Distribution of random Cantor sets on Tubes

Abstract

We show that there exist $(d-1)$ - Ahlfors regular compact sets $E\subset {\mathbb{R}}^d,d\ge 2$ 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.

15 pages, 3 figures

References in corpus (4)

• The dimension of projections of fractal percolations
• Sets which are not tube null and intersection properties of random measures
• On the tube-occupancy of sets in ${\mathbb{R}}^d$
• Projections of Mandelbrot percolation in higher dimensions

Cited by in corpus (3)

• Covering the Sierpiński carpet with tubes
• The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals
• A class of random Cantor sets