The exact Power Law for Buffon's needle landing near some Random Cantor Sets

Abstract: 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 in [P. Mattila, Orthogonal projections, Riesz capacities, and Minkowski content, Indiana Univ. Math. J. 39 (1990), no. 1, 185-198] that the lower bound of the Favard length of ${\cal D}_n(\omega)$ is 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 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 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 $e^{-c\sqrt {\log n}}$.