JournalsrmiVol. 36, No. 2pp. 537–548

The exact power law for Buffon’s needle landing near some random Cantor sets

  • Shiwen Zhang

    University of Minnesota, Minneapolis, USA
The exact power law for Buffon’s needle landing near some random Cantor sets cover

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Abstract

In this paper, we study the Favard length of some random Cantor sets of Hausdorff dimension 1. We start with a unit disk in the plane and replace the unit disk by 44 disjoint subdisks (with equal distance to each other) of radius 1/41/4 inside and tangent to the unit disk. By repeating this operation in a self-similar manner and adding a random rotation in each step, we can generate a random Cantor set D(ω){\cal D}(\omega). Let Dn{\cal D}_n be the nn-th generation in the construction, which is comparable to the 4n4^{-n}-neighborhood of D{\cal D}. We are interested in the decay rate of the Favard length of these sets Dn{\cal D}_n as nn\to\infty, which is the likelihood (up to a constant) that "Buffon's needle'' dropped randomly will fall into the 4n4^{-n}-neighborhood of D{\cal D}. It is well known that the lower bound of the Favard length of Dn(ω){\cal D}_n(\omega) is a constant multiple of n1n^{-1}. We show that the upper bound of the Favard length of Dn(ω){\cal D}_n(\omega) is Cn1C n^{-1} for some C>0C > 0 in the average sense. We also prove the a similar linear decay for the Favard length of Dnd(ω){\cal D}^d_n(\omega), which is the dnd^{-n}-neighborhood of a self-similar random Cantor set with degree dd greater than 44. Notice that in the non-random case where the self-similar set has degree greater than 44, the best known result for the decay rate of the Favard length is eclogne^{-c\sqrt {\log n}}.

Cite this article

Shiwen Zhang, The exact power law for Buffon’s needle landing near some random Cantor sets. Rev. Mat. Iberoam. 36 (2019), no. 2, pp. 537–548

DOI 10.4171/RMI/1138