On the Asymptotic Uniformity of the Quantization Error for Moran Measures on ℝ¹

Abstract

Let E be a Moran set on ℝ¹ associated with a bounded closed interval J and two sequences $(n_k)_{k=1}^\infty$ and $(\mathcal{C}_k=(c_{k,j})_{j=1}^{n_k})_{k\geq1}$. Let µ be the Moran measure on E associated with a sequence $(\mathcal{P}_k)_{k\geq1}$ of positive probability vectors with $(\mathcal{P}_k)=(p_{k,j})_{j=1}^{n_k}$, k ≥ 1. We assume that

$$\begin{array}{*{20}{c}} {\mathop {\inf }\limits_{k \geqslant 1} \mathop {\min }\limits_{1 \leqslant j \leqslant {n_k}} {c_{k,j}} > 0,}&{\mathop {\inf }\limits_{k \geqslant 1} \mathop {\min }\limits_{1 \leqslant j \leqslant {n_k}} {p_{k,j}} > 0.} \end{array}$$

.

For every n ≥ 1, let α_n be an n optimal set in the quantization for µ of order r ∈ (0, ∞) and $\{P_a(\alpha_n)\}_{a\in{\alpha_n}}$ an arbitrary Voronoi partition with respect to α_n. We write

$$\begin{array}{c}I_a(\alpha, \mu):=\int_{P_a(\alpha_n)}d(x, \alpha_n)^rd\mu(x), \;\;a\in\alpha_n;\\{\underline{J}(\alpha_n, \mu) := \mathop {\min }\limits_{a\in\alpha_n} I_a(\alpha, \mu)}, \;\;\overline{J}(\alpha_n, \mu) := \mathop {\max }\limits_{a\in\alpha_n}I_a(\alpha, \mu).\end{array}$$

We show that $\underline{J}(\alpha_n, \mu)$, $\overline{J}(\alpha_n, \mu)$ and ${\rm{e}}_{n,r}^r(\mu)-{\rm{e}}_{n+1,r}^r(\mu)$ are of the same order as $\frac{1}{n}{\rm{e}}_{n,r}^r(\mu)$, where ${\rm{e}}_{n,r}^r(\mu):=\int d(x, \alpha_n)^rd\mu(x)$ is the nth quantization error for µ of order r. In particular, for the class of Moran measures on ℝ¹, our result shows that a weaker version of Gersho’s conjecture holds.