Lower bounds for mask polynomials with many 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 $Φ_{s_1},\dots,Φ_{s_k}$ 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.

Figures (4)

• Figure 1: The cyclotomic divisors of $A$ and the cyclotomic divisors of $A'$ .
• Figure 2: When $N$ is maximal, the truncation of $A$ relative to $S_A^* \cup \{N\}$ has a full block of cyclotomic divisors below an unsupported divisor $\Phi_{M'} (X)$.
• Figure 3: Green cross points are prime power divisors, blue circle points are $(T2)$ divisors, and the red diamond point is the unsupported divisor. After truncation, we have four complete blocks of divisors.
• Figure 4: An example of a configuration of cyclotomic divisors where $N = p^{\gamma_1} q^{\gamma_2}$ satisfies $\gamma_1 \not\in [\alpha_1, \beta_1]$. Notice that, in this scenario, the blocks $\mathcal{B}_3$ and $\mathcal{B}_4$ are necessarily empty. As before: green cross points are prime power divisors, blue circle points are $(T2)$ divisors and the red diamond point is an unsupported divisor.

Theorems & Definitions (79)

• lemma 1
• proof
• theorem 1.1
• theorem 1.2
• theorem 1.3
• definition 1
• lemma 2
• proof
• theorem 1.4
• theorem 1.5