{"paper":{"title":"Asymptotic local uniformity of the quantization error for Ahlfors-David probability measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Sanguo Zhu","submitted_at":"2017-08-25T09:07:46Z","abstract_excerpt":"Let $\\mu$ be an Ahlfors-David probability measure on $\\mathbb{R}^q$, namely, there exist some constants $s_0>0$ and $\\epsilon_0,C_1,C_2>0$ such that \\[ C_1\\epsilon^{s_0}\\leq\\mu(B(x,\\epsilon))\\leq C_2\\epsilon^{s_0},\\;\\epsilon\\in(0,\\epsilon_0),\\;x\\in{\\rm supp}(\\mu). \\] For $n\\geq 1$, let $\\alpha_n$ be an $n$-optimal set for $\\mu$ of order $r$ and $(P_a(\\alpha_n))_{a\\in\\alpha_n}$ an arbitrary Voronoi partition with respect to $\\alpha_n$. The $n$th quantization error $e_{n,r}(\\mu)$ for $\\mu$ of order $r$ is given by $e^r_{n,r}(\\mu):=\\int d(x,\\alpha_n)^rd\\mu(x)$. Write \\[ I_a(\\alpha,\\mu):=\\int_{P_a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.07657","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}