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}

[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

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.

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 :

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},
     publisher = {Elsevier},
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     year = {2010},
     doi = {10.1016/j.crma.2010.04.006},
     language = {en},
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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/

Bibliographie
Cité par

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Dimitris Vardakis; Alexander Volberg The Buffon's needle problem for random planar disk-like Cantor sets, Journal of Mathematical Analysis and Applications, Volume 529 (2024) no. 2, p. 127622 | DOI:10.1016/j.jmaa.2023.127622
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Александр Львович Вольберг; Alexander L'vovich Vol'berg; Владимир Яковлевич Эйдерман; Vladimir Yakovlevich Eiderman Неоднородный гармонический анализ: 16 лет развития, Успехи математических наук, Volume 68 (2013) no. 6(414), p. 3 | DOI:10.4213/rm9556

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