Generalized Continued Logarithms and Related Continued Fractions

TL;DR

This paper introduces a new generalization of continued logarithms for any base, proves their convergence, analyzes their distribution, and extends classical constants like Khinchine's constant to this new framework.

Contribution

It presents a novel type III continued logarithm for arbitrary bases, proves their convergence, and generalizes classical constants and properties from continued fractions.

Findings

All type III continued logarithms converge.

Rational numbers have finite type III continued logarithms.

Khinchine's constant is extended to continued logarithms with an elementary closed form.

Abstract

We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base $b$. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinchine's constant from simple continued fractions to continued logarithms, and show that these logarithmic Khinchine constants have an elementary closed form. Finally, we show that simple continued fractions are the limiting case of our continued logarithms, and briefly consider how we could generalize past continued logarithms.