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

TL;DR

This paper investigates the optimal quantization of probability measures supported on Cantor dusts in , determining optimal sets and errors for all n, and analyzing the quantization dimension and coefficient.

Contribution

It provides explicit solutions for optimal n-means and quantization errors for measures on Cantor dusts, and shows the non-existence of the quantization coefficient.

Findings

Explicit optimal n-means and quantization errors for all n.

Quantization dimension is known, but the quantization coefficient does not exist.

The measure's support is on Cantor dusts generated by similarity mappings.

Abstract

Let $P$ be a Borel probability measure on $\mathbb R^2$ supported by the Cantor dusts generated by a set of $4^u,\ u\geq 1$, 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 $n\geq 2$. In addition, it is shown that though the quantization dimension of the measure $P$ is known, the quantization coefficient for $P$ does not exist.