Algebraic approximations to linear combinations of S-units

Abstract

Let $Γ\subset \bar{\Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers, let $\hat{I}{\pm }_1,\dots ,\hat{I}{\pm }_m$ be non-zero algebraic numbers, and let $\epsilon >0$ be fixed. In this paper, we prove that there exist only finitely many tuples $(u_1,\dots ,u_m,q,p)\in \hat{I}{“}^m\times {\mathbb{Z}}^2$ with $d=[\mathbb{Q}(u_1,\dots ,u_m):\mathbb{Q}]$ such that for any two tuples $(u_1,\dots ,u_m)$ and $(u_1^{\prime },\dots ,u_m^{\prime })$, we have $\frac{u_{i_1}}{u_{i_2}}\ne \frac{u_{i_1}^{\prime }}{u_{i_2}^{\prime }}$ for $1\le i_1\ne i_2\le m$ and it is stable under Galois conjugation over $\Q$, $\max \{|\hat{I}{\pm }_{1}q{u}_1|,\dots ,|\hat{I}{\pm }_{m}q{u}_m|\}>1$, the tuple $(\hat{I}{\pm }_{1}q{u}_1,\dots ,\hat{I}{\pm }_{m}q{u}_m)$ is not pseudo-Pisot and \[0< \left|\sum_{i=1}^m α_iq u_i - p\right|<\frac{1}{\left(\prod_{i=1}^mH( u_i)\right)^{\varepsilon} |q|^{md+\varepsilon}},\] 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.