I'm looking for a reference for the following result, which is a generalization of the classical theorem of Dirichlet on the approximability of real irrationals by rational numbers:


Let $k$ be a number field, $O$ its ring of integers, $v$ an infinite place of $k$, $\alpha$ any element of the completion $k\_v$. Let $\|\cdot\|\_v$ be the usual absolute value (or its square, if $v$ is a complex place). Let $H$ denote the multiplicative height function relative to $k$ -- that is, for any element $x\in k$, let $H(x)=\prod\_w \max(1,\|x\|\_w)$, where the product is over all places $w$ of $k$. Then there is a positive real constant $C$ depending only on $k$ such that


$$\|\alpha-x\|\_v < \frac{C}{H(x)^2}$$


for infinitely many $x\in k$.


I think I can prove this, but I am surely not the first. If anyone can tell me a good place to point to for this result, I'd be very grateful -- thanks!



Reference wanted for generalized Dirichlet's Theorem
It seems like this book has the reference you need.


Schmidt, Wolfgang M. Diophantine approximation. Lecture Notes in Mathematics, 785. Springer, Berlin, 1980.


Chapter VIII, Theorem 2A.


It's available on the Springlink website.


Hope it helps.


