## Abstract

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.

Original language | English |
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Pages (from-to) | 61-72 |

Number of pages | 12 |

Journal | St. Petersburg Mathematical Journal |

Volume | 22 |

Issue number | 1 |

Publication status | Published - Feb 2011 |