THE POWER LAW FOR THE BUFFON NEEDLE PROBABILITY OF THE FOUR-CORNER CANTOR SET

F. Nazarov, Y. Peres, A. Volberg

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

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 languageEnglish
Pages (from-to)61-72
Number of pages12
JournalSt. Petersburg Mathematical Journal
Volume22
Issue number1
Publication statusPublished - Feb 2011

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