Ahlfors-David概率测度量化误差的渐近局部一致性
Asymptotic local uniformity of the quantization error for Ahlfors-David probability measures
摘要：
设$μ$是$\mathbb{R}^q$上的Ahlfors-David概率测度，即存在一些常数$s_0>0$和$ε_0,C_1,C_2>0$使得\[ C_1ε^{s_0}\leqμ(B(x,ε))\leq C_2ε^{s_0},\;ε\in(0,ε_0),\;x\in{\rm supp}(μ). \]对于$n\geq 1$，设$α_n$是$r$阶的$μ$的$n$ -最优集，而$(P_a(α_n))_{a\inα_n}$是关于$α_n$的任意Voronoi划分.阶数为$r$的$μ$的$n$ th量化误差$e_{n,r}(μ)$由$e^r_{n,r}(μ):=\int d(x,α_n)^rdμ(x)$给出。写\[ I_a(α,μ):=\int_{P_a(α_n)}d(x,α_n)^rdμ(x),\;a\inα_n. \]我们证明了，$\underline{J}(α_n,μ):=\min_{a\inα_n}I_a(α,μ)$，$\overline{J}(α_n,μ):=\max_{a\inα_n}I_a(α,μ)$和误差差$e^r_{n,r}(μ)-e^r_{n+1,r}(μ)$与$\frac{1}{n}e^r_{n,r}(μ)$具有相同的顺序。这与格拉夫和卢施吉的工作一起，得出了所有上述三个量与$n^{-(1+\frac{r}{s_0})}$具有相同的数量级。
英文摘要：
Let $μ$ be an Ahlfors-David probability measure on $\mathbb{R}^q$, namely, there exist some constants $s_0>0$ and $ε_0,C_1,C_2>0$ such that \[ C_1ε^{s_0}\leqμ(B(x,ε))\leq C_2ε^{s_0},\;ε\in(0,ε_0),\;x\in{\rm supp}(μ). \] For $n\geq 1$, let $α_n$ be an $n$-optimal set for $μ$ of order $r$ and $(P_a(α_n))_{a\inα_n}$ an arbitrary Voronoi partition with respect to $α_n$. The $n$th quantization error $e_{n,r}(μ)$ for $μ$ of order $r$ is given by $e^r_{n,r}(μ):=\int d(x,α_n)^rdμ(x)$. Write \[ I_a(α,μ):=\int_{P_a(α_n)}d(x,α_n)^rdμ(x),\;a\inα_n. \] We prove that, $\underline{J}(α_n,μ):=\min_{a\inα_n}I_a(α,μ)$, $\overline{J}(α_n,μ):=\max_{a\inα_n}I_a(α,μ)$ and the error difference $e^r_{n,r}(μ)-e^r_{n+1,r}(μ)$ are of the same order as $\frac{1}{n}e^r_{n,r}(μ)$. This, together with Graf and Luschgy's work, yields that all the above three quantities are of the same order as $n^{-(1+\frac{r}{s_0})}$.
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