Continued fractions over non-Euclidean imaginary quadratic rings

DOI10.1016/J.JNT.2022.06.004arXiv1908.00121WikidataQ114156446 ScholiaQ114156446MaRDI QIDQ6322993

Publication date: 31 July 2019

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.

Mathematics Subject Classification ID

Quadratic extensions (11R11) Continued fractions and generalizations (11J70) Continued fractions (11A55) Continued fraction calculations (number-theoretic aspects) (11Y65) Multiplicative structure; Euclidean algorithm; greatest common divisors (11A05) Approximation by numbers from a fixed field (11J17)

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