Kevin Hare, Department of Pure Mathematics, University of Waterloo
"Continued Logarithms"
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