Buffon's needle landing near Besicovitch irregular self-similar sets

TL;DR

This paper estimates the Favard length decay rate of neighborhoods of self-similar Cantor sets, providing a general bound applicable to all such sets, with results indicating a slower decay than power estimates.

Contribution

It introduces a universal Favard length estimate for any self-similar Besicovitch set, extending previous specific power decay results to a broader class.

Findings

Favard length decays at a rate worse than any power for self-similar sets.

The decay rate relates to the regularity of zeros of certain exponential sums.

The method applies broadly but yields less sharp estimates than previous specialized results.

Abstract

In this paper we get an estimate of Favard length of an arbitrary neighbourhood of an arbitrary self-similar Cantor set. Consider $L$ closed disjoint discs of radius $1/L$ inside the unit disc. By using linear maps of smaller disc onto the unit disc we can generate a self-similar Cantor set $G$. Then $\G=\bigcap_n\G_n$. 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 $\G_n$ decays to zero as a function of $n$. The quantitative results for the Favard length problem were obtained by Peres-Solomyak and Tao; in the latter paper a general way of making a quantitative statement from the Besicovitch theorem is considered. But being rather general, this method does not give a good estimate for self-similar structures such as $\G_n$. Indeed, vastly improved estimates have been proven in these cases: in the paper of Nazarov-Peres-Volberg, it was shown that for 1/4 corner Cantor set one has $p<1/6$, such that $Fav(\K_n)\leq\frac{c_p}{n^{p}}$, and in Laba-Zhai and Bond-Volberg the same type power estimate was proved for the product Cantor sets (with an extra tiling property) and for the Sierpinski gasket $S_n$ for some other $p>0$. In the present work we give an estimate that works for {\it any} Besicovitch set which is self-similar. However estimate is worse than the power one. The power estimate still appears to be related to a certain regularity property of zeros of a corresponding linear combination of exponents (we call this property {\it analytic tiling}).