The power law for the Buffon needle probability of the fourcorner 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), 6172
MSC (2010):
Primary 28A80; Secondary 28A75, 60D05, 28A78
Published electronically:
November 16, 2010
MathSciNet review:
2641082
Fulltext PDF
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Abstract: Let be the th generation in the construction of the middlehalf 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), http://dx.doi.org/10.1090/S000299041960104430
 2.
Guy
David, Analytic capacity, CalderónZygmund operators, and
rectifiability, Publ. Mat. 43 (1999), no. 1,
3–25. MR
1697514 (2000e:30044), http://dx.doi.org/10.5565/PUBLMAT_43199_01
 3.
K.
J. Falconer, The geometry of fractal sets, Cambridge Tracts in
Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
(88d:28001)
 4.
Peter
W. Jones and Takafumi
Murai, Positive analytic capacity but zero Buffon needle
probability, Pacific J. Math. 133 (1988), no. 1,
99–114. MR
936358 (89m:30050)
 5.
Richard
Kenyon, Projecting the onedimensional Sierpinski gasket,
Israel J. Math. 97 (1997), 221–238. MR 1441250
(98i:28002), http://dx.doi.org/10.1007/BF02774038
 6.
Jeffrey
C. Lagarias and Yang
Wang, Tiling the line with translates of one tile, Invent.
Math. 124 (1996), no. 13, 341–365. MR 1369421
(96i:05040), http://dx.doi.org/10.1007/s002220050056
 7.
Joan
Mateu, Xavier
Tolsa, and Joan
Verdera, The planar Cantor sets of zero
analytic capacity and the local 𝑇(𝑏)theorem, J. Amer. Math. Soc. 16 (2003), no. 1, 19–28 (electronic). MR 1937197
(2003k:30041), http://dx.doi.org/10.1090/S0894034702004010
 8.
Pertti
Mattila, Orthogonal projections, Riesz capacities, and Minkowski
content, Indiana Univ. Math. J. 39 (1990),
no. 1, 185–198. MR 1052016
(91d:28018), http://dx.doi.org/10.1512/iumj.1990.39.39011
 9.
Yuval
Peres, Károly
Simon, and Boris
Solomyak, Selfsimilar sets of zero Hausdorff measure and positive
packing measure, Israel J. Math. 117 (2000),
353–379. MR 1760599
(2001g:28017), http://dx.doi.org/10.1007/BF02773577
 10.
Yuval
Peres and Boris
Solomyak, How likely is Buffon’s needle to fall near a planar
Cantor set?, Pacific J. Math. 204 (2002), no. 2,
473–496. MR 1907902
(2003h:28010), http://dx.doi.org/10.2140/pjm.2002.204.473
 11.
Terence
Tao, A quantitative version of the Besicovitch projection theorem
via multiscale analysis, Proc. Lond. Math. Soc. (3)
98 (2009), no. 3, 559–584. MR 2500864
(2010f:28009), http://dx.doi.org/10.1112/plms/pdn037
 1.
 A. S. Besicovitch, Tangential properties of sets and arcs of infinite linear measure, Bull. Amer. Math. Soc. 66 (1960), 353359. MR 0120335 (22:11090)
 2.
 G. David, Analytic capacity, CalderónZygmund operators, and rectifiability, Publ. Mat. 43 (1999), 325. 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), 99114. MR 0936358 (89m:30050)
 5.
 R. Kenyon, Projecting the onedimensional Sierpinski gasket, Israel J. Math. 97 (1997), 221238. MR 1441250 (98i:28002)
 6.
 J. C. Lagarias and Y. Wang, Tiling the line with translates of one tile, Invent. Math. 124 (1996), 341365. 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), 1928. MR 1937197 (2003k:30041)
 8.
 P. Mattila, Orthogonal projections, Riesz capacities, and Minkowski content, Indiana Univ. Math. J. 39 (1990), 185198. MR 1052016 (91d:28018)
 9.
 Y. Peres, K. Simon, and B. Solomyak, Selfsimilar sets of zero Hausdorff measure and positive packing measure, Israel J. Math. 117 (2000), 353379. 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), 473496. 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. 128. Proc. London Math. Soc. (3) 98 (2009), 559584. 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:
http://dx.doi.org/10.1090/S106100222010011336
PII:
S 10610022(2010)011336
Keywords:
Favard length,
fourcorner 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 DMS0501067 (Nazarov and Volberg) and DMS0605166 (Peres).
Dedicated:
Dedicated to Victor Petrovich Havin on the occasion of his 75th birthday
Article copyright:
© Copyright 2010 American Mathematical Society
