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

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 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.