The Favard length of product Cantor sets

Nazarov, Peres and Volberg proved recently that the Favard length of the $n$-th iteration of the four-corner Cantor set is bounded from above by $n^{-c}$ for an appropriate $c$. We generalize this result to all product Cantor sets whose projection in some direction has positive 1-dimensional measure.

💡 Research Summary

The paper studies the decay of the Favard length (also known as the Buffon needle probability) for a broad class of planar product Cantor sets. The Favard length of a planar set (E\subset\mathbb{R}^{2}) is defined as the average, over all directions (\theta\in