Convergence order of the quantization error for self-affine measures on Lalley-Gatzouras carpets

TL;DR

This paper investigates the asymptotic behavior of quantization errors for self-affine measures on Lalley-Gatzouras carpets, establishing bounds on quantization coefficients and generalizing previous results.

Contribution

It introduces new methods to bound quantization errors and constructs auxiliary measures, extending prior work from Bedford-McMullen to Lalley-Gatzouras carpets.

Findings

Quantization coefficients are bounded away from zero and infinity.

The results generalize previous work on Bedford-McMullen carpets.

New techniques involve bounding errors and applying Prohorov's theorem.

Abstract

Let $E$ be a Lalley-Gatzouras carpet determined by a set of contractive affine mappings $\{f_{ij}\}_{(i,j)\in G}$. We study the asymptotics of quantization error for the self-affine measures $\mu$ on $E$. We prove that the upper and lower quantization coefficient for $\mu$ are both bounded away from zero and infinity in the exact quantization dimension. This significantly generalizes the previous work concerning the quantization for self-affine measures on Bedford-McMullen carpets. The new ingredients lie in the method to bound the quantization error for $\mu$ from below and that to construct auxiliary measures by applying Prohorov's theorem.