The power law for the Buffon needle probability of the four-corner Cantor set
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- by F. Nazarov, Y. Peres and A. Volberg
- St. Petersburg Math. J. 22 (2011), 61-72
- DOI: https://doi.org/10.1090/S1061-0022-2010-01133-6
- Published electronically: November 16, 2010
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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.References
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Bibliographic Information
- F. Nazarov
- Affiliation: Department of Mathematics, University of Wisconsin
- MR Author ID: 233855
- Email: nazarov@math.wisc.edu
- Y. Peres
- Affiliation: Microsoft Research Redmond – and – Departments of Statistics and Mathematics, University of California, Berkeley
- MR Author ID: 137920
- Email: peres@microsoft.com
- A. Volberg
- Affiliation: Department of Mathematics, Michigan State University – and – the University of Edinburgh, United Kingdom
- Email: volberg@math.msu.edu, a.volberg@ed.ac.uk
- 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).
- © Copyright 2010 American Mathematical Society
- Journal: St. Petersburg Math. J. 22 (2011), 61-72
- MSC (2010): Primary 28A80; Secondary 28A75, 60D05, 28A78
- DOI: https://doi.org/10.1090/S1061-0022-2010-01133-6
- MathSciNet review: 2641082
Dedicated: Dedicated to Victor Petrovich Havin on the occasion of his 75th birthday