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 $\hat{I}{¦}_{s_1},\dots ,\hat{I}{¦}_{s_k}$ 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 $k\ge 2$; 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.

Minor revisions, update of author information; accepted for publication by Advances in Mathematics