Buffon's needle estimates for rational product Cantor sets

Abstract

Let $S_\infty=A_\infty\times B_\infty$ be a self-similar product Cantor set in the complex plane, defined via $S_\infty=\bigcup_{j=1}^L T_j(S_\infty)$, where $T_j:\C\to\C$ have the form $T_j(z)=\frac1{L}z+z_j$ and $\{z_1,...,z_L\}=A+iB$ for some $A,B\subset\rr$ with $|A|,|B|>1$ and $|A||B|=L$. Let $S_N$ be the $L^{-N}$-neighbourhood of $S_\infty$, or equivalently (up to constants), its $N$-th Cantor iteration. We are interested in the asymptotic behaviour as $N\to\infty$ 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 $CN^{-p/\log\log N}$ 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.