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

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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 D(ω). Let Dₙ be the n-th generation in the construction, which is comparable to the 4⁻ⁿ-neighborhood of D. We are interested in the decay rate of the Favard length of these sets Dₙ as n→∞, which is the likelihood (up to a constant) that "Buffon's needle" dropped randomly will fall into the 4⁻ⁿ-neighborhood of 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 Dₙ(ω) is constant multiple of n⁻¹. We show that the upper bound of the Favard length of Dₙ(ω) is C n⁻¹ for some C>0 in the average sense. We also prove the similar linear decay for the Favard length of Dᵈₙ(ω) 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√ n.