# Patterns in Random Fractals

## TL;DR

This paper investigates the presence of geometric configurations like angles, distances, and simplices in fractal percolation sets, establishing dimension thresholds for their almost sure existence and extending percolation theory to dependent structures.

## Contribution

It characterizes the dimension thresholds for various geometric configurations in fractal percolation sets and extends percolation results to settings with long-range dependencies.

## Key findings

- Identifies dimension thresholds for configurations in fractal percolation sets.
- Extends percolation theory to dependent structures with algebraic varieties.
- Analyzes intersections with transversal planes to determine configuration existence.

## Abstract

We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemer\'{e}di theorems for random discrete sets, we also consider the corresponding problem for sets of positive $\nu$-measure, where $\nu$ is the natural measure on $A$. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating the intersections of the products of $m$ independent realizations of $A$ with transversal planes and, more generally, algebraic varieties, and extend some well known features of independent percolation on trees to a setting with long-range dependencies.