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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 rationa 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 $\|\cdot\|_v$ denote 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 $H(x)=\prod_w \max\{1,\|x\|_w\}$, max(1,\|x\|_w)$, where the product is over all places$w$of$k$. Then there are infinitely many elements$x\in k$and is a positive real constant$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! 3 Tex improvements (change \{ to \\{, and displayed main equation) 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 rationa 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$\|\cdot\|_v$denote 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$H(x)=\prod_w \max{1,\|x\|_w}$, max\{1,\|x\|_w\}$, where the product is over all places $w$ of $k$. Then there are infinitely many elements $x\in k$ and a positive real constant $C$ depending only on $k$ such that

$\|\alpha-x\|_v$\|\alpha-x\|_v < C/H(x)^2$\frac{C}{H(x)^2}$$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! 2 edited body 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 rationa 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$\|\cdot\|_v$denote 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 are infinitely many elements$x\in O$k$ and a positive real constant $C$ depending only on $k$ such that

$\|\alpha-x\|_v < C/H(x)^2$

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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