Asymptotic local uniformity of the quantization error for Ahlfors-David probability measures

Let $\mu$ 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 $$C_1{ϵ}^{s_0}\le \mu (B(x,ϵ))\le C_2{ϵ}^{s_0},ϵ\in (0,{ϵ}_0),x\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu )\mathrm{.}$$ For $n\ge 1$, 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 $e_{n,r}^r(\mu ):=\int d(x,{\alpha }_n{)}^{r}d\mu (x)$. Write $$I_a(\alpha ,\mu ):={\int }_{P_a({\alpha }_n)}d(x,{\alpha }_n{)}^{r}d\mu (x),a\in {\alpha }_n\mathrm{.}$$ We prove that, $\underset{‾}{J}({\alpha }_n,\mu ):={\min }_{a\in {\alpha }_n}{I}_a(\alpha ,\mu )$, $\bar{J}({\alpha }_n,\mu ):={\max }_{a\in {\alpha }_n}{I}_a(\alpha ,\mu )$ and the error difference $e_{n,r}^r(\mu )-e_{n+1,r}^r(\mu )$ are of the same order as $\frac{1}{n}{e}_{n,r}^r(\mu )$. This, together with Graf and Luschgy’s work, yields that all the above three quantities are of the same order as $n^{-(1+\frac{r}{s_0})}$.