Geodesic
A Note on Transcendental Power Series Mapping the Set of Rational Numbers into Itself
Communications in Mathematics, Tome 25 (2017) no. 1, pp. 1-4 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this note, we prove that there is no transcendental entire function $f(z)\in \mathbb{Q} [[z]]$ such that $f(\mathbb{Q} )\subseteq \mathbb{Q}$ and $\mathop{\rm den} f(p/q)=F(q)$, for all sufficiently large $q$, where $F(z)\in \mathbb{Z} [z]$.
In this note, we prove that there is no transcendental entire function $f(z)\in \mathbb{Q} [[z]]$ such that $f(\mathbb{Q} )\subseteq \mathbb{Q}$ and $\mathop{\rm den} f(p/q)=F(q)$, for all sufficiently large $q$, where $F(z)\in \mathbb{Z} [z]$.
Classification : 11J81
Keywords: Liouville numbers; Mahler's question; power series 
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Marques, Diego; Silva, Elaine. A Note on Transcendental Power Series Mapping the Set of Rational Numbers into Itself. Communications in Mathematics, Tome 25 (2017) no. 1, pp. 1-4. http://geodesic.mathdoc.fr/item/COMIM_2017_25_1_a0/

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