How likely is Buffon’s needle to fall near a planar Cantor set?

Let Λ be a compact planar set of positive finite one-dimensional Hausdorff measure. Suppose that the intersection of Λ with any rectifiable curve has zero length. Then a theorem of Besicovitch (1939) states that the orthogonal projection of Λ on almost all lines has zero length. Consequently, the probability p(Λ,ɛ) that a needle dropped at random will fall withindistance ɛ from Λ, tends to zero with ɛ. However, existing proofs do not yield any explicit upper bound tending to zero for p(Λ,ɛ), even in the simplest cases, e.g., whenΛ = K 2 is the Cartesian square of the middle-half Cantor set K. Inthis paper we establish such a bound for a class of selfsimilar sets Λ that includes K 2. We also determine the order of magnitude of p(Λ,ɛ) for certain stochastically self-similar sets Λ. Determining the order of magnitude of p(K 2,ɛ) is an unsolved problem.