Assouad-like dimensions of random Moran measures

Abstract

In this paper, we determine the almost sure values of the $\Phi$-dimensions of random measures supported on random Moran sets that satisfy a uniform separation condition. The $\Phi$-dimensions are intermediate Assouad-like dimensions, the (quasi-)Assouad dimensions and $\theta$-Assouad spectrum being special cases. Their values depend on the size of $\Phi$, with one size coinciding with the Assouad dimension and the other coinciding with the quasi-Assouad dimension. We give many applications, including to equicontractive self-similar measures and $1$-variable random Moran measures such as Cantor-like measures with probabilities that are uniformly distributed. We can also deduce the $\Phi$-dimensions of the underlying random sets.