The Favard length of product Cantor sets

TL;DR

This paper extends the understanding of Favard length decay rates to a broad class of product Cantor sets, showing they diminish polynomially under certain projection conditions.

Contribution

It generalizes previous results by Nazarov, Peres, and Volberg to all product Cantor sets with positive measure projections in some direction.

Findings

Favard length of these sets decays polynomially with iteration number

The decay rate is bounded above by a power law

Applicable to a wide class of product Cantor sets with positive measure projections

Abstract

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.