Vilmos Komornik, University of Strasbourg
"Topological and fractal properties of non-integer base expansions"
Given
a
base
$1<q\le
2$,
following
R\'enyi
(1957)
we
consider
expansions
of
the
form
\begin{equation*}
x=\sum_{i=1}^{\infty}\frac{c_i}{q^i}
\end{equation*}
with
digits
$c_i$
belonging
to
$\{0,1\}$. In
case
$q=2$
of
the
familiar
binary
expansions
every
$x\in
[0,1]$
has
an
expansion,
and
this
is
unique
except
the
dyadic
rational
numbers
that
have
two
expansions.
The case $1<q<2$ is radically different: almost every $x\in [0,\frac{1}{q-1}]$ has a continuum of expansions. However, there may still be many numbers having a unique expansion, and the purpose of this talk is to investigate the sets $U_q$ of such "univoque'' numbers.
We determine the values of $q$ for which $U_q$ is closed or even a Cantor set. These values have unexpected number-theoretical properties. Moreover, we determine the Hausdorff dimension of the sets $U_q$. It turns out to depend on $q$ in a quite intricate way.
Our survey is mostly based on joint research with P. Erdos, I. Joo, P. Loreti, M. de Vries, D. Kong and W. Li.
MC 5417