Quantization dimension of probability measures

The quantization dimension of a probability measure defined on a metric compact space $X$ is known not to exceed the box dimension of its support. It is proved that on any metric compact space of box dimension $\dim_BX=a\leq\infty$, for arbitrary two numbers $b\in[0,a]$ and $c\in[b,a]$ there is a probability measure such that its lower quantization dimension is $b$ and its upper quantization dimension is $c$.
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