A note on transcendental analytic functions with rational coefficients mapping Q into itself

Abstract

In this note, the main focus is on a question about transcendental entire functions mapping Q into Q (which is related to a Mahler’s problem). In particular, we prove that, for any t > 0, there is no a transcendental entire function f ∈ Q[[z]] such that f(Q) ⊆ Q and whose denominator of f(p/q) is O(qt), for all rational numbers p/q, with q sufficiently large.