Fuglede’s conjecture holds on cyclic groups $\mathbb{Z}_{pqr}$

Fuglede’s conjecture holds on cyclic groups $\mathbb{Z}_{pqr}$, Discrete Analysis 2019:14, 14 pp.

A conjecture of Fuglede from 1974 states that a measurable set $E\subset \mathbb R^n$ of positive Lebesgue measure has a set of translates that tile $\mathbb R^n$ 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 $E$. The conjecture is known to be false in dimensions 3 and higher, with counterexamples due to Tao, Kolountzakis, Matolcsi, Farkas, Révész, and Móra. Nonetheless, there are important special cases for which the conjecture has been confirmed. Fuglede proved it under the assumption that either the spectrum or the translation set is a lattice. The conjecture has also been proved for convex bodies in $\mathbb R^n$, first by Iosevich, Katz and Tao for $n=2$, and, very recently, by Lev and Matolcsi (building on the earlier work by Greenfeld and Lev) for $n\geq 3$.

For general non-convex sets in dimensions 1 and 2, the conjecture remains open in both directions. In dimension 1, the “tiling implies spectrum” part would follow if one can prove that all finite sets that tile the integers (or, equivalently, finite Abelian groups) by translations must satisfy certain conditions formulated by Coven and Meyerovitz. In the converse direction, Lai and Wang proved recently that a spectrum in dimension 1 must be a (possibly translated and rescaled) subset of the rational numbers. Therefore the remaining key questions focus on tiling and spectral properties of subsets of finite groups.

In this article, Ruxi Shi proves that the conjecture is true in both directions in cyclic groups of order $N=pqr$, where $p,q,r$ are distinct primes. In this case, the “tiling implies spectrum” direction follows immediately from the earlier work on the subject. Specifically, an argument due to Coven and Meyerowitz shows that if $A$ tiles $\mathbb{Z}_N$ and $N$ is a product of distinct primes, then $A$ also tiles with period $|A|$, so in particular the translation set is a lattice. However, the “spectrum implies tiling” question is much harder. While there are earlier articles (by Malikiosis-Kolountzakis and Kiss-Malikiosis-Somlai-Viser) resolving certain 2-prime cases, there is a significant increase in difficulty between 2-prime and 3-prime settings. For example, the 2-prime results rely on a structure theorem (due to R’edei, de Bruijn, and Schoenberg) for vanishing sums of $N$-th roots of unity. If $N$ has three or more distinct prime factors, the structure forced by that theorem becomes much more complicated, and in particular it becomes more difficult to prove certain quantitative results needed here.

After this paper was accepted for publication, Gábor Somlai extended the result here to groups of order $\mathbb Z_{p^2qr}$, where $p,q,r$ are distinct primes.