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

Abstract

Let $\Cant_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $\K_n = \Cant_n \times \Cant_n$ 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 $\exp(- c\log_* n)$, 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 Nazarov-Peres-Volberg paper (arxiv:math 0801.2942) the power estimate from above was obtained. The exponent in this paper was less than 1/6 but could have been slightly improved. On the other hand, a simple estimate shows that from below we have the estimate $\frac{c}{n}$. Here we apply the idea from papers of Nets Katz (MRL (1996), 527-536) and Bateman-Katz (arXiv:math/0609187v1 2006) to show that the estimate from below can be in fact improved to $c \frac{\log n}{n}$. This is in drastic difference from the case of {\em random} Cantor sets studied by Peres and Solomyak in Pacific J. Math. 204 (2002), 473-496.