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: $$int limits _{P_a(alpha_n)}{rm d}(x, a)^rdmu(x) asymp frac{1}{n}V_{n,r}(mu),quad {rm for,all},a in alpha_n.$$