On transcendental entire functions mapping [formula omitted] into itself

Abstract

In this paper, we study a question proposed by Marques and Moreira about transcendental entire functions mapping Q into itself. In particular, we prove that there is no a transcendental entire function f such that f(Q)⊆Q and the denominator of f(p/q) is O(q), for all rational number p/q, with q sufficiently large.