# Asymptotic order of the geometric mean error for self-affine measures on Bedford-McMullen carpets

## TL;DR

This paper investigates the asymptotic behavior of the geometric mean error in quantization for self-affine measures on Bedford-McMullen carpets, establishing its order as inversely proportional to the Hausdorff dimension.

## Contribution

It proves that under a separation condition, the geometric mean error decays at a rate of $n^{-1/s_0}$, linking quantization error to Hausdorff dimension for these measures.

## Key findings

- Geometric mean error decays as $n^{-1/s_0}$.
- Order of error established under separation condition.
- Connects quantization error rate to Hausdorff dimension.

## Abstract

Let $E$ be a Bedford-McMullen carpet associated with a set of affine mappings $\{f_{ij}\}_{(i,j)\in G}$ and let $\mu$ be the self-affine measure associated with $\{f_{ij}\}_{(i,j)\in G}$ and a probability vector $(p_{ij})_{(i,j)\in G}$. We study the asymptotics of the geometric mean error in the quantization for $\mu$. Let $s_0$ be the Hausdorff dimension for $\mu$. Assuming a separation condition for $\{f_{ij}\}_{(i,j)\in G}$, we prove that the $n$th geometric error for $\mu$ is of the same order as $n^{-1/s_0}$.