Scientific publications

It is known that the quantization dimension of a probability measure on a metric compact space does not exceed the box-dimension of its support. In this connection, the following question naturally arises about intermediate values of quantization dimensions. Let (X,ρ) be a metric compact with box-dimension dim_BX=d. Is it true that for any a∈[0,d] there exists a probability measure μ with support supp(μ)=X for which the quantization dimension D(μ) is a? In this paper we consider a special case of this question concerning the existence of measures whose quantization dimension takes the largest possible value, which is equal to dim_BX. An estimate is obtained for the lower quantization dimension of a probability measure μ satisfying the condition μ(B(x,ε)) ≥ cε^γ for any point x∈X, where c and γ are positive constants (Theorem 1). This estimate implies the existence of the desired measures on weakly homogeneous compact spaces. Theorem 1 also implies the equality D(μ)=dim_BX for uniformly distributed measures (in the sense of the terminology adopted in geometric measure theory) and probability measures of compact metric Ahlfors spaces.