On a problem posed by Mahler (Q2788672)

Denote the set of Liouville numbers by \(\mathcal L\). In [Introduction à la théorie des nombres transcendants et des propriétés arithmétiques des fonctions. Paris: Gauthier-Villars (1906; JFM 37.0237.02)] \textit{E. Maillet} proved that if \(f\) is a nonconstant rational function with rational coefficients, then \(f(\mathcal L)\subset\mathcal L\). In [Bull. Aust. Math. Soc. 29, 101--108 (1984; Zbl 0517.10001)] \textit{K. Mahler} asked whether there exist nonconstant entire transcendental functions for which this is true. NEWLINENEWLINEFor large parametrized classes of Liouville numbers, the authors construct such functions, and they show the following. Let \(\Phi\) be the set of all functions \(\psi: \mathbb R_{\geq 2} \to \mathbb R_{\geq 2}\) which are nondecreasing and satisfy \(\lim_{x \to \infty} \psi(x) = \infty\). Let \(\varphi\in \Phi\) be an arbitrary fixed function. Then there exist uncountably many entire transcendental functions \(f(z)=c_0+c_1 z+\cdots\) with nonzero rational numbers \(c_j\), and for any nonnegative integer \(s\) with the following propertiesNEWLINENEWLINE{\parindent= 6mm \begin{itemize}\item[1)] \(f^{(s)}(0)\in\mathbb Q\), \item[2)] \(f^{(s)}(\mathbb Q\backslash\{0\})\subseteq\mathcal L\), and \item[3)] \(f^{(s)}(\mathcal L_\varphi^\ast)\subseteq\mathcal L\).NEWLINENEWLINE\end{itemize}} Suitable functions \(f\) can be explicitly constructed. Here, \(\mathcal L_\varphi^\ast\) is the set of \(\zeta\in\mathcal L\), such that the following estimate holds: \(-\log\|\zeta q\|/\log q\geq N\) has an integer solution \(q=q(N)\) with \(q\leq q\leq\varphi(N)\). Concerning results on related topics, the main result is to show that for any nonzero rational number \(q\), let \(f_q(z)=z^q\). Then there exist uncountably many \(\zeta\), some of which can be explicitly constructed, such that \(f_q(\zeta)\in\mathcal L\) if and only if \(q\in\mathbb Z\).