  Let $C_n$ be the $n$-th generation in the construction of the middle-half
Cantor set. The Cartesian square $K_n$ of $C_n$ 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 $K_n$ is essentially the average length of the
projections of $K_n$, also known as the Favard length of $K_n$. A classical
theorem of Besicovitch implies that the Favard length of $K_n$ 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 $n$). We obtain a power law bound by
combining analytic and combinatorial ideas.
