Algebraic approximations to linear combinations of S-units

Abstract

Let $\Gamma\subset \bar{\Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers, let $\alpha_1,\ldots,\alpha_m$ be non-zero algebraic numbers, and let $\varepsilon >0$ be fixed. In this paper, we prove that there exist only finitely many tuples $(u_1, \ldots, u_m, q, p)\in \Gamma^m\times\mathbb{Z}^2$ with $d = [\mathbb{Q}(u_1, \ldots, u_m):\mathbb{Q}]$ such that for any two tuples $(u_1,\ldots,u_m)$ and $(u'_1,\ldots,u'_m)$, we have $\frac{u_{i_1}}{u_{i_2}}\neq \frac{u'_{i_1}}{u'_{i_2}}$ for $1\leq i_1\neq i_2\leq m$ and it is stable under Galois conjugation over $\Q$, $\max\{|\alpha_1 qu_1|, \ldots, |\alpha_m qu_m|\}>1$, the tuple $(\alpha_1qu_1, \ldots, \alpha_mq u_m)$ is not pseudo-Pisot and \[0< \left|\sum_{i=1}^m \alpha_iq u_i - p\right|<\frac{1}{\left(\prod_{i=1}^mH( u_i)\right)^{\varepsilon} |q|^{md+\varepsilon}},\] 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.