Abstract:

Let $E$ be a Moran set on ${{\Bbb R}} ^{1}$ associated with a bounded closed interval $J$ and two sequences $(n_{k})^{\propto}_{k=1}$ and ${\cal C}_{k}=((c_{k, j})_{j=1}^{n_{k}})_{k\geq 1}$ of numbers. Let $\mu$ be the Moran measure on $E$ determined by a sequence $({\cal P}_{k})_{k\geq1}$ of positive probability vectors. For every $n\geq 1$, let $C_{n}(\mu)$ denote the collection of all the $n$-optimal sets for $\mu$ with respect to the geometric mean error; let $\alpha_n\in C_n(\mu)$ and $\{P_{a}(\alpha_{n})\}_{a\in\alpha_{n}}$ be an arbitrary Voronoi partition with respect to $\alpha_n$. We prove that $\min\limits_{a\in\alpha_n}\mu (P_{a}(\alpha_{n})), \max\limits_{a\in\alpha_n}\mu (P_{a}(\alpha_{n}))\asymp n^{-1}.$ For each $a\in\alpha_{n}$, we show that the set $P_{a}(\alpha_{n})$ contains a closed interval of radius $d_{2}|P_{a}(\alpha_{n})\cap E|$ which is centered at $a$, where $d_{2}$ is a constant and $|B|$ denotes the diameter of a set $B\subset{{\Bbb R}} ^1$. Let $e_n(\mu)$ denote the $n$th geometric mean error for $\mu$ and $\hat{e}_n(\mu):=\log e_n(\mu)$. We show that $\hat{e}_n(\mu)-\hat{e}_{n+1}(\mu)\asymp n^{-1}$.

Key words: Geometric mean error, Optimal Voronoi partition, Moran measure