Asymptotic uniformity of the quantization error of self-similar measures

Let??? be a self-similar measure on ${\mathbb{R}^d}$ associated with a family of contractive similitudes {S ₁, . . . , S _N } and a probability vector {p ₁, . . . , p _N }. Let ${(\alpha_n)_{n=1}^\infty}$ be a sequence of n-optimal sets for??? of order r. For each n, we denote by ${\{P_a(\alpha_n) : a \in \alpha_n\}}$ a Voronoi partition of ${\mathbb{R}^d}$ with respect to ?? _n . Under the strong separation condition for {S ₁, . . . , S _N }, we show that the nth quantization error of??? of order ${r \in 1, \infty)}$ satisfies the following asymptotic uniformity property: $$\int \limits _{P_a(\alpha_n)}{\rm d}(x, a)^rd\mu(x) \asymp \frac{1}{n}V_{n,r}(\mu),\quad {\rm for\,all}\,a \in \alpha_n.$$