Publications

Scientific publications

А.В. Иванов.
О вероятностных мерах с максимальной размерностью квантования
A.V. Ivanov. On probability measures with a maximum of quantization dimension // Transactions of Karelian Research Centre of Russian Academy of Science. No 7. Mathematical Modeling and Information Technologies. 2020. P. 57–61
Keywords: quantization dimension; box-dimension; weakly homogeneous compact space; Ahlfors space
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 dimBX=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 dimBX. An estimate is obtained for the lower quantization dimension of a probability measure μ satisfying the condition μ(B(x,ε)) ≥ cεγ 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)=dimBX for uniformly distributed measures (in the sense of the terminology adopted in geometric measure theory) and probability measures of compact metric Ahlfors spaces.
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Last modified: July 1, 2020