Buffon needle lands in $\epsilon$-neighborhood of a 1-Dimensional Sierpinski Gasket with probability at most $|\log \epsilon {|}^{-c}$

Abstract

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, $f_{\theta }$, where $f_{\theta }(x)$ is the number of connected components of the partial fractal set that orthogonally project in the $\theta$ direction to cover $x$. In \cite{NPV}, 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 \cite{BV}, 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 \cite{NPV} to the gasket: we sketch them here. We also formulate a result about all self-similar sets of dimension 1.

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