The Power Law For Buffon's Needle Landing Near the Sierpinski Gasket

Abstract

In this paper we get a power estimate from above of the probability that Buffon's needle will land within distance 3^{-n} of Sierpinski's gasket of Hausdorff dimension 1. In comparison with the case of 1/4 corner Cantor set considered in Nazarov, Peres, and the second author: we still need the technique of arXiv:0801.2942 for splitting the directions to good and bad ones, but the case of Sierpinski gasket is considerably more generic and lacks symmetry, resulting in a need for much more careful estimates of zeros of the Fourier transform of Cantor measure.