关于Moran集上加倍测度的渐近量化误差
On the asymptotic quantization error for the doubling measures on Moran sets
摘要：
研究了一类Moran集$E\subset\mathbb{R}^q$上支持的加倍概率测度$μ$的量化误差.对于每个$n\geq 1$，令$α_n$为$r$阶的$μ$的任意$n$ -最优集，而$\{P_a(α_n)\}_{a\inα_n}$为关于$α_n$的任意Voronoi划分.我们用$I_a(α_n,μ)$表示积分$\int_{P_a(α_n)}d(x,a)^rdμ(x)$，并定义\begin{eqnarray*} \underline{J}(α_n,μ):=\min\limits_{a\inα_n}I_a(α_n,μ),\; \overline{J}(α_n,μ):=\max\limits_{a\inα_n}I_a(α_n,μ). \end{eqnarray*}。令$e_{n,r}(μ)$表示阶为$r$的$μ$的第$n$个量化误差。假设$E$的开集条件的一个版本，我们证明了\[ \underline{J}(α_n,μ),\overline{J}(α_n,μ)\asymp\frac{1}{n}e_{n,r}^r(μ). \]这个结果表明，对于Moran集$E$上的加倍测度，Gersho猜想的一个弱版本成立。
英文摘要：
We study the quantization errors for the doubling probability measures $μ$ which are supported on a class of Moran sets $E\subset\mathbb{R}^q$. For each $n\geq 1$, let $α_n$ be an arbitrary $n$-optimal set for $μ$ of order $r$ and $\{P_a(α_n)\}_{a\inα_n}$ an arbitrary Voronoi partition with respect to $α_n$. We denote by $I_a(α_n,μ)$ the integral $\int_{P_a(α_n)}d(x,a)^rdμ(x)$ and define \begin{eqnarray*} \underline{J}(α_n,μ):=\min\limits_{a\inα_n}I_a(α_n,μ),\; \overline{J}(α_n,μ):=\max\limits_{a\inα_n}I_a(α_n,μ). \end{eqnarray*} Let $e_{n,r}(μ)$ denote the $n$th quantization error for $μ$ of order $r$. Assuming a version of the open set condition for $E$, we prove that \[ \underline{J}(α_n,μ),\overline{J}(α_n,μ)\asymp\frac{1}{n}e_{n,r}^r(μ). \] This result shows that, for the doubling measures on Moran sets $E$, a weak version of Gersho's conjecture holds.
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