On the Arithmetic Behavior of Liouville Numbers under Rational Maps

TL;DR

This paper extends previous results on how Liouville numbers behave under rational functions, identifying a broader class of Liouville numbers that map into $U_m$-numbers, thus deepening understanding of their arithmetic properties.

Contribution

The paper generalizes Alnia extit{c}ik's result by characterizing a larger class of Liouville numbers that are mapped into $U_m$-numbers under rational functions.

Findings

Identifies a broader class of Liouville numbers with the same mapping property.

Shows the set of such Liouville numbers is sharp in a certain sense.

Extends the understanding of the arithmetic behavior of Liouville numbers.

Abstract

In 1972, Alnia\c{c}ik proved that every strong Liouville number is mapped into the set of $U_m$-numbers, for any non-constant rational function with coefficients belonging to an $m$-degree number field. In this paper, we generalize this result by providing a larger class of Liouville numbers (which, in particular, contains the strong Liouville numbers) with this same property (this set is sharp is a certain sense).