Fuglede's conjecture on cyclic groups of order $p^{n}q$

Abstract

We show that the spectral set conjecture by Fuglede holds in the setting of cyclic groups of order $p^{n}q$, where $p$, $q$ are distinct primes and $n\ge 1$. This means that a subset $E$ of such a group $G$ tiles the group by translation ($G$ can be partitioned into translates of $E$) if and only if there exists an orthogonal basis of $L^2(E)$ consisting of group characters. The main ingredient of the present proof is the structure of vanishing sums of roots of unity of order $N$, where $N$ has at most two prime divisors; the extension of this proof to the case of cyclic groups of order $p^n{q}^m$ seems therefore feasible. The only previously known infinite family of cyclic groups, for which Fuglede's conjecture is verified in both directions, is that of cyclic $p$-groups, i.e. ${\mathbb{Z}}_{p^n}$.

16 pages

Cited by in corpus (5)

• On the discrete Fuglede and Pompeiu problems
• On the structure of spectral and tiling subsets of cyclic groups
• Fuglede's conjecture holds on cyclic groups ${\mathbb{Z}}_{pqr}$
• Formal duality in finite cyclic groups
• On product spectral sets and functional tiles in ${\mathbb{Q}}_p^d$