Dirichlet Approximation over a Number Field

For any $alpha in mathbb{R}$ and a parameter $Q$, we can write $alpha = a/q + theta$, for integers $a, q$ with $q leq Q$, and real $theta$ with $|theta|leq (qQ)^{-1}$, a simple application of the Dirichlet approximation theorem. I'm looking for a similar statement for number fields.

Setup: $K$ is a fixed number field of degree $n$ over $mathbb{Q}$, with ring of integers $O_K$. $omega_1, dots , omega_n$ is a fixed $mathbb{Z}$-basis for $O_K$.
$sigma_i, dots sigma_{n}$ are the distinct embeddings of $K$.

$V$ is the $n$- dimensional commutative $mathbb{R}$-algebra $K otimes_mathbb{Q} mathbb{R}$.

We define a distance function $| cdot |$ on $V$ as follows:

$$|x| = |x_1 omega_1 + cdots + x_n omega_n| = maxlimits_{i} | x_i |.$$ I would just like to point out that I am not necessarily attached to this distance function, if you can say anything sensible using another distance function, then I am interested.

Precise Statement: Given $alpha in V$ and a parameter $Q$, is it always possible to find $lambda, mu in O_K$, such that $|mu| ll Q$ and

$$|alpha - dfrac{lambda}{mu}| ll dfrac{1}{Q |mu|}?$$

Equivalent Statement: can we find $gamma in K$ such that $mathcal{N}=textrm{N}(bf{a}_gamma) ll Q^n$, and

$$|alpha - gamma| ll dfrac{1}{Q mathcal{N}^{1/n}},$$ where $bf{a}_gamma$ is the denominator ideal of $gamma$?

Note that it is easy to find an analogous statement to Dirichlet's original theorem, ie $exists lambda, mu in O_K$, such that $|mu| ll Q$ and

$$|alpha mu - lambda| ll dfrac{1}{Q},$$ by an application of the pigeonhole principle, but unlike in $mathbb{R}$, we cannot just divide through by $mu$ at this point, as the only decent bound (that I know of) for $|mu^{-1}|$ is $$|mu^{-1}| ll dfrac{|mu|^{n-1}}{textrm{N}(mu)}$$ .

Does anyone have a reference for dealing with the fractional form like this? The closest I have managed to find was a generalisation to number fields of the Thue-Siegel-Roth theorem by LeVeque.

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