On a problem posed by Mahler

TL;DR

This paper investigates whether entire transcendental functions can preserve Liouville numbers, constructing such functions for large classes and analyzing their derivatives, thus addressing a problem posed by Mahler.

Contribution

It constructs entire transcendental functions that preserve Liouville numbers and their derivatives, expanding understanding of their behavior under analytic functions.

Findings

Constructed entire transcendental functions preserving Liouville numbers.

Showed derivatives of these functions also preserve Liouville numbers.

Analyzed the image of Liouville numbers under specific rational power functions.

Abstract

E. Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Kurt Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that it can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were constructed (though with no information on derivatives). More generally, we study the image of Liouville numbers under analytic functions, with particular attention to $f(z)=z^{q}$ where $q$ is a rational number.