A Note on Transcendental Power Series Mapping the Set of Rational Numbers into Itself

Abstract

In this note, we prove that there is no transcendental entire function f(z) ∈ ℚ[[z]] such that f(ℚ) ⊆ ℚ and den f(p/q) = F(q), for all sufficiently large q, where F(z) ∈ ℤ[z].

On approximation constants for Liouville numbers

“…Problem 1 in context of [15,Corollary 2.2], were obtained in [9,14,16]. Besides Maillet's result, it is known that any "reasonable" function enjoys the weaker property that, while not all, many Liouville numbers are mapped to Liouville numbers.…”