Buffon needle lands in $\epsilon$-neighborhood of a 1-Dimensional Sierpinski Gasket with probability at most $|\log\epsilon |^{-c}
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...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
27.12.2009
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Subjects | |
Online Access | Get full text |
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Summary: | 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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DOI: | 10.48550/arxiv.0912.5095 |