Asymptotic uniformity of the quantization error for Moran measures on $\mathbb{R}^1$

Abstract

Let $E$ be a Moran set on $\mathbb{R}^1$ associated with a 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 $\mu$ be the infinite product measure (Moran measure) on $E$ associated with a sequence $(\mathcal{P}_k)_{k\geq1}$ of positive probability vectors with $\mathcal{P}_k=(p_{k,j})_{j=1}^{n_k},k\geq 1$. We assume that \[ \inf_{k\geq1}\min_{1\leq j\leq n_k}c_{k,j}>0,\;\inf_{k\geq1}\min_{1\leq j\leq n_k}p_{k,j}>0. \] For every $n\geq 1$, let $\alpha_n$ be an $n$ optimal set in the quantization for $\mu$ of order $r\in(0,\infty)$ and $\{P_a(\alpha_n)\}_{a\in\alpha_n}$ an arbitrary Voronoi partition with respect to $\alpha_n$. For every $a\in\alpha_n$, we write $I_a(\alpha,\mu):=\int_{P_a(\alpha_n)}d(x,\alpha_n)^rd\mu(x)$ and \[ \underline{J}(\alpha_n,\mu):=\min_{a\in\alpha_n}I_a(\alpha,\mu),\; \overline{J}(\alpha_n,\mu):=\max_{a\in\alpha_n}I_a(\alpha,\mu). \] We show that $\underline{J}(\alpha_n,\mu),\overline{J}(\alpha_n,\mu)$ and $e^r_{n,r}(\mu)-e^r_{n+1,r}(\mu)$ are of the same order as $\frac{1}{n}e^r_{n,r}(\mu)$, where $e^r_{n,r}(\mu):=\int d(x,\alpha_n)^rd\mu(x)$ is the $n$th quantization error for $\mu$ of order $r$. In particular, for the class of Moran measures on $\mathbb{R}^1$, our result shows that a weaker version of Gersho's conjecture holds.