Buffon's needle landing near Besicovitch irregular self-similar sets Matthew BondAlexander Volberg 28A8028A7560D0528A78fractalslengthareavolumeother geometric measure theorygeometric probabilitystochastic geometryrandom setsHausdorff and packing measures 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]. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4828 10.1512/iumj.2012.61.4828 en Indiana Univ. Math. J. 61 (2012) 2085 - 2109 state-of-the-art mathematics http://iumj.org/access/