On $C^k$-functions mapping $\mathbb{Q}$ into itself and Mahler's problem on Liouville numbers

Abstract

Liouville numbers form a classical class of transcendental real numbers characterized by exceptionally strong rational approximations. A theorem of Maillet shows that non-constant rational functions with rational coefficients preserve the Liouville property, motivating a question of Mahler on whether analogous phenomena hold for transcendental functions. In this paper, we address this problem for real functions of finite smoothness. For any $\varepsilon>0$, we construct an uncountable set of $C^k$-functions on $\mathbb{R}$, dense with respect to the topology of uniform convergence on compact sets, mapping $\mathbb{Q}$ into itself and satisfying $\operatorname{den}(f(p/q)) \le q^{2k+\varepsilon}$, and deduce that such functions preserve Liouville numbers. In contrast, we prove a rigidity result about a $C^{2k+1}$-function mapping $\mathbb{Q}$ into itself and satisfying $\operatorname{den}(f(p/q)) \ll q^k$.