Quantization for uniform distributions of Cantor dusts on ${\mathbb{R}}^2$

Abstract

Let $P$ be a Borel probability measure on ${\mathbb{R}}^2$ supported by the Cantor dusts generated by a set of $4^u,u\ge 1$, contractive similarity mappings satisfying the strong separation condition. For this probability measure, we determine the optimal sets of $n$-means and the $n$th quantization errors for all $n\ge 2$. In addition, it is shown that though the quantization dimension of the measure $P$ is known, the quantization coefficient for $P$ does not exist.