Spectral sets in $\mathbb{Z}_{p^2qr}$ tile

Spectral sets in $\mathbb{Z}_{p^2qr}$ tile, Discrete Analysis 2023:5, 10 pp.

Fuglede’s conjecture (1974) asserts that a measurable set $E\subset \mathbb R^n$ of positive Lebesgue measure tiles $\mathbb R^n$ by translations if and only if the space $L^2(E)$ admits an orthonormal basis of exponential functions $\{ e^{2\pi i \lambda\cdot x}:\ \lambda\in\Lambda\}$. The set $\Lambda$ is called a spectrum for $E$. In its full generality, the conjecture is false in dimensions 3 and higher, with counterexamples due to Tao, Kolountzakis, Matolcsi, Farkas, Révész, and Móra. On the other hand, there are important special cases when the conjecture is known to be true. Fuglede proved it under the assumption that either the spectrum or the translation set is a lattice. For convex bodies in $\mathbb R^n$, the conjecture was proved by Iosevich, Katz and Tao for $n=2$, Greenfeld and Lev for $n=3$, and Lev and Matolcsi for $n\geq 4$.

In the last few years there has been considerable interest in variants of Fuglede’s conjecture for finite abelian groups. The case of finite cyclic groups is closely related to Fuglede’s conjecture in $\mathbb R$, as follows. The “tiling implies spectrum” part of the conjecture on the real line has been reduced (through the work of Lagarias, Wang, and Łaba) to proving that all finite sets that tile the integers by translations must satisfy certain algebraic conditions formulated by Coven and Meyerowitz. This problem, and all progress made on it, translates directly to the cyclic group setting. There is no similar reduction available in the other direction (spectrality implies tiling). However, if one can prove that spectral subsets of certain types of cyclic groups must tile, this can be translated back to a partial result in $\mathbb R$ under additional assumptions on both the set in question and on its spectrum.

The main result of the present article is that Fuglede’s conjecture holds on $\mathbb Z_{p^2 qr}$, where $p,q,r$ are distinct primes. The proof relies on many of the methods used previously in this research area, in particular the Rédei-de Bruijn-Schoenberg theorem on the structure of vanishing sums of roots of unity. The main new difficulties in this case, and the novelty of this paper, lie in a combination of two obstacles. One is that the structure of vanishing $N$-th roots of unity is considerably more complicated when $N$ has 3 or more distinct prime factors. The other is that, due to the $p^2$ factor, the problem has a multiscale structure. This requires a multiscale cuboid analysis using the “cube rule” (Proposition 3.3 in the paper) and increases the number of cases that must be considered.