论文标题
关于代数数的同时近似
On simultaneous approximation of algebraic numbers
论文作者
论文摘要
令$γ\ subset \ bar {\ mathbb q}^{\ times} $为有限生成的代数数字。令$α_1，\ ldots，α_r\ in \ bar {\ mathbb q}^\ times $为代数数字，是$ \ mathbb {q} $ - 线性独立，让$ε> 0 $为给定的实际数字。我们在本文中证明的主要结果之一是如下；只有有限的许多元组$（u，q，p_1，\ ldots，p_r）\inγ\ times \ times \ mathbb {z}^{r+1} $，带有$ d = [\ mathbb {q}（q}（q}（u）） $α_iq u $不是\ {1，\ ldots，r \} $和$$ 0 <|α_jqu-p_j | <\ frac {1} {1} {整数$ j = 1，2，\ ldots，r $，其中$ h（u）$是绝对的高度。特别是，当$ r = 1 $时，[3]中的Corvaja和Zannier证明了此结果。作为我们结果的应用，我们还证明了超越标准，它概括了[4]中Hančl，Kolouch，Pulcerová和štěpnička的结果。证据依赖于Corvaja和Zannier的工作的巧妙使用子空间定理以及基本思想。
Let $Γ\subset \bar{\mathbb Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers. Let $α_1,\ldots,α_r\in\bar{\mathbb Q}^\times$ be algebraic numbers which are $\mathbb{Q}$-linearly independent and let $ε>0$ be a given real number. One of the main results that we prove in this article is as follows; There exist only finitely many tuples $(u, q, p_1,\ldots,p_r)\inΓ\times\mathbb{Z}^{r+1}$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ for some integer $d\geq 1$ satisfying $|α_i q u|>1$, $α_i q u$ is not a pseudo-Pisot number for some integer $i\in\{1, \ldots, r\}$ and $$ 0<|α_j qu-p_j|<\frac{1}{H^ε(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 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.
