Assouad-like dimensions of a class of random Moran measures II -- non-homogeneous Moran sets

Abstract

In this paper, we determine the almost sure values of the $\hat{I}¦$-dimensions of random measures $\hat{I}¼$ supported on random Moran sets in ${\mathbb{R}}^d$ that satisfy a uniform separation condition. This paper generalizes earlier work done on random measures on homogeneous Moran sets \cite{HM} to the case of unequal scaling factors. The $\hat{I}¦$-dimensions are intermediate Assouad-like dimensions with the (quasi-)Assouad dimensions and the $\hat{I}¸$-Assouad spectrum being special cases. The almost sure value of ${\dim }_{\hat{I}}¦\hat{I}¼$ exhibits a threshold phenomena, with one value for ``large'' $\hat{I}¦$ (with the quasi-Assouad dimension as an example of a ``large'' dimension) and another for ``small'' $\hat{I}¦$ (with the Assouad dimension as an example of a ``small'' dimension). We give many applications, including where the scaling factors are fixed and the probabilities are uniformly distributed. The almost sure $\hat{I}¦$ dimension of the underlying random set is also a consequence of our results.

We added the details for the lower dimension to Lemma 2, which requires the doubling condition on the dimension function