How likely is BuffonΓÇÖs needle to meet a Cantor square?

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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 1/(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. It is still an open problem to determine the exact rate of decay of this average. Until recently, the only explicit upper bound exp(- log_* n) was due to Peres and Solomyak. (log_* n is the number of times one needs to take log to obtain a number less than 1, when starting from n). We obtain a much better bound by combining analytic and combinatorial ideas. This is joint work with Y. Peres and  A. Volberg.



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