On simultaneous approximation of algebraic numbers

Veekesh Kumar, R. Thangadurai

Published 2020-01-02Version 1

Let $\Gamma\subset \overline{\mathbb Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers. Let $\alpha_1,\ldots,\alpha_r\in\overline{\mathbb Q}^\times$ be algebraic numbers which are $\mathbb{Q}$-linearly independent with $1$ and let $\epsilon>0$ and $c>0$ be given real numbers. In this paper, we prove that there exist only finitely many tuple $(u, q, p_1,\ldots,p_r)\in\Gamma\times\mathbb{Z}^{r+1}$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ such that $|\alpha_i q u|>c$, $\alpha_i q u$ is not a $c$-pseudo-Pisot number for some $i$ and $$ 0<|\alpha_j qu-p_j|<\frac{1}{H^\epsilon(u)q^{\frac{d}{r}+\epsilon}} $$ for $1\leq j\leq r$, where $H(u)$ denotes the absolute Weil height. When $r=1$, we recover the main theorem of Corvaja and Zannier. Also, we prove a more general version of the main theorem in the paper of Corvaja and Zannier. The proofs relies on the subspace theorem and the idea of the work of Corvaja and Zannier with suitable modifications.