Geodesic
The power law for the Buffon needle probability of the four-corner Cantor set
Algebra i analiz, Tome 22 (2010) no. 1, pp. 82-97.

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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.
Keywords: Favard length, four-corner Cantor set, Buffon's needle. 
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F. Nazarov; Y. Peres; A. Volberg. The power law for the Buffon needle probability of the four-corner Cantor set. Algebra i analiz, Tome 22 (2010) no. 1, pp. 82-97. http://geodesic.mathdoc.fr/item/AA_2010_22_1_a3/

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