J.C. Saunders, Department of Pure Mathematics, University of Waterloo
"Problems in Combinatorial and Analytic Number Theory"
We
focus
on
three
problems
in
combinatorial
and
analytic
number
theory.
The
first
problem
studies
the
random
Fibonacci
tree,
which
is
an
infinite
binary
tree
with
non-negative
integers
at
each
node.
The
root
consists
of
the
number
1 with
a
single
child,
also
the
number
1.
We
define
the
tree
recursively
in
the
following
way:
if
x is
the
parent
of
y,
then
y has
two
children,
namely
|x-y| and
x+y.
This
tree
was
studied
by
Benoit
Rittaud
who
proved
that
any
pair
of
integers
a,b that
are
coprime
occur
as
a
parent-child
pair
infinitely
often.
We
extend
his
results
by
determining
the
probability
that
a
random
infinite
walk
in
this
tree
contains
exactly
one
pair
(1,1),
that
being
at
the
root
of
the
tree.
Also,
we
give
tight
upper
and
lowerbounds
on
the
number
of
occurrences
of
any
specific
coprime
pair
(a,b) at
any
given
fixed
depth
in
the
tree.
The
second
problem
studies
sieve
methods
in
combinatorics.
We
apply
the
Turán
sieve
and
the
simple
sieve
developed
by
Ram
Murty
and
Yu-Ru
Liu
to
study
problems
in
random
graph
theory.
More
specifically,
we
obtain
bounds
on
the
probability
of
a
graph
having
diameter
2 (or
diameter
3 in
the
case
of
bipartite
graphs).
A
surprising
feature
revealed
in
these
results
is
that
the
Turán
sieve
and
the
simple
sieve
"almost
completely''
complement
each
other.
The
third
problem
studies
the
Mahler
measure
of
a
polynomial
with
integer
coefficients.
There
are
many
results
that
provide
a
lowerbound
on
the
Mahler
measure
of
certain
classes
of
polynomials
by
Chris
Smyth,
Peter
Borwein,
Kevin
Hare,
and
Michael
Mossinghoff
to
name
but
a
few.
Here
we
give
a
lowerbound
of
the
Mahler
measure
on
a
set
of
polynomials
that
are
"almost"
reciprocal.
Here
"almost"
reciprocal
means
that
the
outermost
coefficients
of
each
polynomial
mirror
each
other
in
proportion,
while
this
pattern
breaks
down
for
the
innermost
coefficients.
MC 5403