On the Range of the Quantization Dimension of Probability Measures on a Metric Compactum
Aleksandr Vladimirovich Ivanov · Siberian Mathematical Journal · 2022
The quantization dimension of a probability measure on a metric compactum $ X $ does not exceed the box dimension of the support of the measure. We prove the following intermediate value theorem for the upper quantization dimension: If $ X $ is a metric compact space whose upper box dimension is equal to $ a\leq\infty $ then for every real $ b $ such that $ 0\leq b\leq a $ there exists a probability measure on $ X $ whose support is $ X $ and whose upper quantization dimension is $ b $ .