IMSc Webinar

On simultaneous approximation of algebraic numbers

Veekesh Kumar

IMSc

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 talk we will discuss the following result, which is a joint work with R. Thangadurai: {\it 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 we take $r=1$, we get Roth's type inequality $|\alpha qu-p|<\frac{1}{H^\epsilon(u)}\frac{1}{q^{d+\epsilon}}$, which was proved by Corvaja and Zannier in 2004.


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