The power law for the Buffon needle probability of the four-corner Cantor set

Abstract

Let $\mathcal {C}_n$ be the $n$th generation in the construction of the middle-half Cantor set. The Cartesian square $\mathcal {K}_n$ of $\mathcal {C}_n$ consists of $4^n$ squares of side length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $\mathcal {K}_n$ is essentially the average length of the projections of $\mathcal {K}_n$, also known as the Favard length of $\mathcal {K}_n$. A classical theorem of Besicovitch implies that the Favard length of $\mathcal {K}_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\exp (- c\log _* n)$, due to Peres and Solomyak ($\log _* n$ is the number of times one needs to take the log to obtain a number less than $1$, starting from $n$). In the paper, a power law bound is obtained by combining analytic and combinatorial ideas.