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_∞=ⱼ₌₁L Tⱼ(S_∞), where Tⱼ:C→C have the form Tⱼ(z)=1 Lz+zⱼ and ₁,…,zL\=A+iB for some A,B⊂R 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/loglog 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.