On the closeness of approach of complex rational fractions to a complex irrational number

Abstract

is satisfied by an infinite number of real rational fractions, p/q, when k = 1. Later Hermite gave a method, based on binary quadratic forms, of constructing an infinite suite of fractions satisfying the inequality when k 1 / V73. The problem of the minimum value of k was solved by Hurwitzt, who showed that if k = 1 / 1/5 an infinite number of fractions satisfy the inequality, whereas if k < 1 / VW5 there is an t in every interval of the real axis for which the inequality holds for only a finite number of rational fractions. Proofs of Hurwitz' theorem have been given by Borelt, by Humbert? and by the present authorll. In the present paper we propose to investigate the analogous problem in the complex domain. Let w be any complex irrational number and consider the inequality