IUMJ

Title: Buffon's needle landing near Besicovitch irregular self-similar sets

Authors: Matthew Bond and Alexander Volberg

Issue: Volume 61 (2012), Issue 6, 2085-2109

Abstract:

In this paper, we get an upper estimate of the Favard length (sometimes called Buffon needle probability) of an arbitrary neighborhood of a big class of self-similar Cantor set of dimension $1$. Consider $L$ disjoint closed discs of radius $1/L$ inside the unit disc. By using linear maps of the disc onto the smaller discs, we can generate a self-similar Cantor set $\mathcal{G}$. Let $\mathcal{G}_n$ be the union of all possible images of the unit disc under $n$-fold compositions of the similarity maps. One may then ask the rate at which the Favard length---the average over all directions of the length of the orthogonal projection onto a line in that direction---of these sets $\mathcal{G}_n$ decays to zero as a function of $n$. Previous quantitative results for the Favard length problem were obtained by Peres-Solomyak [Y. Peres and B. Solomyak, \textit{How likely is Buffon's needle to fall near a planar Cantor set?}, Pacific J. Math. \textbf{204} (2002), no. 2, 473--496] and Tao [T. Tao, \textit{A quantitative version of the Besicovitch projection theorem via multiscale analysis}, Proc. Lond. Math. Soc. (3) \textbf{98} (2009), no. 3, 559--584]; in the latter paper, a general way of making a quantitative statement from the Besicovitch theorem is considered. But since it is rather general, this method does not give a good estimate for self-similar structures such as $\mathcal{G}_n$. In the present work, we prove the estimate $\Fav(\mathcal{G}_n)\leq e^{-c\sqrt{\log n}}$. While this estimate is vastly improved compared to [Y. Peres and B. Solomyak, \textit{How likely is Buffon's needle to fall near a planar Cantor set?}, Pacific J. Math. \textbf{204} (2002), no. 2, 473--496] and [T. Tao, \textit{A quantitative version of the Besicovitch projection theorem via multiscale analysis}, Proc. Lond. Math. Soc. (3) \textbf{98} (2009), no. 3, 559--584], it is worse than the power estimate $\Fav(\mathcal{G}_n)\leq C/n^p$ proved for specific sets $\mathcal{G}_n$ with additional product structures in Nazarov-Peres-Volberg [F.\:L. Nazarov, Y. Peres, and A. Volberg, \textit{The power law for the Buffon needle probability of the four-corner Cantor set}, Algebra i Analiz \textbf{22} (2010), no. 1, 82--97], \L aba-Zhai [I. \L aba and K. Zhai, \textit{The Favard length of product Cantor sets}, Bull. Lond. Math. Soc. \textbf{42} (2010), no. 6, 997--1009], and Bond-\L aba-Volberg [M. Bond, I. \L aba, and A. Volberg, \textit{Buffon's needle estimates for rational product Cantor sets}, http://arxiv.org/abs/arXiv:1109.1031, 1--38].