# Transcendence Measures for some $U_m$-numbers related to Liouville's constant

## TL;DR

This paper proves that multiplying or adding an algebraic number by the Liouville constant results in a $U$-number with a specific type, and provides bounds on certain transcendence measures related to these numbers.

## Contribution

It establishes that algebraic numbers combined with the Liouville constant produce $U$-numbers with predictable types and bounds, advancing understanding of their transcendence properties.

## Key findings

- Sum and product with Liouville constant produce $U$-numbers.
- Type of resulting $U$-numbers equals the degree of the algebraic number.
- Provides explicit bounds on transcendence measures for these numbers.

## Abstract

In this note, we shall prove that the sum and the product of an algebraic number $\alpha$ by the \textit{Liouville constant} $L=\sum_{j=1}^{\infty}10^{-j!}$ is a $U$-number with type equals to the degree of $\alpha$ (with respect to $\mathbb{Q}$). Moreover, we shall have that $\max\{w^{\ast}_n(\alpha L),w^{\ast}_n(\alpha + L)\}\leq 2m^2n+m-1$, for $n=1,...,m-1$.