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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!
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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 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!
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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 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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