Lower bounds for mask polynomials with many cyclotomic divisors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912976924049408 |
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| author | Kiss, Gergely Łaba, Izabella Marshall, Caleb Somlai, Gábor |
| author_facet | Kiss, Gergely Łaba, Izabella Marshall, Caleb Somlai, Gábor |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_11672 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower bounds for mask polynomials with many cyclotomic divisors Kiss, Gergely Łaba, Izabella Marshall, Caleb Somlai, Gábor Number Theory Combinatorics 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. |
| title | Lower bounds for mask polynomials with many cyclotomic divisors |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2507.11672 |