Asymptotic uniformity of the quantization error for the Ahlfors-David probability measures

Abstract

Let μ be an Ahlfors-David probability measure on ℝ^q; therefore, there exist some constants s₀ > 0 and ϵ₀, C₁, C₂ > 0 such that ${C_1}{\varepsilon ^{s_0}} \leqslant \mu \left( {B\left( {x,\varepsilon } \right)} \right) \leqslant {C_2}{\varepsilon ^{s_0}}$ for all ϵ ∈ (0, ϵ₀) and x ∈ supp(μ). For n ⩾ 1, let α_n be an n-optimal set for μ of order r; furthermore, let $\left\{ {{P_a}\left( {{\alpha _n}} \right)} \right\}_{a \in {\alpha _n}}$ be an arbitrary Voronoi partition with respect to α_n. The n-th quantization error e_n,r(μ) for μ of order r can be defined as e^r_n,r (μ):= ∫ d(x,α_n)^rdμ(x). We define ${I_a}\left( {{\alpha _n},\mu } \right):={\smallint _{{P_a}\left( {{\alpha _n}} \right)}}d{\left( {x,{\alpha _n}} \right)^r}d\mu \left( x \right)$, α ∈ α_n, and prove that, the three quantities

$$\underline J \left( {{\alpha _n},\mu } \right) := \mathop {\min }\limits_{a \in {\alpha _n}} \,{I_a}\left( {{\alpha _n},\mu } \right),\,\,\,\,\,\,\,\,\,\,\overline J \left( {{\alpha _n},\mu } \right) := \mathop {\max }\limits_{a \in {\alpha _n}} \,{I_a}\left( {{\alpha _n},\mu } \right),\,\,\,\,\,\,\,\,\,\,\,e_{n,r}^r\left( \mu \right) - e_{n + 1,\,r}^r\left( \mu \right)$$

are of the same order as that of $\frac{1} {n}e_{n,r}^r\left( \mu \right)$. Thus, our result exhibits that, a weak version of Gersho’s conjecture holds true for the Ahlfors-David probability measures on ℝ^q.