Let $C_n$ be the $n$-th generation in the construction of the middle-halfCantor set. The Cartesian square $K_n$ of $C_n$ consists of $4^n$ squares ofside-length $4^{-n}$. The chance that a long needle thrown at random in theunit square will meet $K_n$ is essentially the average length of theprojections of $K_n$, also known as the Favard length of $K_n$. A classicaltheorem 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. Untilrecently, the only explicit upper bound was $\exp(- c\log_* n)$, due to Peresand Solomyak. ($\log_* n$ is the number of times one needs to take log toobtain a number less than 1 starting from $n$). We obtain a power law bound bycombining analytic and combinatorial ideas.
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