On the Intermediate Values of the Lower Dimension of Quantization

It is well known that the lower dimension $\underline{D}(\mu)$ of quantization of a Borel probability measure $\mu$ given on a metric compactum $(X,\rho)$ does not exceed the lower capacitive dimension $\underline{\dim}_BX$ of $X$. We prove the following theorem on the intermediate values of the lower dimension of quantization of probability measures: for any nonnegative number $a$ smaller that the dimension $z\underline{\dim}_BX$ of the compactum $X$, there exists a probability measure $\mu_a$ on $X$ with support $X$ such that $\underline{D}(\mu_a)=a$. The number $z\underline{\dim}_BX$ characterizes the asymptotic behavior of the lower capacitive dimension of closed $\varepsilon$-neighborhoods of zero-dimensional, in the sense of $\dim_B$, closed subsets of $X$ as $\varepsilon\to 0$. For a wide class of metric compacta, the equality $z\underline{\dim}_BX=\underline{\dim}_BX$ holds.