Optimal points for a probability distribution on a Cantor set

Given a probability measure $P$ on a compact subset
of  ${\mathbb R}^d$ and a natural number $n$, the {\it{$n$th quantization error of
$P$} is defined to be

$$V_n=\inf_{\ga} \int \min_{a\in\ga} \|x-a\|^2 dP(x),$$

where the infimum is taken over all subsets $\alpha$ of ${\mathbb R}^d$ with
card $\alpha\leq n$, and $\| \cdot\|$ denotes the Euclidean norm on ${\mathbb
R}^d$. A set $\alpha$ for which the infimum is
achieved is called a {\it {$n$-optimal set}. 
The {\it {Quantization dimension} for the probability measure $P$
is defined by

$$D(P)=\lim_{n\to \infty} \frac{2\log n}{-\log V_n},$$

and corresponds to the rate how fast $V_n$ goes to zero as $n$ tends
to infinity.

In this talk, we consider the Cantor set equipped with the natural
homogeneous probability measure on it, and discuss  the quantization
errors of the measure and $n$-optimal
sets for $n \geq 1$, and the quantization dimension.  Some open
problems in the area will be pointed out.