On intermediate values of quantization dimensions of idempotent measures

The quantization dimension $\dim_{\mathcal F}(\xi)$ is defined for any point $\xi$ of spaces of the form $\mathcal F(X)$, where $\mathcal F$ is a half-normal metrizable functor and $X$ is a metric compact space. An example of a quantization dimension is the classical box dimension $\dim_B$ of closed subsets of a compact space $X$. In this work, the functor $I$ of idempotent measures or Maslov measures is considered as $\mathcal F$. It is known that, for any idempotent measure $\mu\in I(X)$, its (upper and lower) quantization dimensions do not exceed the upper and lower box dimensions, respectively, of the space $X$. These inequalities motivate the question about intermediate values of the quantization dimensions of idempotent measures. The following theorem is proved: on any metric compact space $X$ of dimension $\dim_BX=a<\infty$, for any numbers $c\in[0,a]$ and $b\in[0,a/2)\cap[0 ,c]$, there is an idempotent measure whose lower quantization dimension is $b$ and whose upper quantization dimension is $c$. As follows from this theorem, if a metric compact space $X$ has positive box dimension, then $X$ always has an idempotent measure with a positive lower quantization dimension. Moreover, it is known that a similar statement for the box dimension is not true in the general case, since there exists a metric compact space whose box dimension is $1$ such that all of its proper closed subsets are zero-dimensional in the sense of the lower box dimension.