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The power law for the Buffon needle probability of the four-corner Cantor set

Authors: F. Nazarov, Y. Peres and A. Volberg
Original publication: Algebra i Analiz, tom 22 (2010), nomer 1.
Journal: St. Petersburg Math. J. 22 (2011), 61-72
MSC (2010): Primary 28A80; Secondary 28A75, 60D05, 28A78
Published electronically: November 16, 2010
MathSciNet review: 2641082
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Abstract: Let $ \mathcal{C}_n$ be the $ n$th generation in the construction of the middle-half Cantor set. The Cartesian square $ \mathcal{K}_n$ of $ \mathcal{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 $ \mathcal{K}_n$ is essentially the average length of the projections of $ \mathcal{K}_n$, also known as the Favard length of $ \mathcal{K}_n$. A classical theorem of Besicovitch implies that the Favard length of $ \mathcal{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 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.

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Additional Information

F. Nazarov
Affiliation: Department of Mathematics, University of Wisconsin

Y. Peres
Affiliation: Microsoft Research Redmond – and – Departments of Statistics and Mathematics, University of California, Berkeley

A. Volberg
Affiliation: Department of Mathematics, Michigan State University – and – the University of Edinburgh, United Kingdom

Keywords: Favard length, four-corner Cantor set, Buffon’s needle
Received by editor(s): October 20, 2008
Published electronically: November 16, 2010
Additional Notes: The research of the authors was supported in part by NSF grants DMS-0501067 (Nazarov and Volberg) and DMS-0605166 (Peres).
Dedicated: Dedicated to Victor Petrovich Havin on the occasion of his 75th birthday
Article copyright: © Copyright 2010 American Mathematical Society