Functional tilings and the Coven-Meyerowitz tiling conditions

Functional tilings and the Coven-Meyerowitz tiling conditions, Discrete Analysis 2025:18, 20 pp.

A finite set \(A\subset\mathbb Z\) tiles \(\mathbb Z\) by translations if \(\mathbb Z\) can be covered by a union of disjoint translates of \(A\). In other words, there is an infinite set \(T\subset \mathbb Z\) such that every \(x\in\mathbb Z\) can be uniquely represented as \(x=a+t\), with \(a\in A\), \(t\in T\). For example, \(A=\{0,2\}\) and \(A=\{0,4,8\}\) tile \(\mathbb Z\); \(A=\{0,1,3\}\) does not.

It is well known (and due to Newman) that all such tilings must be periodic, with \(T=B+M\mathbb Z\) for some \(M\in\mathbb N\) and \(B\subset\mathbb Z\) finite. We write this as \(A\oplus B=\mathbb Z_M\), a factorization of the cyclic group \(\mathbb Z_M\). This may be further reformulated in terms of polynomials, as follows. We may assume that \(A,B\subset\{0,1,\dots\}\). Define the mask polynomials
\[A(X)=\sum_{a\in A}X^a,\ B(X)=\sum_{b\in B}X^b.\]
Then the tiling condition \(A\oplus B=\mathbb Z_M\) is equivalent to
\[ A(X)B(X)=1+X+\dots+X^{M-1}\ \mod (X^M-1). \]
Cyclotomic polynomials \(\Phi_s\) are defined as the irreducible factors of \(X^M-1\), so that
\[ X^M-1=\prod_{s|M}\Phi_s(X). \]
Thus, if \(A\oplus B=\mathbb Z_M\), then every \(\Phi_s\) with \(s|M\) and \(s\neq 1\) divides at least one of \(A(X)\) and \(B(X)\).

In an influential 1998 paper, Coven and Meyerowitz proposed certain conditions (T1) and (T2) governing the distribution of these cyclotomic factors between \(A\) and \(B\), and proved that both of these conditions hold for every tile \(A\) whose cardinality \(|A|\) has at most two distinct prime factors. The statement that these conditions must in fact hold for all finite tiles has become known in the literature as the Coven-Meyerowitz conjecture. If true, the conjecture would have a number of striking consequences, including good estimates on minimal tiling periods, strong structural conditions on all finite tiles, and the confirmation of one direction of Fuglede’s spectral set conjecture in dimension 1. However, despite recent progress by Łaba and Londner, the conjecture in its full generality remains open.

One possible direction of investigation is to consider more general variants of the conjecture in settings where counterexamples might be easier to find. This could serve as a stepping stone towards counterexamples to the original conjecture, or, if the conjecture is true, it could provide insight as to which features of the tiling problem must be important in any proof. This is the route taken by the present paper. The authors consider an extension of the tiling problem where, instead of sets \(A\) and \(B\), one considers two functions \(f,g:\mathbb Z\to[0,\infty)\) such that \(g\) is \(M\)-periodic, \(f\) is supported in \(\{0,1,\dots,M-1\}\), and \(f*g(x)=1\) for all \(x\in\mathbb Z\). They prove that the appropriately generalized Coven-Meyerowitz conjecture is false in that setting, even if the functions \(f\) and \(g\) satisfy the natural analogue of the two-prime condition. This is an important insight that is likely to play a role in the further development of this research area.