On simultaneous approximation of algebraic numbers

Abstract

Let $\Gamma \subset {\bar{\mathbb{Q}}}^{\times }$ be a finitely generated multiplicative group of algebraic numbers. Let ${\alpha }_1,\dots ,{\alpha }_r\in {\bar{\mathbb{Q}}}^{\times }$ 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 $(u,q,p_1,\dots ,p_r)\in \Gamma \times {\mathbb{Z}}^{r+1}$ with $d=[\mathbb{Q}(u):\mathbb{Q}]$ for some integer $d\ge 1$ satisfying $|{\alpha }_{i}qu|>1$, ${\alpha }_{i}qu$ is not a pseudo-Pisot number for some integer $i\in \{1,\dots ,r\}$ and $$0<|{\alpha }_{j}qu-p_j|<\frac{1}{H^{\epsilon }(u)|q{|}^{\frac{d}{r}+\epsilon }}$$ for all integers $j=1,2,\dots ,r$, 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 Hančl, Kolouch, Pulcerová and Štěpnička in [4]. The proofs rely on the clever use of the subspace theorem and the underlying ideas from the work of Corvaja and Zannier.

arXiv commentAccepted for publication in 'Mathematika'