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

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 \cite{NPV} the power estimate from above was obtained. The exponent in \cite{NPV} 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 \cite{katz}, \cite{BK} to show that the estimate from below can be in fact improved to $c\,\frac{\log n}{n}$. This is in drastic contrast to the case of {\em random} Cantor sets studied in \cite{PS}.

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Published 1 January 2010