Quasi-Assouad dimensions for random measures supported on $[0,1]^d$

TL;DR

This paper introduces a probability distribution on measures in [0,1]^d and demonstrates that almost all such measures have infinite upper and zero lower quasi-Assouad dimensions, revealing extreme dimensional behaviors.

Contribution

It establishes a probabilistic framework showing typical measures have extreme Assouad dimension values, extending understanding of fractal dimensions in measure spaces.

Findings

Almost all measures have infinite upper quasi-Assouad dimension.

Almost all measures have zero lower quasi-Assouad dimension.

Results extend to other Assouad-like dimensions.

Abstract

We introduce a probability distribution on $\mathcal{P}([0,1]^d)$, the space of all Borel probability measures on $[0,1]^d$. Under this distribution, almost all measures are shown to have infinite upper quasi-Assouad dimension and zero lower quasi-Assouad dimension (hence the upper and lower Assouad dimensions are almost surely infinite or zero). We also indicate how the results extend to other Assouad-like dimensions.