Continued fractions over non-Euclidean imaginary quadratic rings

TL;DR

This paper introduces a generalized continued fraction algorithm applicable to any imaginary quadratic field, extending classical properties like convergence, approximation, and periodicity beyond Euclidean cases.

Contribution

It presents a novel continued fraction algorithm for all imaginary quadratic fields, not limited to Euclidean cases, maintaining key classical properties.

Findings

Exponential convergence of the algorithm

Retention of best approximation qualities

Periodicity for quadratic irrationals

Abstract

We propose and study a generalized continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field, the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued fractions are shown to be retained, including exponential convergence, best-of-the-second-kind approximation quality (up to a constant), periodicity of quadratic irrational expansions, and polynomial time complexity.