How likely is Buffon’s needle to meet a Cantor square?
- Fedor Nazarov | Michigan State University
Let Cn be the n’th generation in the construction of the middle-half Cantor set. The Cartesian square Kn of Cn consists of 4n squares of side-length 1/(4n). 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. It is still an open problem to determine the exact rate of decay of this average.
Until recently, the only explicit upper bound exp(- log_* n) was due to Peres and Solomyak. (log_* n is the number of times one needs to take log to obtain a number less than 1, when starting from n). We obtain a much better bound by combining analytic and combinatorial ideas.
This is joint work with Y. Peres and A. Volberg.
Speaker Details
Fedor Nazarov obtained his PhD in 1993 from Leningrad State University under the guidance of V. Havin. He is well known for his ability to crack hard problems in analysis, and for his joint work with Volberg and Treil. Prof. Nazarov received the Salem Prize (the highest prize in Analysis) in 1999.
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Jeff Running
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