Quantization dimension of probability measures (Q6640553)

In quantization theory, the quantization dimensions (the upper one \(\overline{D}(\mu)\) and the lower one \(\underline{D}(\mu)\)) of a probability measure \(\mu\) are studied. These dimensions characterize the asymptotic rate at which this measure is approximated by measures with finite support. The quantization dimensions of a probability measure \(\mu\) defined on a metric compact space \((X,\rho)\) are known not to exceed the corresponding box dimensions \(\dim_B\) of its support. Inspired by this investigation, in this paper the author proves the main result which states that on any metric compact space \(X\) of box dimension \(\dim_BX=a \leqslant\infty\), for two arbitrary numbers \(b\in [0,a]\) and \(c\in [b,a]\) there is a probability measure \(\mu\) such that its lower quantization dimension \(\underline{D}(\mu)\) is \(b\) and its upper quantization dimension \(\overline{D}(\mu)\) is \(c\).