  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}).
