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

TL;DR

This paper estimates the probability decay rate that Buffon's needle lands near the Sierpinski gasket, extending previous work on similar fractal sets and employing Fourier analysis techniques.

Contribution

It provides a power law estimate for Buffon's needle landing probability near the Sierpinski gasket, adapting advanced Fourier analysis methods to a more complex fractal.

Findings

Established an upper bound power estimate for landing probability

Extended Fourier analysis techniques to less symmetric fractals

Demonstrated the applicability of splitting directions in complex fractal analysis

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.