Liam Orovec, Department of Pure Mathematics, University of Waterloo
"Unique Representations of Real Numbers in Non-Integer Bases"
When
looking
at
the
representation
of
numbers
in
non-integer
bases,
β-expansions,
we
often
find
an
infinite
number
of
expansions
for
any
given
real
number
under
any
given
base.
We
look
at
finding,
given
a
fixed
positive
real
number
x,
the
smallest
base
qs(x)
for
which
x
has
a
unique
qs(x)-expansion.
Beginning
with
x
=
1
we
find
the
ever
present
Thue-Morse
sequence
will
be
helpful
throughout
the
talk.
Having
found
our
constant
qKL
=
qs(1),
the
Komornik-Loreti
constant,
we
will
explore
when
qs(x)
<
qKL.
The
majority
of
this
talk
will
follow
the
Results
due
to
Derong
Kong
which
covers
the
case
where
are
expansions
have
only
digits
0
and
1,
in
what
time
that
remains
we
will
look
at
generalizing
these
results
for
larger
alphabets.
MC 5479