Revista Matemática Iberoamericana


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Published online first: 2019-12-17
DOI: 10.4171/rmi/1138

The exact power law for Buffon’s needle landing near some random Cantor sets

Shiwen Zhang[1]

(1) University of Minnesota, Minneapolis, USA

In this paper, we study the Favard length of some random Cantor sets of Hausdorff dimension 1. We start with a unit disk in the plane and replace the unit disk by $4$ disjoint subdisks (with equal distance to each other) of radius $1/4$ inside and tangent to the unit disk. By repeating this operation in a self-similar manner and adding a random rotation in each step, we can generate a random Cantor set ${\cal D}(\omega)$. Let ${\cal D}_n$ be the $n$-th generation in the construction, which is comparable to the $4^{-n}$-neighborhood of ${\cal D}$. We are interested in the decay rate of the Favard length of these sets ${\cal D}_n$ as $n\to\infty$, which is the likelihood (up to a constant) that "Buffon's needle'' dropped randomly will fall into the $4^{-n}$-neighborhood of ${\cal D}$. It is well known that the lower bound of the Favard length of ${\cal D}_n(\omega)$ is a constant multiple of $n^{-1}$. We show that the upper bound of the Favard length of ${\cal D}_n(\omega)$ is $C n^{-1}$ for some $C > 0$ in the average sense. We also prove the a similar linear decay for the Favard length of ${\cal D}^d_n(\omega)$, which is the $d^{-n}$-neighborhood of a self-similar random Cantor set with degree $d$ greater than $4$. Notice that in the non-random case where the self-similar set has degree greater than $4$, the best known result for the decay rate of the Favard length is $e^{-c\sqrt {\log n}}$.

Keywords: Fractals, orthogonal projection, other geometric measure theory

Zhang Shiwen: The exact power law for Buffon’s needle landing near some random Cantor sets. Rev. Mat. Iberoam. Electronically published on December 17, 2019. doi: 10.4171/rmi/1138 (to appear in print)