Continued fractions over non-Euclidean imaginary quadratic rings
arXiv:1908.00121
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.
arXiv commentAlgorithm 2 (a subroutine for Algorithm 1, which is the main continued fraction algorithm) has been added in version 4. This gives the continued fraction algorithm polynomial complexity as stated in Theorem 4.6