On the Intermediate Values of the Lower Quantization Dimension

Abstract

It is well known that the lower quantization dimension \(\underline{D}(\mu)\) of a Borel probability measure \(\mu\) given on a metric compact set \((X,\rho)\) does not exceed the lower box dimension \(\underline{\dim}_BX\) of \(X\). We prove the following intermediate value theorem for the lower quantization dimension of probability measures: for any nonnegative number \(a\) smaller that the dimension \(z\underline{\dim}_BX\) of the compact set \(X\), 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 box 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 compact sets, the equality \(z\underline{\dim}_BX=\underline{\dim}_BX\) holds.