Dirichlet Approximation over a Number Field - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T22:42:58Z http://mathoverflow.net/feeds/question/17722 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17722/dirichlet-approximation-over-a-number-field Dirichlet Approximation over a Number Field sobe86 2010-03-10T13:35:02Z 2010-03-10T13:43:06Z <p>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. I'm looking for a similar statement for number fields.</p> <p><strong>Setup</strong>: $K$ is a fixed number field of degree $n$ over $\mathbb{Q}$, with ring of integers $O_K$. $\omega_1, \dots , \omega_n$ is a fixed $\mathbb{Z}$-basis for $O_K$. $\sigma_i, \dots \sigma_{n}$ are the distinct embeddings of $K$.</p> <p>$V$ is the $n$- dimensional commutative $\mathbb{R}$-algebra $K \otimes_\mathbb{Q} \mathbb{R}$. </p> <p>We define a distance function $| \cdot |$ on $V$ as follows: $$|x| = |x_1 \omega_1 + \cdots + x_n \omega_n| = \max\limits_{i} | x_i |.$$ 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.</p> <p><strong>Precise Statement</strong>: 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 $$|\alpha - \dfrac{\lambda}{\mu}| \ll \dfrac{1}{Q |\mu|}?$$</p> <p><strong>Equivalent Statement</strong>: can we find $\gamma \in K$ such that <code>$\mathcal{N}=\textrm{N}(\bf{a}_\gamma) \ll Q^n$</code>, and $$|\alpha - \gamma| \ll \dfrac{1}{Q \mathcal{N}^{1/n}},$$ where <code>$\bf{a}_\gamma$</code> is the denominator ideal of $\gamma$?</p> <p>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, as the only decent bound (that I know of) for $|\mu^{-1}|$ is $$|\mu^{-1}| \ll \dfrac{|\mu|^{n-1}}{\textrm{N}(\mu)}$$.</p> <p>Does anyone have a reference for dealing with the fractional form like this? The closest I have managed to find was a generalisation to number fields of the Thue-Siegel-Roth theorem by LeVeque. </p> http://mathoverflow.net/questions/17722/dirichlet-approximation-over-a-number-field/17725#17725 Answer by Felipe Voloch for Dirichlet Approximation over a Number Field Felipe Voloch 2010-03-10T13:43:06Z 2010-03-10T13:43:06Z <p>Schmidt, Wolfgang M. Diophantine approximation. Lecture Notes in Mathematics, 785. Springer, Berlin, 1980.</p>