On simultaneous approximation of algebraic numbers

Let Γ ⊂ Q ¯ × $\Gamma \subset \overline{\mathbb {Q}}^{\times }$ be a finitely generated multiplicative group of algebraic numbers. α 1 , … r ∈ $\alpha _1,\ldots ,\alpha _r\in {Q}}^\times$ numbers which are $\mathbb {Q}$ -linearly independent and let ε > 0 $\epsilon >0$ given real number. One the main results that we prove in this article is as follows: There exist only many tuples ( u q p ) Z + $(u, q, p_1,\ldots ,p_r)\in \Gamma \times \mathbb {Z}^{r+1}$ with d = [ : ] $d [\mathbb {Q}(u):\mathbb {Q}]$ for some integer ⩾ $d\geqslant 1$ satisfying | i $|\alpha _i u|>1$ u$ not pseudo-Pisot number { } $i\in \lbrace 1, \ldots r\rbrace$ < j − H \begin{equation*} \hspace*{4pc}0<|\alpha _j qu-p_j|<\frac{1}{H^\epsilon (u)|q|^{\frac{d}{r}+\varepsilon }} \end{equation*} all integers 2 $j 2,\ldots r$ where $H(u)$ absolute Weil height. In particular, when $r =1$ result was proved by Corvaja Zannier [Acta Math. 193 (2004), 175–191]. As an application our result, also transcendence criterion generalizes Hančl, Kolouch, Pulcerová, Štěpnička [Czech. J. 62 (2012), no. 3, 613–623]. The proofs rely on clever use subspace theorem underlying ideas from work Zannier.