Vanishing sums of roots of unity and the Favard length of self-similar product sets

Abstract: We improve a special case of the Lam-Leung lower bound on the number of elements in a vanishing sum of $N$-th roots of unity. Using this result, we extend the Favard length estimates due to Bond, {\L}aba, and Volberg to a new class of rational product Cantor sets in $\mathbb{R}^2$.