Fuglede’s conjecture on cyclic groups of order $p^nq$

Fuglede’s conjecture on cyclic groups of order $p^nq$, Discrete Analysis 2017:12, 16 pp.

A conjecture of Fuglede from 1974 states 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 $E$.)

We now know that the conjecture is false in dimensions 3 and higher, with counterexamples due to Tao, Kolountzakis, Matolcsi, Farkas, Révész, and Móra. Nonetheless, there are important special cases where the conjecture has been confirmed, such as convex bodies in $\mathbb R^2$ (due to Iosevich, Katz and Tao), and the case where either the spectrum or the translation set is a lattice (Fuglede). Research on Fuglede’s conjecture has also established a broader family of correspondences between geometric and harmonic-analytic properties of sets that are of interest in their own right.

For general non-convex sets in dimensions 1 and 2, the conjecture remains open in both directions. The main focus so far has been on the corresponding discrete problems and their links to questions in combinatorial number theory and factorization of Abelian groups. On one hand, the “tiling implies spectrum” part in dimension 1 would follow from an independently made conjecture of Coven and Meyerovitz on characterizing finite sets that tile the integers by translations. On the other hand, given that the higher-dimensional counterexamples are based on adapting finite fields constructions to the continuous setting, it is tempting to try to disprove the conjecture in lower dimensions in a similar manner.

In this paper, the authors prove that the conjecture is true in both directions in cyclic groups of order $N=p^nq$, where $p$ and $q$ are distinct primes. The “tiling implies spectrum” direction follows immediately from the earlier work of Coven-Meyerowitz and Łaba. The main new contribution is the resolution of the more difficult “spectrum implies tiling” question in the case under consideration.

The proof is based on the existing results on the structure of vanishing sums of roots of unity. This is relevant because if $A\subset \mathbb Z_N$ is spectral, then the orthogonality relations between the exponential functions in the spectrum can be expressed in terms of the zeros of the mask polynomial $A(x)=\sum_{a\in A} x^a$ on the unit circle. Lam and Leung proved that any vanishing sum of roots of unity of order $p^n q^m$ can be expressed as a linear combination of (roots of unity forming) rotated regular $p$-gons and $q$-gons, with positive coefficients. In the present paper, Kolountzakis and Malikiokis use this to analyze the structure of spectral sets $A$ in $\mathbb Z_{p^nq}$, proving that such sets must satisfy the Coven-Meyerowitz tiling conditions and therefore must tile the group.

The use of the Lam-Leung theorem is an exciting new development in this area. It is possible that this method will extend to $\mathbb Z_N$ with $N=p^nq^m$, where the results of Coven-Meyerowitz, Łaba, and Lam-Leung continue to hold. For general cyclic groups, the situation is less clear, since vanishing sums of roots of unity of arbitrary order can be much more complicated. (There is a more general structure theorem in this setting, due to Rédei, de Bruijn, and Schoenberg, but the coefficients in that theorem need not be positive.) Nonetheless, it is likely that the link between the two phenomena can be explored further.