Buffon’s needle estimates for rational product Cantor sets
Buffon’s needle estimates for rational product Cantor sets
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 $T_j:\Bbb{C}\to\Bbb{C}$ have the form $T_j(z)={1\over L}z+z_j$ and $\{z_1,\ldots,z_L\}=A+iB$ for some $A,B\subset\Bbb{R}$ with $|A|,|B|>1$ and $|A||B|=L$. Let $S_N$ be the $L^{-N}$-neighborhood of $S_\infty$, or equivalently (up to constants), its $N$-th Cantor iteration. We …