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,𝜖) that
a needle dropped at random will fall within distance 𝜖 from Λ, tends to
zero with 𝜖. However, existing proofs do not yield any explicit upper bound
tending to zero for p(Λ,𝜖), even in the simplest cases, e.g., when Λ = K2
is the Cartesian square of the middle-half Cantor set K. In this paper we
establish such a bound for a class of self-similar sets Λ that includes K2. We
also determine the order of magnitude of p(Λ,𝜖) for certain stochastically
self-similar sets Λ. Determining the order of magnitude of p(K2,𝜖) is an unsolved
problem.
