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arXiv · 2604.08818

Values of algebraic functions at Liouville numbers

Yuri Bilu·Diego Marques
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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.

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Yuri Bilu, Diego Marques. 2026-04-09. Values of algebraic functions at Liouville numbers. https://arxiv.org/abs/2604.08818

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