On the Intermediate Values of the Lower Quantization Dimension.

It is well known that the lower quantization dimension of a Borel probability measure given on a metric compact set does not exceed the lower box dimension of . We prove the following intermediate value theorem for the lower quantization dimension of probability measures: for any nonnegative number smaller that the dimension of the compact set , there exists a probability measure on with support such that . The number characterizes the asymptotic behavior of the lower box dimension of closed -neighborhoods of zero-dimensional, in the sense of , closed subsets of as . For a wide class of metric compact sets, the equality holds. [ABSTRACT FROM AUTHOR]