Weaker quantization dimension results for self-similar measures

TL;DR

This paper explores the quantization dimension of self-similar measures under relaxed separation conditions, providing new insights into approximation errors in measure spaces.

Contribution

It introduces weaker conditions for quantization dimension results and analyzes approximation of measure spaces using geometric mean error.

Findings

Quantization dimension results hold under sub-IFS separation conditions.

Analysis of approximation of probability measure spaces with geometric mean error.

Extension of quantization theory to less restrictive self-similar measure conditions.

Abstract

In this paper, we investigate the quantization dimension of self-similar measures, particularly when the IFS does not satisfy the separation condition, but the sub-IFS at some level satisfies the separation condition. Further, we study the approximation of the space of Borel probability measures $\mathcal{P}(\mathbb{R}^m)$ with respect to the geometric mean error, i.e., the quantization dimension of order zero.