Buffon needle lands in epsilon-neighborhood of a 1-dimensional Sierpinski Gasket with probability at most vertical bar log epsilon vertical bar(-c)

Matthew Bond, Alexander Volberg

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Abstract / Description of output

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 LP norms of the "projection multiplicity" functions, f(theta), where f(theta)(x) is the number of connected components of the partial fractal set that orthogonally project in the 0 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 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. (C) 2010 Academic des sciences. Published by Elsevier Masson SAS. All rights reserved.

Original languageEnglish
Pages (from-to)653-656
Number of pages4
JournalComptes Rendus Mathématique
Volume348
Issue number11-12
DOIs
Publication statusPublished - Jun 2010

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