Convergence order of the quantization error for self-affine measures on Lalley-Gatzouras carpets
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911583221841920 |
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| author | Zhu, Sanguo |
| author_facet | Zhu, Sanguo |
| contents | 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 $μ$ on $E$. We prove that the upper and lower quantization coefficient for $μ$ 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 $μ$ from below and that to construct auxiliary measures by applying Prohorov's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06875 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence order of the quantization error for self-affine measures on Lalley-Gatzouras carpets Zhu, Sanguo Classical Analysis and ODEs Metric Geometry 28A80, 28A75 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 $μ$ on $E$. We prove that the upper and lower quantization coefficient for $μ$ 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 $μ$ from below and that to construct auxiliary measures by applying Prohorov's theorem. |
| title | Convergence order of the quantization error for self-affine measures on Lalley-Gatzouras carpets |
| topic | Classical Analysis and ODEs Metric Geometry 28A80, 28A75 |
| url | https://arxiv.org/abs/2508.06875 |