The Buffon's needle problem for random planar disk-like Cantor sets

TL;DR

This paper investigates the decay rate of the probability that a Buffon needle lands close to a random planar Cantor set, establishing a universal decay rate of order 1/log(1/δ) across different models of randomness.

Contribution

It proves that a third model of random Cantor sets exhibits the same decay rate as previous models, suggesting a universal behavior for such sets.

Findings

Decay rate of Buffon needle probability is of order 1/log(1/δ)

Different models of random Cantor sets share the same decay order

Supports the hypothesis that 'reasonable' random Cantor sets have similar Favard length decay

Abstract

We consider a model of randomness for self-similar Cantor sets of finite and positive $1$-Hausdorff measure. We find the sharp rate of decay of the probability that a Buffon needle lands $\delta$-close to a Cantor set of this particular randomness. Two quite different models of randomness for Cantor sets, by Peres and Solomyak, and by Shiwen Zhang, appear to have the same order of decay for the Buffon needle probability: $\frac{c}{\log\frac{1}{\delta}}$. In this note, we prove the same rate of decay for a third model of randomness, which asserts a vague feeling that any "reasonable" random Cantor set of positive and finite length will have Favard length of order $\frac{c}{\log\frac{1}{\delta}}$ for its $\delta$-neighbourhood. The estimate from below was obtained long ago by Mattila.