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.

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 $n=1,...,m-1$.