The power law for the Buffon needle probability of the four-corner Cantor set

Abstract

Let Cn be the nth generation in the construction of the middle-half Cantor set. The Cartesian square Kn of Cn consists of 4n squares of side length 4-n. The chance that a long needle thrown at random in the unit square will meet Kn is essentially the average length of the projections of Kn, also known as the Favard length of Kn. A classical theorem of Besicovitch implies that the Favard length of Kn 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*ss 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. © 2010 American Mathematical Society.