Power Laws for the Favard Length Problem in $\mathbb{R}^d$

Abstract

We prove a power law for the asymptotic decay of the Favard length of neighbourhoods of certain self-similar sets in $\mathbb{R}^d$ with $d \geq 2$. These self-similar sets are generalizations of the so-called four-corner Cantor set to higher dimensions, as well as to a more general class of rational digit sets. When $d \geq 3$, our estimates are the first such non-trivial asymptotic upper bounds for the Favard length problem. The extension to a new class of digit sets (which is new even when $d = 2$, but holds for $d \geq 2$ generally) uses the work of G. Kiss, I. Laba, G. Somlai and the author on vanishing sums of roots of unity and divisibility by many cyclotomic polynomials.