The geometry of fractal percolation

TL;DR

This paper investigates the geometric properties of fractal percolation, extending classical projection theorems to all directions and more general projections, with results on measure and interior properties of projections.

Contribution

It provides stronger versions of Marstrand's theorem for fractal percolation, applicable to all directions and general projections, including radial ones, with new measure and interior results.

Findings

Projections of fractal percolation have positive measure in all directions.

Projections with Hausdorff dimension > 1 have nonempty interior.

Classical projection theorems are extended to more general settings.

Abstract

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: {itemize} the statements work for all directions, not almost all, the statements are true for more general projections, for example radial projections onto a circle, in the case $\dim_H >1$, each projection has not only positive Lebesgue measure but also has nonempty interior. {itemize}