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arXiv · 2508.06875

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

Sanguo Zhu
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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.

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Sanguo Zhu. 2025-08-09. Convergence order of the quantization error for self-affine measures on Lalley-Gatzouras carpets. https://doi.org/10.1016/j.jmaa.2026.130956

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