Lower bounds for mask polynomials with many cyclotomic divisors

TL;DR

This paper investigates the minimal size of sets with mask polynomials divisible by multiple cyclotomic polynomials, revealing complex structural behaviors with implications for various problems in harmonic analysis and number theory.

Contribution

It extends understanding of the structure of sets with multiple cyclotomic divisors, showing that simple fibered structures are not always optimal, unlike the single divisor case.

Findings

Proves that fibered structures are not always optimal for multiple divisors

Identifies special cases where simple structures suffice

Highlights the complexity of minimal sets with multiple cyclotomic divisors

Abstract

Given a nonempty set $A \subset \mathbb{N}\cup\{0\}$, define the mask polynomial $A(X)=\sum_{a\in A} X^a$. Suppose that there are $s_1,\dots,s_k\in\nn\setminus\{1\}$ such that the cyclotomic polynomials $\Phi_{s_1},\dots,\Phi_{s_k}$ divide $A(X)$. What is the smallest possible size of $A$? For $k=1$, this was answered by Lam and Leung in 2000. Less is known about the case when $k\geq 2$; in particular, one may ask whether (similarly to the $k=1$ case) the optimal configurations have a simple ``fibered" structure on each scale involved. We prove that this is true in a number of special cases, but false in general, even if further strong structural assumptions are added. Results of this type are expected to have a broad range of applications, including Favard length of product Cantor sets, Fuglede's spectral set conjecture, and the Coven-Meyerowitz conjecture on integer tilings.