Some remarks on absolute continuity and quantization of probability measures (Q879076)

Let \(\mathcal M\) be the set of all probability measures on \(({\mathbb R}^d, {\mathcal B})\). A function \(\dim: {\mathcal M} \to {\mathbb R}\) is said to be increasing (respectively decreasing) if for all \(\nu, \mu \in {\mathcal M}\) and \(\nu \ll \mu\) implies \(\dim \nu \leq \dim \mu\) (respectively \(\dim \nu \geq \dim \mu\)). It can be seen that the upper (lower) Hausdorff and packing dimensions are increasing (decreasing). In the paper under review, the author proves that the upper and lower quantization dimensions of order \(r\) are not monotone, and provides sufficient conditions in terms of the so-called vanishing rates such that \(\nu \ll \mu\) implies \(\overline{D}_r(\nu) \leq \overline{D}_r(\mu)\). The author studies the set of quantization dimensions of measures which are absolutely continuous with respect to \(\mu \in {\mathcal M}\), and proves that the infimum on this set coincides with the lower packing dimension \(\dim_{\text{P}} \mu\), of \(\mu\). Moreover, this infimum can be attained provided that \(\dim_{\text{P}}^* \mu = \dim_{\text{P}} \mu.\) As an application, the author also determines the quantization dimension of a class of measures which are absolutely continuous with respect to some self-similar measure \(\mu\).