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

Abstract

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 $\{p_n/q_n\}$ 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 $q\in \mathfrak{E}$ such that $1\le |q|\le |q_n|$ and any $p\in \mathfrak{E}$, $|qz-p|\ge \frac{1}{2}|q_{n}z-p_n|$. This enables us to conclude that $z\in \mathbb{C}$ is badly approximable, in terms of Eisenstein integers, if and only if the corresponding sequence of partial quotients is bounded.

arXiv comment6 pages