Quantization for uniform distributions on stretched Sierpi\'nski triangles

TL;DR

This paper investigates the optimal quantization of a uniform distribution on a stretched Sierpiński triangle, determining optimal sets and errors for all n, and analyzing the quantization dimension and coefficient.

Contribution

It explicitly computes optimal n-means and quantization errors for all n on a stretched Sierpiński triangle, and shows the non-existence of the quantization coefficient.

Findings

Optimal n-means and errors are determined for all n ≥ 2.

Quantization dimension exists but the quantization coefficient does not.

The measure exhibits unique quantization properties due to its fractal structure.

Abstract

In this paper, we have considered a uniform probability distribution supported by a stretched Sierpi\'nski triangle. For this probability measure, the optimal sets of $n$-means and the $n$th quantization errors are determined for all $n\geq 2$. In addition, it is shown that the quantization coefficient for such a measure does not exist though the quantization dimension exists.