Buffon's needle estimates for rational product Cantor sets

Let $S_{\mathrm{\infty }}=A_{\mathrm{\infty }}\times B_{\mathrm{\infty }}$ be a self-similar product Cantor set in the complex plane, defined via $S_{\mathrm{\infty }}={\bigcup }_{j=1}^L{T}_j(S_{\mathrm{\infty }})$, where $T_j:\text{C}\to \text{C}$ have the form $T_j(z)=\frac{1}{L}z+z_j$ and $\{z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_L\}=A+iB$ for some $A,B\subset \text{rr}$ with $\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid }>1$ and $\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }=L$. Let $S_N$ be the $L^{-N}$-neighbourhood of $S_{\mathrm{\infty }}$, or equivalently (up to constants), its $N$-th Cantor iteration. We are interested in the asymptotic behaviour as $N\to \mathrm{\infty }$ of the {\it Favard length} of $S_N$, 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 $S_N$ is bounded from above by $C{N}^{-p\mathrm{/}\log \log N}$ 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, {\L}aba-Zhai, and Bond-Volberg.