Distribution of random Cantor sets on Tubes

TL;DR

This paper constructs specific random Cantor sets in Euclidean space that have controlled intersection properties with tubes, answering a question about the distribution of such fractal sets in geometric measure theory.

Contribution

It introduces a new probabilistic construction of Cantor sets with prescribed intersection bounds with tubes, resolving a question posed by T. Orponen.

Findings

Existence of (d-1)-Ahlfors regular sets with bounded intersection ratios with tubes

Construction of random Cantor sets satisfying the intersection property

Resolution of a geometric measure theory question by Orponen

Abstract

We show that there exist $(d-1)$ - Ahlfors regular compact sets $E \subset \mathbb{R}^{d}, d\geq 2$ such that for any $t< d-1$, we have \[ \sup_T \frac{\mathcal{H}^{d-1}(E\cap T)}{w(T)^t}<\infty \] where the supremum is over all tubes $T$ with width $w(T) >0$. This settles a question of T. Orponen. The sets we construct are random Cantor sets, and the method combines geometric and probabilistic estimates on the intersections of these random Cantor sets with affine subspaces.