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

TL;DR

This paper investigates the probability that a Buffon needle lands near a 1D Sierpinski gasket, showing it decreases at a rate proportional to a negative power of the logarithm of the neighborhood size.

Contribution

It extends quantitative projection results to the Sierpinski gasket, providing bounds on the likelihood of a needle landing near the fractal.

Findings

Probability decays as | log psilon|^{-c} for small neighborhoods

Adapts techniques from previous work on Cantor sets to the Sierpinski gasket

Provides a general framework for self-similar sets of dimension 1

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.