Kevin Hare, Department of Pure Mathematics, University of Waterloo
Let $1 \leq \alpha \in \mathbb{R}$. Let $y_0 = \alpha$ and
recursively define $a_n = \lfloor \log_2 y_n \rfloor$. If
$y_n - 2^{a_n} = 0$ then terminate. Otherwise set
\[
y_{n+1} = \frac{2^{a_n} }{y_n - 2^{a_n}}
\]
and recurse. This produces the binary (base 2) continued logarithm
for $y_0$:
\[
y_0 = 2^{a_0} + \frac{2^{a_0}}{2^{a_1} + \frac{2^{a_1}}{2^{a_2} + \frac{2^{a_2}}{2^{a_3} + \dots}}}.
\]
These binary continued logarithms were introduced explicitly by
Gosper in his appendix on Continued Fraction Arithmetic. These
were further studied by Borwein et. al. extending classical
continued fraction recurrences for binary continued logs and
investigating the distribution of aperiodic binary continued
logarithm terms for quadratic irrationals --- such as cannot occur
for simple continued fractions.
In this talk we discuss some research work generalizing and
continued many of these results.
M3 3103