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arXiv · 1703.07672

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

S. G. Dani
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Abstract

Let $\frak E$ denote be the ring of Eisenstein integers. Let $z\in \mathbb C$ and $p_n,q_n \in \frak E$ 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 $q\in \frak E$ such that $1\leq |q|\leq |q_n|$ and any $p\in \frak E$, $|qz-p|\geq \frac12 |q_nz-p_n|$. 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.

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S. G. Dani. 2017-03-22. Convergents as approximants in continued fraction expansions of complex numbers with Eisenstein integers. https://arxiv.org/abs/1703.07672

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