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

In this talk, we study the Favard length of some planar random Cantor sets of Hausdorff dimension one. We start with a unit disk  and replace it by 4 randomly distributed quarter subdisks. By repeating this operation in a self-similar, independent manner, we generate a random Cantor set D. Let Dn be the n-th generation in the construction. We are interested in the decay rate of the Favard length of  Dn as n → ∞, which is the likelihood  that “Buffon’s needle” dropped randomly will fall into the small neighborhood of D. It is well known that the lower bound of the Favard length of Dn is constant multiple of 1/n . We show that the upper bound of the Favard length of Dn is also constant multiple of 1/n in the average sense.