How Likely Is Buffon ' S Needle to Fall near Aplanar Cantor Set ?

@inproceedings{Peres2002HowLI,
  title={How Likely Is Buffon ' S Needle to Fall near Aplanar Cantor Set ?},
  author={Yuval Peres},
  year={2002}
}
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, ) that a needle dropped at random will fall within distance from Λ, tends to zero with . However, existing proofs do not yield any explicit upper bound tending to zero for p(Λ, ), even… CONTINUE READING