Values of algebraic functions at Liouville 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.

arXiv commentUpdated following referee's suggestions