The quasi-Assouad dimension for stochastically self-similar sets

TL;DR

This paper proves that for a broad class of stochastically self-similar sets, the quasi-Assouad dimension almost surely equals the Hausdorff dimension, under certain conditions, extending understanding of fractal dimensions in random sets.

Contribution

It establishes the almost sure equality of quasi-Assouad and Hausdorff dimensions for self-similar random recursive sets under mild assumptions, including the uniform open set condition.

Findings

Quasi-Assouad dimension equals Hausdorff dimension almost surely.

Results apply to Mandelbrot percolation and similar random fractals.

Discussion includes random homogeneous and V-variable sets.

Abstract

The class of stochastically self-similar sets contains many famous examples of random sets, e.g. Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and $V$-variable sets and the removal of overlap conditions.