Mahler's Problem on Liouville Numbers

Abstract

A classical theorem of Maillet states that every nonconstant rational function with rational coefficients maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether there exists a transcendental entire function with the same property. We resolve this question in the negative. More generally, we prove that every real-analytic function on an interval with this property is the restriction of a rational function in $\mathbb{R}(x)$. The local result is quantitative: for every nonrational real-analytic function, every nonempty open subinterval of its domain contains a Liouville number whose image has irrationality exponent at most $100$. The proof rests on a two-height counting estimate that treats source and target denominators independently. An adaptive determinant argument, uniform Wronskian sublevel bounds, and Farey separation establish the estimate; a nested-interval construction yields the quantitative result.