On simultaneous approximation of algebraic numbers

Abstract

Let $\Gamma\subset \bar{\mathbb Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers. Let $\alpha_1,\ldots,\alpha_r\in\bar{\mathbb Q}^\times$ be algebraic numbers which are $\mathbb{Q}$-linearly independent and let $\epsilon>0$ be a given real number. One of the main results that we prove in this article is as follows; There exist only finitely many tuples $(u, q, p_1,\ldots,p_r)\in\Gamma\times\mathbb{Z}^{r+1}$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ for some integer $d\geq 1$ satisfying $|\alpha_i q u|>1$, $\alpha_i q u$ is not a pseudo-Pisot number for some integer $i\in\{1, \ldots, r\}$ and $$ 0<|\alpha_j qu-p_j|<\frac{1}{H^\epsilon(u)|q|^{\frac{d}{r}+\varepsilon}} $$ for all integers $j = 1, 2,\ldots, r$, where $H(u)$ is the absolute Weil height. In particular, when $r =1$, this result was proved by Corvaja and Zannier in [3]. As an application of our result, we also prove a transcendence criterion which generalizes a result of Han\v{c}l, Kolouch, Pulcerov\'a and \v{S}t\v{e}pni\v{c}ka in [4]. The proofs rely on the clever use of the subspace theorem and the underlying ideas from the work of Corvaja and Zannier.