Conformal equivalence of countable dense sets

Abstract

In [1, p. 297, problem 24], Erd6s asks: Does there exist an entire functionf, not of the form f(z) = ao+aiz, such that the number f(x) is rational or irrational according as x is rational or irrational? More generally, if A and B are two denumerable, dense sets, does there exist an entire function which maps A onto B? The following theorem settles the second part of this question as it is stated.