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

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 $\max \{w_n^{\ast }(\alpha L),w_n^{\ast }(\alpha +L)\}\le 2m^{2}n+m-1$, for $n=1,...,m-1$.

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