On a stronger version of a question proposed by K. Mahler (Q1788109)

Let \({\overline{\mathbb{Q}}}\) be the field of algebraic numbers. In a previous paper [\textit{D. Marques} and \textit{C. G. Moreira}, Math. Ann. 368, No. 3--4, 1059--1062 (2017; Zbl 1387.11056)], answering a question of [\textit{K. Mahler}, Proc. Symp. Pure Math. 20, 248--274 (1971; Zbl 0213.32703)] the authors proved that there exist uncountably many transcendental entire functions \(f(z)=\sum_{n\ge 0} a_n z^n\) with rational coefficients \(a_n\) such that the image \(f({\overline{\mathbb{Q}}})\) and the preimage \(f^{-1}({\overline{\mathbb{Q}}})\) of \({\overline{\mathbb{Q}}}\) under \(f\) are subsets of \({\overline{\mathbb{Q}}}\). The main result of the paper under review is the following generalization. Let \(X\) and \(Y\) be countable subsets of \({\mathbb{C}}\) that are dense and closed for complex conjugation. Suppose that either both \(X\cap {\mathbb{R}}\) and \(Y \cap {\mathbb{R}}\) are dense in \( {\mathbb{R}}\) or both intersections are the empty set. Assume also that if \(0 \in X\) , then \(Y \cap {\mathbb{Q}}\ne \emptyset\). Then, there are uncountably many transcendental entire functions \(f(z)=\sum_{n\ge 0} a_n z^n\) with rational coefficients \(a_n\) and such that \(f(X) = Y\), \(f^{-1}(Y ) = X\) and \(f'(\alpha)\ne 0\), for all \(\alpha\in X\).