Convergents as approximants in continued fraction expansions of complex numbers with Eisenstein integers

Let $\mathfrak{E}$ denote be the ring of Eisenstein integers. Let $z\in \mathbb{C}$ and $p_n,q_n\in \mathfrak{E}$ be such that $\{p_n\mathrm{/}{q}_n\}$ is the sequence of convergents corresponding to the continued fraction expansion of $z$ with respect to the nearest integer algorithm. Then we show that for any $q\in \frak E$suchthat$1\leq |q|\leq |q_n|$andany$p\in \frak E$,$|qz-p|\geq \frac12 |q_nz-p_n|$\mathrm{.}Thisenablesustoconcludethat$z\in \mathbb C$ is badly approximable, in terms of Eisenstein integers, if and only if the corresponding sequence of partial quotients is bounded.