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}$

Matthew Bond; Alexander Volberg

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}

Presented by: Gilles Pisier

Matthew Bond ¹; Alexander Volberg ^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, Volume 348 (2010) no. 11-12, pp. 653-656.

Abstract
Résumé

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.

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.

Received: 2009-12-12
Accepted: 2010-03-12
Published online: 2010-06-01

DOI: 10.1016/j.crma.2010.04.006

Author's affiliations:

Matthew Bond ¹; Alexander Volberg ^{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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     author = {Matthew Bond and Alexander Volberg},
     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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Matthew Bond; Alexander Volberg. 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, Volume 348 (2010) no. 11-12, pp. 653-656. doi : 10.1016/j.crma.2010.04.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.006/

References
Cited by

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

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

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

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

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

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