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

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 $1\mathrm{/}L$ 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 $\text{G}={\bigcap }_n{\text{G}}_n$. 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 ${\text{G}}_n$ 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 ${\text{G}}_n$. 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 $p<1\mathrm{/}6$, such that $Fav({\text{K}}_n)\le \frac{c_p}{n^p}$, 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}).