Dirichlet proved a classical theorem about approximating irrational real numbers with rational numbers, saying that for any irrational real number $\alpha$, you can find infinitely many rational numbers $a/b$ such that $|a/b-\alpha|<1/b^2$. I asked a question previously ("Reference wanted for generalized Dirichlet's Theorem") for a generalization of this to arbitrary number fields and archimedean valuations, and was very kindly directed to an excellent reference for this.

Like Oliver Twist, I want more. I was hoping that someone might be able to point the way to a reference to the following result:

Let $k$ be a number field, $S=\{v_1,\ldots,v_n\}$ a finite set of places of $k$.

For each place $v$ of $k$, let $k_v$ be the corresponding completion of $k$. Let $d_v$ be the local degree at $v$ -- that is, let $d_v=[k_v:Q_p]$, where $v$ lies over $p$. (If $v$ is archimedean, define $d_v=1$ for real places and 2 for complex ones.)

For each $v\in S$, choose an element $a_v\in k_v$ that is algebraic over $k$.

Fix an algebraic closure $\overline{k}$ of $k$, and for each $v\in S$ choose a valuation of $\overline{k}$ that lies over $v$.

Normalize the absolute values $|\cdot|_v$ so that $\prod_v |x|_v^{d_v}=1$ for all $x\in k$.

For each $v\in S$, choose a positive real number $e_v$ such that $e=\sum d_ve_v\leq 2$.

Then there exist infinitely many elements $x\in k$ such that for all $v\in S$, we have $|x-a_v|_v\leq H(x)^{-e_v}$, where $H(x)$ denotes the multiplicative height of $x$ relative to $k$.

I'm pretty sure I can prove this, but surely someone else has thought of this already, and hopefully published it somewhere. If anyone has any suggestions as to where I might find this result, I'd be very grateful -- thanks!


A statement similar to what you are asking (but for $k = Q$) is in my paper with Tomanov, http://arxiv.org/pdf/math/0506510.pdf (section 10.2) -- we could not find a reference for that either.

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Just came across your question randomly while I was searching for something else on Google- here is a quick but hopefully correct response:

You might want to take a look at the paper "Diophantine approximation in projective space" by Choi and Vaaler. Their one dimensional projective theorems are a non-Archimedean version of the classical Dirichlet's Theorem, like what you are looking for, but (I think) without the $e_v$ variables.

The important thing however is the proof of their result, which relies on geometry of numbers in the Adeles over number fields (see Section 3 of "On the decomposition of vectors over number fields" by Burger and Vaaler). Once you understand this machinery, I would guess that it would be fairly straightforward to incorporate the weights $e_v$ to change the shapes of the boxes involved. Hope that helps.

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