The Buffon's needle problem for random planar disk-like Cantor sets

Abstract

We consider a model of randomness for self-similar Cantor sets of finite and positive $1$-Hausdorff measure. We find the sharp rate of decay of the probability that a Buffon needle lands $\delta$-close to a Cantor set of this particular randomness. Two quite different models of randomness for Cantor sets, by Peres and Solomyak, and by Shiwen Zhang, appear to have the same order of decay for the Buffon needle probability: $\frac{c}{\log\frac{1}{\delta}}$. In this note, we prove the same rate of decay for a third model of randomness, which asserts a vague feeling that any "reasonable" random Cantor set of positive and finite length will have Favard length of order $\frac{c}{\log\frac{1}{\delta}}$ for its $\delta$-neighbourhood. The estimate from below was obtained long ago by Mattila.