Buffon’s needle estimates for rational product Cantor sets

Let S_=A_ B_ be a self-similar product Cantor set in the complex plane, defined via S_=₉=₁L Tⱼ (S_), where Tⱼ: C have the form Tⱼ (z) =1 Lz+zⱼ and \z₁, , zL\=A+iB for some A, B with |A|, |B|>1 and |A||B|=L. Let SN be the L^-N-neighborhood of S_, or equivalently (up to constants), its N-th Cantor iteration. We are interested in the asymptotic behavior as N of the Favard length of SN, defined as the average (with respect to direction) length of its 1-dimensional projections. If the sets A and B are rational and have cardinalities at most 6, then the Favard length of SN is bounded from above by CN^-p/ N for some p>0. The same result holds with no restrictions on the size of A and B under certain implicit conditions concerning the generating functions of these sets. This generalizes the earlier results of Nazarov-Perez-Volberg, aba-Zhai, and Bond-Volberg.