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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!

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For $k=Q$ the statement is false: a non-integral $\alpha$ cannot be approximated well with integral $x$. Dirichlet's Theorem is concerned with approximation by rational numbers, not integers. –  GH from MO May 7 '12 at 18:35
    
I changed the $O$ to $k$. –  Joe Silverman May 7 '12 at 19:02
    
Then $O$ is not needed in the third line either. –  GH from MO May 7 '12 at 20:32

1 Answer 1

up vote 4 down vote accepted

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.

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Thanks! Technically, the case for a complex place is left as an exercise in this reference rather than proven as a theorem, but I don't think that's a big deal. –  David McKinnon May 8 '12 at 19:00

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