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<trans data-src="For any $\alpha \in \mathbb{R}$ and a parameter $Q$, we can write $\alpha = a/q + \theta$, for integers $a, q$ with $q \leq Q$, and real $\theta$ with $|\theta|\leq (qQ)^{-1}$, a simple application of the Dirichlet approximation theorem.  ">对于任何$\alpha\in\mathbb｛R｝$和参数$Q$，我们可以写$\alpha=a/Q+\theta$，对于整数$a，Q$和$Q\leq Q$，以及实数$\theta$和$|\theta|\leq（qQ）^｛-1｝$，这是狄利克雷近似定理的一个简单应用。</trans><trans data-src="I'm looking for a similar statement for number fields.">我正在为数字字段寻找类似的语句。</trans><trans data-src="**Setup**: $K$ is a fixed number field of degree $ n $ over $ \mathbb{Q} $, with ring of integers $O_K$. $">**Setup**：$K$是一个次数为$n$超过$\mathbb{Q}$的固定数字字段，具有整数$O_K$的环$</trans><trans data-src="\omega_1, \dots , \omega_n$ is a fixed $\mathbb{Z}$-basis for $O_K$.">\omega_1，\dots，\omega_n$是$O_K$的固定$\mathbb{Z}$-基。</trans><trans data-src="$\sigma_i, \dots \sigma_{n}$ are the distinct embeddings of $K$.">$\sigma_i、\dots\sigma_{n}$是$K$的不同嵌入。</trans><trans data-src="$V$ is the $n$- dimensional commutative $\mathbb{R}$-algebra $K \otimes_\mathbb{Q} \mathbb{R}$. ">$V$是$n$维交换$\mathbb{R}$-代数$K\otimes_\mathbb{Q}\mathbb2{R}$。</trans><trans data-src="We define a distance function $| \cdot |$ on $V$ as follows: ">我们在$V$上定义了距离函数$|\cdot|$，如下所示：</trans><trans data-src="$$|x| = |x_1 \omega_1 + \cdots + x_n \omega_n| = \max\limits_{i} | x_i |.$$  ">$$|x|=|x_1\omega_1+\cdots+x_n\omega_n|=\max\limits_{i}|x_i|.$$</trans><trans data-src="I would just like to point out that I am not necessarily attached to this distance function, if you can say anything sensible using another distance function, then I am interested.">我想指出的是，我不一定喜欢这个距离函数，如果你能用另一个距离函数说一些合理的话，那么我很感兴趣。</trans><trans data-src="**Precise Statement**: Given $\alpha \in V$ and a parameter $Q$, is it always possible to find $\lambda, \mu \in O_K$, such that $|\mu| \ll Q$ and $$|\alpha - \dfrac{\lambda}{\mu}| \ll \dfrac{1}{Q |\mu|}?$$">**精确语句**：给定V$中的$\alpha\和一个参数$Q$，总是有可能在O_K$中找到$\lambda，\mu\，比如$|\mu|\llQ$和$$|\alpha-\dfrac{\lambda}{\mu}|\ll\dfrac{1}{Q|\mu}$$</trans><trans data-src="**Equivalent Statement**: can we find $\gamma \in K$ such that $\mathcal{N}=\textrm{N}(\bf">**等价语句**：我们能否在K$中找到$\gamma\，使$\mathcal{N}=\textrm{N}（\bf</trans><trans data-src="{a}_">{一}_</trans><trans data-src="\gamma) \ll Q^n$, and $$|\alpha - \gamma| \ll \dfrac{1}{Q \mathcal{N}^{1/n}}, $$ where $\bf">\gamma）\ll Q^n$和$$|\alpha-\gamma|\ll\dfrac{1}{Q\mathcal{n}^{1/n}}，$$其中$\bf</trans><trans data-src="{a}_">{一}_</trans><trans data-src="\gamma$ is the denominator ideal of $\gamma$?">\gamma$是$\gamma$的理想分母吗？</trans><trans data-src="Note that it is easy to find an analogous statement to Dirichlet's original theorem, ie $\exists \lambda, \mu \in O_K$, such that $|\mu| \ll Q$ and $$|\alpha \mu - \lambda| \ll \dfrac{1}{Q},$$ by an application of the pigeonhole principle, but unlike in $\mathbb{R}$, we cannot just divide through by $\mu$ at this point,">注意，通过鸽子洞原理的应用很容易找到与Dirichlet的原始定理类似的语句，即O_K$中的$\exists\lambda，\mu，例如$|\mu|\ll Q$和$$|\alpha\mu-\lambda|\ll\dfrac{1}{Q}，$$，但与$\mathbb{R}$不同的是，此时我们不能只除以$\mu$，</trans><trans data-src=" as the only decent bound (that I know of) for $|\mu^{-1}|$ is $$|\mu^{-1}| \ll \dfrac{|\mu|^{n-1}}{\textrm{N}(\mu)}$$.">据我所知，$|\mu^{-1}|$的唯一合理界限是$$|\mu^{-1{|ll\dfrac{|\mu|^{n-1}}{\textrm{n}（\mu）}$$。</trans><trans data-src="Does anyone have a reference for dealing with the fractional form like this?  ">有人可以参考一下处理这样的分数形式吗？</trans><trans data-src="The closest I have managed to find was a generalisation to number fields of the Thue-Siegel-Roth theorem by LeVeque.">我找到的最接近的是LeVeque对Thue-Siegel-Roth定理的数字域的推广。</trans>

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