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arXiv · 2507.11672

Lower bounds for mask polynomials with many cyclotomic divisors

Gergely Kiss·Izabella Łaba·Caleb Marshall·Gábor Somlai
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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.

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BibTeXRIS

Gergely Kiss, Izabella Łaba, Caleb Marshall, Gábor Somlai. 2026-03-22. Lower bounds for mask polynomials with many cyclotomic divisors. https://arxiv.org/abs/2507.11672

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