Lower bounds for mask polynomials with many cyclotomic divisors

Abstract: We study finite subsets and multisets of cyclic groups \(\mathbb{Z}_M\) whose mask polynomials have prescribed cyclotomic divisors. More precisely, if \(A\subseteq \mathbb{Z}_M\), we consider its mask polynomial $$ A(X)=\sum_{a\in A} X^a \qquad \text{in } \mathbb{Z}[X]/(X^M-1), $$ and ask how divisibility by selected cyclotomic polynomials constrains the size and structure of \(A\). This question is motivated by its connections with translational tilings, the Coven--Meyerowitz conjecture, and one-dimensional Fuglede-type problems. We prove new lower bounds for the cardinality of such sets and develop several structural tools, including \\ & a truncation method and a multiscale extension of the de Bruijn--Rédei--Schoenberg theorem. These results show that the expected fibre-type extremal configurations do not always give the correct minimum once the prescribed cyclotomic divisors become sufficiently complicated. At the same time, in the two-dimensional case and in several further special situations, the lower bounds agree with the natural fibre constructions. This is joint work with I. Łaba, C. Marshall, and G. Somlai.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)