Values of algebraic functions at Liouville numbers

TL;DR

This paper investigates when algebraic functions evaluated at Liouville numbers produce $U_m$-numbers, extending LeVeque's 1953 results and exploring conditions under which algebraic functions yield $U_m$-values.

Contribution

It generalizes previous findings by establishing that algebraic functions of degree $m$ often produce $U_m$-numbers at all $ ext{L}$-numbers under broad conditions.

Findings

Algebraic functions of degree m often take $U_m$-values at $ ext{L}$-numbers.

The study extends LeVeque's 1953 results on Liouville numbers.

Under general assumptions, the property holds for all $ ext{L}$-numbers.

Abstract

In 1953 LeVeque proved the existence of $U_m$-numbers by showing that for some specially defined Liouville number $\lambda$, the $m$th root $\lambda^{1/m}$ is in $U_m$. In this article we study the following question: let $u$ be an algebraic function of degree $m$ and $\lambda$ a Liouville number; under which conditions is $u(\lambda)$ a $U_m$-number? We consider a more refined notion of $\mathcal{L}$-numbers, and show that, under very general assumptions, an algebraic function of degree $m$ takes $U_m$-values at all $\mathcal{L}$-numbers.