Distribution of random Cantor sets on tubes

Changhao Chen Department of Mathematical Sciences, University of Oulu

Classical Analysis and ODEs mathscidoc:1701.05006

Arkiv for Matematik, 54, (1), 39-54, 2014.10
We show that there exist $(d-1)$ -Ahlfors regular compact sets $E \subset\mathbb{R}^{d}$ , $d\geq2$ 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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@inproceedings{changhao2014distribution,
  title={Distribution of random Cantor sets on tubes},
  author={Changhao Chen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203641713871092},
  booktitle={Arkiv for Matematik},
  volume={54},
  number={1},
  pages={39-54},
  year={2014},
}
Changhao Chen. Distribution of random Cantor sets on tubes. 2014. Vol. 54. In Arkiv for Matematik. pp.39-54. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203641713871092.
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