Buffon needle lands in ϵ-neighborhood of a 1-dimensional Sierpinski Gasket with probability at most

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, fθ, where fθ(x) is the number of connected components of the partial fractal set that orthogonally project in the θ 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 , 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.

RésuméOn donne une estimation de la probabilité pour que l'aiguille de Buffon soit ϵ-proche d'un ensemble de Cantor–Sierpinski. On trouve une majoration de cette probabilité en , où c est une constante strictement positive, cette constante n'est pas connue de mannière précise, mais l'estimation est optimale.