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

Let μ be an Ahlfors-David probability measure on Rq;therefore,there exist some constants s0> 0 and ε0,C1,C2> 0 such that C1εs0≤μ(B(x,ε))≤C2εs0 for all ε∈(0,ε0) and x ∈ supp(μ).For n≥ 1,let αn be an n-optimal set for μ of order r;furthermore,let {Pa(αn)}a∈αn be an arbitrary Voronoi partition with respect to αn.The n-th quantization error en,r(μ) for μ of order r can be defined as en,rr(μ):=∫ d(x,αn)r dμ(x).We define Ia(αn,μ):=∫Pa(αn) d(x,αn)r dμ(x),a ∈αn,and prove that,the three quantities ■ are of the same order as that of 1/nen,rr(μ).Thus,our result exhibits that,a weak version of Gersho’s conjecture holds true for the Ahlfors-David probability measures on Rq.