Fractal percolation is unrectifiable

TL;DR

This paper proves that certain fractal percolation sets are almost surely purely unrectifiable for all dimensions above a specific threshold, highlighting their complex geometric structure.

Contribution

It establishes a threshold $eta_0$ such that fractal percolation is almost surely purely $eta$-unrectifiable for all $eta>eta_0$, advancing understanding of fractal geometry.

Findings

Fractal percolation is almost surely purely unrectifiable above a certain dimension.

Existence of a threshold $eta_0$ depending on parameters.

Fractal percolation exhibits complex geometric properties.

Abstract

We show that there exists $0<\alpha_0<1$ (depending on the parameters) such that the fractal percolation is almost surely purely $\alpha$-unrectifiable for all $\alpha>\alpha_0$.