Superoptimal continued fractions

TL;DR

This paper introduces algorithms for superoptimal continued fractions of irrationals, which yield highly accurate rational approximations with rapid convergence, advancing the understanding of continued fraction expansions.

Contribution

It defines and develops algorithms for superoptimal continued fractions, extending prior work on optimal continued fractions with improved approximation properties.

Findings

Convergents provide arbitrarily good rational approximations.

Convergents converge arbitrarily quickly.

Algorithms generalize and improve upon existing optimal continued fraction methods.

Abstract

Motivated by the optimal continued fractions studied independently by Selenius and Bosma, we define and introduce algorithms producing superoptimal continued fraction expansions of irrationals. The convergents of these expansions simultaneously provide arbitrarily good rational approximations and converge arbitrarily quickly.