The following problem came up on a mailing list that I subscribe to:

If $\alpha$ is irrational we can find (using continued fractions) infinitely many rational fractions $p/q$ such that $|q \alpha - p| \le C q^{-1}$ for some absolute constant $C >0$. However, suppose that one also restricts $p/q$ to be the norm of an algebraic number in $\mathbb{Q}(\sqrt{-1})$, is the statement still true? If so, is there a good algorithm to find all such?

Of course one can ask this where $\mathbb{Q}(\sqrt{-1})$ is replaced by any number field. I know that there a number of results about approximation by elements of $\Gamma$ a *finitely* generated subgroup of $\mathbb{C}$ but that doesn't apply here.