Asymptotic order of the quantization errors for self-affine measures on Bedford-McMullen carpets

TL;DR

This paper establishes the asymptotic behavior of quantization errors for self-affine measures on Bedford-McMullen carpets, showing they decay at a specific rate independent of horizontal fibers, with positive finite quantization coefficients.

Contribution

It proves that the quantization coefficients are positive and finite for all orders, and that the quantization error scales as k^{-1/s_r}, independent of horizontal fibers.

Findings

Quantization coefficients are positive and finite.

Quantization error scales as k^{-1/s_r}.

Result is independent of horizontal fibers.

Abstract

Let $E$ be a Bedford-McMullen carpet determined by a set of affine mappings $(f_{ij})_{(i,j)\in G}$ and $\mu$ a self-affine measure on $E$ associated with a probability vector $(p_{ij})_{(i,j)\in G}$. We prove that, for every $r\in(0,\infty)$, the upper and lower quantization coefficient are always positive and finite in its exact quantization dimension $s_r$. As a consequence, the $k$th quantization error for $\mu$ of order $r$ is of the same order as $k^{-\frac{1}{s_r}}$. In sharp contrast to the Hausdorff measure for Bedford-McMullen carpets, our result is independent of the horizontal fibres of the carpets.