Algebraic approximations to linear combinations of S-units

Let $Γ\subset \bar{\Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers, let $α_1,\ldots,α_m$ be non-zero algebraic numbers, and let $\varepsilon >0$ be fixed. In this paper, we prove that there exist only finitely many tuples $(u_1, \ldots, u_m, q, p)\in Γ^m\times\mathbb{