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arXiv · 1410.1183

Distribution of random Cantor sets on Tubes

Changhao Chen
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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.

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Changhao Chen. 2014-10-05. Distribution of random Cantor sets on Tubes. https://doi.org/10.1007/s11512-015-0219-x

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