Scott Lindstrom, CARMA, University of Newcastle
Continued Logarithms and Associated Continued Fractions.
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. For another formulation of the generalization to base b, we obtain finite termination for rationals and discover a theoretical recursion for building the analogous distributions.
This is a joint project with Jonathan Borwein, Neil Calkin, and Andrew Mattingly