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 $({\mathcal{C}}_k=(c_{k,j}{)}_{j=1}^{n_k}{)}_{k\ge 1}$. Let $\mu$ be the infinite product measure (Moran measure) on $E$ associated with a sequence $({\mathcal{P}}_k{)}_{k\ge 1}$ of positive probability vectors with ${\mathcal{P}}_k=(p_{k,j}{)}_{j=1}^{n_k},k\ge 1$. 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 $n\ge 1$, 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 $I_a(\alpha ,\mu ):={\int }_{P_a({\alpha }_n)}d(x,{\alpha }_n{)}^{r}d\mu (x)$ and \[ \underline{J}(α_n,μ):=\min_{a\inα_n}I_a(α,μ),\; \overline{J}(α_n,μ):=\max_{a\inα_n}I_a(α,μ). \] We show that $\underline{J}({\alpha }_n,\mu ),\overline{J}({\alpha }_n,\mu )$ and $e_{n,r}^r(\mu )-e_{n+1,r}^r(\mu )$ are of the same order as $\frac{1}{n}{e}_{n,r}^r(\mu )$, where $e_{n,r}^r(\mu ):=\int d(x,{\alpha }_n{)}^{r}d\mu (x)$ 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.