Let C_n be the nth generation in the construction of the middle-half Cantor set. The Cartesian square K_n of Cn 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 the 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.
展开▼
