Continued logarithms and associated continued fractions

Title: Continued logarithms and associated continued fractions
Creator: Borwein, Jonathan M.; Calkin, Neil J.; Lindstrom, Scott B.; Mattingly, Andrew
Relation: Experimental Mathematics Vol. 26, Issue 4, p. 412-429
Publisher Link: http://dx.doi.org/10.1080/10586458.2016.1195307
Publisher: Taylor & Francis
Resource Type: journal article
Date: 2017
Description: We investigate some of the connections between continued fractions and continued logarithms. We study the binary continued logarithms as introduced by Bill Gosper and explore two generalizations of the continued logarithm to base b. We show convergence for them using equivalent forms of their corresponding continued fractions. Through numerical experimentation, we discover that, for one such formulation, the exponent terms have finite arithmetic means for almost all real numbers. This set of means, which we call the logarithmic Khintchine numbers, has a pleasing relationship with the geometric means of the corresponding continued fraction terms. While the classical Khintchine's constant is believed not to be related to any naturally occurring number, we find surprisingly that the logarithmic Khintchine numbers are elementary.
Subject: continued logarithms; continued fractions; Khintchine's constant; Gauss-Kuzmin distribution; logarithmic Khintchine numbers
Identifier: http://hdl.handle.net/1959.13/1387456
Identifier: uon:32609
Identifier: ISSN:1058-6458
Language: eng
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