On the asymptotic quantization error for the doubling measures on Moran sets

We study the quantization errors for the doubling probability measures $\mu$ which are supported on a class of Moran sets $E\subset {\mathbb{R}}^q$. For each $n\ge 1$, let ${\alpha }_n$ be an arbitrary $n$-optimal set for $\mu$ of order $r$ and $\{P_a({\alpha }_n){\}}_{a\in {\alpha }_n}$ an arbitrary Voronoi partition with respect to ${\alpha }_n$. We denote by $I_a({\alpha }_n,\mu )$ the integral ${\int }_{P_a({\alpha }_n)}d(x,a{)}^{r}d\mu (x)$ and define \begin{eqnarray*} \underline{J}(\alpha_n,\mu):=\min\limits_{a\in\alpha_n}I_a(\alpha_n,\mu),; \overline{J}(\alpha_n,\mu):=\max\limits_{a\in\alpha_n}I_a(\alpha_n,\mu). \end{eqnarray*} Let $e_{n,r}(\mu )$ denote the $n$th quantization error for $\mu$ of order $r$. Assuming a version of the open set condition for $E$, we prove that $$\underset{‾}{J}({\alpha }_n,\mu ),\bar{J}({\alpha }_n,\mu )\asymp \frac{1}{n}{e}_{n,r}^r(\mu )\mathrm{.}$$ This result shows that, for the doubling measures on Moran sets $E$, a weak version of Gersho’s conjecture holds.