A Note on Lacunary Power Series With Rational coefficients

Abstract

In this note, we prove that for any ${\it\nu}>0$, there is no lacunary entire function $f(z)\in \mathbb{Q}[[z]]$ such that $f(\mathbb{Q})\subseteq \mathbb{Q}$ and $\text{den}f(p/q)\ll q^{{\it\nu}}$, for all sufficiently large $q$.

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.…”

Lacunary Power Series and 𝑼𝒎-Numbers

“…In particular, are there entire transcendental functions with this property?". That question has interested a lot of mathematicians [11,12,13,14].…”

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

“…For example, to prove this, Marques and Moreira [4] showed the existence of transcendental entire functions f , such that f (Q) ⊆ Q and den f (p/q) < q 8q 2 , for all p/q ∈ Q, with q > 1 (where den z denotes the denominator of the rational number z). Moreover, their proof implies that the Mahler's question has an affirmative answer if the answer to the below question is also 'yes' (see also [ In 2015, Marques, Ramirez and Silva [6] proved that the answer for the previous question is 'no' for lacunary power series in Q[[z]] (see [1] for the definition of lacunary power series as well as some results related to their arithmetic properties). Moreover, their proof also implies that there is no transcendental entire function…”