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 {\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}} .