlinna theory

Abstract

Abstract. We show an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory. IntroductionSince P. Vojta [17] created a dictionary between Nevanlinna theory and Diophantine approximation theory, researchers (e.g. [7], [15], [20], [4] and [22]) from both fields have started to look for more analogues between these two theories. It is known that a famous theorem of Roth in number theory is analogous to a weak form of the second main theorem in Nevanlinna theory, and the Artin-Whaples product formula in number theory can be viewed as an analogue of the first main theorem in Nevanlinna theory. Theoretically speaking, we should be able to find an analogue of any theorem related to the Roth theorem in Diophantine approximation for Nevanlinna theory, and vice versa. An up-to-date account of these matters appears in [2], [18] and [14]. The author [23] has found an analogue of Khinchin's theorem for Nevanlinna theory, which has given an answer to one of S. Lang's questions in [9]. S. Lang also suggested (in a personal conversation) finding an analogue of the continued fractions for Nevanlinna theory. This is an interesting question that has been around for a while. In this paper, we find an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory. Notation and preliminariesFor the convenience of the general readers, we briefly give some definitions and notation in Nevanlinna theory and continued fractions. Standard references are [3] and [12] for Nevanlinna theory, and [5] and [8] for continued fractions.Let f be a meromorphic function on the whole plane and D r = {|z| < r}. Denote the number of poles of f in D r by n(f, ∞, r), and define n(f, a, r) = n(1/(f − a), ∞, r) if a ∈ C. We also let N (f, a, r) = r 0 n(f, a, t) − n(f, a, 0) t dt + n(f, a, 0) log r.