Is there an efficient algorithm for Dirichlet approximation for a given (high-degree) number field and its ring of integers, perhaps analogous to the Euclidean/continued fractions algorithm for the integers?
In more detail, I have the following setup: let $K$ be a fixed number field of degree $n$ with canonical/Minkowski embedding $\sigma \colon K \to \mathbb{C}^n$, and let $O_K$ denote the ring of integers. Assume that a "high quality" (w.r.t. $\sigma$) integral basis of $O_K$ is known. (An example of particular interest is cyclotomic number fields.)
Given some $\omega \in K$, I would like to find (as efficiently as possible) an ideal $J \subset O_K$ of "not too large" norm, and a field element $x \in J^{-1}$ such that $\| \sigma(\omega - x) \|$ is "rather small" relative to $N(J)$ -- significantly smaller than the distance from a typical $\gamma$ to $J^{-1}$.
When $K=\mathbb{Q}$, we of course have the Dirichlet approximation theorem and an efficient continued fractions algorithm, which gives us a sequence of $J=q\mathbb{Z}$, $x=p/q$ such that $| \omega - x | \leq 1/q^2$.
For $K=\mathbb{Q}(i)$ or $K=\mathbb{Q}(\zeta_3)$, whose rings of integers are respectively the Gaussian and Eisenstein integers, there are analogous approximation bounds and continued fractions algorithms due to Hurwitz, Ford, and others.
Dirichlet's approximation theorem generalizes to any number field (depending on exactly how one defines the approximation quality; see this question), but I have not found anything about an efficient approximation algorithm (or a conjecture that the problem is computationally hard). Similarly, there is a great deal of work devoted to showing that various number rings are Euclidean domains (with either the standard norm or some other function), but I haven't seen anything about whether the associated continued fraction algorithms yield good approximations in the sense described above.
