On a question proposed by K. Mahler concerning Liouville numbers

TL;DR

This paper investigates the existence of transcendental entire functions that map an uncountable subset of Liouville numbers into Liouville numbers, extending classical results about rational functions and Liouville numbers.

Contribution

It constructs an uncountable subset of Liouville numbers for which such transcendental entire functions exist, addressing a question posed by Mahler.

Findings

Existence of transcendental entire functions preserving Liouville numbers on an uncountable set

Extension of Maillet's result to transcendental entire functions

Provides a constructive example of such functions

Abstract

In 1906, Maillet proved that given a non-constant rational function $f$, with rational coefficients, if $\xi$ is a Liouville number, then so is $f(\xi)$. Motivated by this fact, in 1984, Mahler raised the question about the existence of transcendental entire functions with this property. In this work, we provide an uncountable subset of Liouville numbers for which there exists a transcendental entire function taking this set into the set of the Liouville numbers.