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

TL;DR

This paper improves bounds on vanishing sums of roots of unity and applies these results to extend Favard length estimates for certain rational product Cantor sets in the plane.

Contribution

It introduces a refined lower bound on vanishing sums of roots of unity and applies it to analyze the Favard length of new classes of rational product Cantor sets.

Findings

Enhanced lower bounds for vanishing sums of roots of unity.

Extended Favard length estimates to new rational product Cantor sets.

Demonstrated applicability of algebraic bounds to geometric measure problems.

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 $\mathbb{R}^2$.