Algebraic approximations to linear combinations of S-units

Let $\mathrm{\Gamma }\subset {\bar{\text{Q}}}^{\times }$ be a finitely generated multiplicative group of algebraic numbers, let ${\alpha }_1,\dots ,{\alpha }_m$ be non-zero algebraic numbers, and let $\epsilon >0$ be fixed. In this paper, we prove that there exist only finitely many tuples $(u_1,\dots ,u_m,q,p)\in {\mathrm{\Gamma }}^m\times {\mathbb{Z}}^2$ with $d=[\mathbb{Q}(u_1,\dots ,u_m):\mathbb{Q}]$ such that for any two tuples $(u_1,\dots ,u_m)$ and $(u_1^{\mathrm{\prime }},\dots ,u_m^{\mathrm{\prime }})$, we have $\frac{u_{i_1}}{u_{i_2}}\mathrm{\ne }\frac{u_{i_1}^{\mathrm{\prime }}}{u_{i_2}^{\mathrm{\prime }}}$ for $1\le i_1\mathrm{\ne }{i}_2\le m$ and it is stable under Galois conjugation over $\text{Q}$, $\max \{\mathrm{\mid }{\alpha }_{1}q{u}_1\mathrm{\mid },\dots ,\mathrm{\mid }{\alpha }_{m}q{u}_m\mathrm{\mid }\}>1$, the tuple $({\alpha }_{1}q{u}_1,\dots ,{\alpha }_{m}q{u}_m)$ is not pseudo-Pisot and $$0<\mid \sum _{i=1}^m{\alpha }_{i}q{u}_i-p\mid <\frac{1}{{(\prod _{i=1}^{m}H(u_i))}^{\epsilon }\mathrm{\mid }q{\mathrm{\mid }}^{md+\epsilon }},$$ where $H(u_i)$ denotes the absolute Weil height. This result extends one of the main results of Corvaja-Zannier \cite{corv}. In addition, we prove a result similar to \cite[Theorem 1.4]{kul} in a more general setting. In our proofs, we exploit the subspace theorem based on the work of Corvaja-Zannier.