Buffon needle lands in <i>ϵ</i>-neighborhood of a 1-dimensional Sierpinski Gasket with probability at most $ {|\mathrm{log}\phantom{\rule{0.2em}{0ex}}ϵ|}^{-c}$

Bond, Matthew; Volberg, Alexander

doi:10.1016/j.crma.2010.04.006

Harmonic Analysis/Analytic Geometry

Buffon needle lands in ϵ-neighborhood of a 1-dimensional Sierpinski Gasket with probability at most

{| \log ϵ |}^{- c}

[Une estimation de la probabilité pour l'aiguille de Buffon de se situer dans un ϵ-voisinage de l'ensemble de Sierpinski]

Présenté par : Gilles Pisier

Bond, Matthew ¹ ; Volberg, Alexander ^1,²

¹ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
² School of Mathematics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, UK

Comptes Rendus. Mathématique, Tome 348 (2010) no. 11-12, pp. 653-656.

Résumé
Abstract

On donne une estimation de la probabilité pour que l'aiguille de Buffon soit ϵ-proche d'un ensemble de Cantor–Sierpinski. On trouve une majoration de cette probabilité en ${| \log ϵ |}^{- c}$ , où c est une constante strictement positive, cette constante n'est pas connue de mannière précise, mais l'estimation est optimale.

In recent years, relatively sharp quantitative results in the spirit of the Besicovitch projection theorem have been obtained for self-similar sets by studying the $L^{p}$ norms of the “projection multiplicity” functions, $f_{θ}$ , where $f_{θ} (x)$ is the number of connected components of the partial fractal set that orthogonally project in the θ direction to cover x. In Nazarov et al. (2008) [4], it was shown that n-th partial 4-corner Cantor set with self-similar scaling factor 1/4 decays in Favard length at least as fast as $\frac{C}{n^{p}}$ , for $p < 1 / 6$ . In Bond and Volberg (2009) [1], this same estimate was proved for the 1-dimensional Sierpinski gasket for some $p > 0$ . A few observations were needed to adapt the approach of Nazarov et al. (2008) [4] to the gasket: we sketch them here. We also formulate a result about all self-similar sets of dimension 1.

Reçu le : 2009-12-12
Accepté le : 2010-03-12
Publié le : 2010-06-01

DOI : 10.1016/j.crma.2010.04.006

Affiliations des auteurs :

Bond, Matthew ¹ ; Volberg, Alexander ^{1, 2}

¹ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
² School of Mathematics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, UK

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     title = {Buffon needle lands in \protect\emph{\ensuremath{\epsilon}}-neighborhood of a 1-dimensional {Sierpinski} {Gasket} with probability at most $ {|\mathrm{log}\phantom{\rule{0.2em}{0ex}}\ensuremath{\epsilon}|}^{-c}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {653--656},
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     doi = {10.1016/j.crma.2010.04.006},
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}

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Bond, Matthew; Volberg, Alexander. Buffon needle lands in ϵ-neighborhood of a 1-dimensional Sierpinski Gasket with probability at most $ {|\mathrm{log}\phantom{\rule{0.2em}{0ex}}ϵ|}^{-c}$. Comptes Rendus. Mathématique, Tome 348 (2010) no. 11-12, pp. 653-656. doi : 10.1016/j.crma.2010.04.006. http://www.numdam.org/articles/10.1016/j.crma.2010.04.006/

Bibliographie
Cité par

[1] Bond, M.; Volberg, A. The Power Law for Buffon's Needle Landing Near the Sierpinski Gasket, 2009 (pp. 1–34) | arXiv

[2] Falconer, K.J. The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, New York, 1986

[3] Laba, I.; Zhai, K. Favard length of product Cantor sets, February 5, 2009 | arXiv

[4] Nazarov, F.; Peres, Y.; Volberg, A. The power law for the Buffon needle probability of the four-corner Cantor set, 2008 (pp. 1–15) | arXiv

[5] Peres, Y.; Solomyak, B. How likely is Buffon's needle to fall near a planar Cantor set?, Pacific J. Math., Volume 204 (2002) no. 2, pp. 473-496

[6] Tao, T. A quantitative version of the Besicovitch projection theorem via multiscale analysis, 18 June 2007 (pp. 1–28) | arXiv

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