Abstract

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.

Authors