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

DOI:
https://doi.org/10.1090/S1061-0022-2010-01133-6

Published electronically:
November 16, 2010

MathSciNet review:
2641082

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the th generation in the construction of the middle-half Cantor set. The Cartesian square of consists of squares of side length . The chance that a long needle thrown at random in the unit square will meet is essentially the average length of the projections of , also known as the Favard length of . A classical theorem of Besicovitch implies that the Favard length of tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was , due to Peres and Solomyak ( is the number of times one needs to take the log to obtain a number less than , starting from ). In the paper, a power law bound is obtained by combining analytic and combinatorial ideas.

**1.**A. S. Besicovitch,*Tangential properties of sets and arcs of infinite linear measure*, Bull. Amer. Math. Soc.**66**(1960), 353-359. MR**0120335 (22:11090)****2.**G. David,*Analytic capacity, Calderón-Zygmund operators, and rectifiability*, Publ. Mat.**43**(1999), 3-25. MR**1697514 (2000e:30044)****3.**K. J. Falconer,*The geometry of fractal sets*, Cambridge Tracts in Math., vol. 85, Cambridge Univ. Press, Cambridge, 1986. MR**0867284 (88d:28001)****4.**P. W. Jones and T. Murai,*Positive analytic capacity but zero Buffon needle probability*, Pacific J. Math.**133**(1988), 99-114. MR**0936358 (89m:30050)****5.**R. Kenyon,*Projecting the one-dimensional Sierpinski gasket*, Israel J. Math.**97**(1997), 221-238. MR**1441250 (98i:28002)****6.**J. C. Lagarias and Y. Wang,*Tiling the line with translates of one tile*, Invent. Math.**124**(1996), 341-365. MR**1369421 (96i:05040)****7.**J. Mateu, X. Tolsa, and J. Verdera,*The planar Cantor sets of zero analytic capacity and the local -theorem*, J. Amer. Math. Soc.**16**(2003), 19-28. MR**1937197 (2003k:30041)****8.**P. Mattila,*Orthogonal projections, Riesz capacities, and Minkowski content*, Indiana Univ. Math. J.**39**(1990), 185-198. MR**1052016 (91d:28018)****9.**Y. Peres, K. Simon, and B. Solomyak,*Self-similar sets of zero Hausdorff measure and positive packing measure*, Israel J. Math.**117**(2000), 353-379. MR**1760599 (2001g:28017)****10.**Y. Peres and B. Solomyak,*How likely is Buffon's needle to fall near a planar Cantor set?*Pacific J. Math.**204**(2002), 473-496. MR**1907902 (2003h:28010)****11.**T. Tao,*A quantitative version of the Besicovitch projection theorem via multiscale analysis*, arXiv:0706.2446v1 [math. CA] 18 Jun 2007, pp. 1-28. Proc. London Math. Soc. (3)**98**(2009), 559-584. MR**2500864 (2010f:28009)**

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Additional Information

**F. Nazarov**

Affiliation:
Department of Mathematics, University of Wisconsin

Email:
nazarov@math.wisc.edu

**Y. Peres**

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

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

DOI:
https://doi.org/10.1090/S1061-0022-2010-01133-6

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