Let C-n, be the nth 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(-clog(*) 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.
|Number of pages||12|
|Journal||St. Petersburg Mathematical Journal|
|Publication status||Published - Feb 2011|