An estimate from below for the Buffon needle probability of the four-corner Cantor set

Let Cn be the n-th generation in the construction of the middle-half Cantor set. The Cartesian square Kn = Cn 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 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 * n is the number of times one needs to take log to obtain a number less than 1 starting from n). In [11] the power estimate from above was obtained. The exponent in On the other hand, a simple estimate shows that from below we have the estimate c n . Here we apply the idea from [4], [1] to show that the estimate from below can be in fact improved to c log n n . This is in drastic contrast to the case of random Cantor sets studied in