Fuglede's conjecture on cyclic groups of order pnq

Romanos Diogenes Malikiosis,Mihail N Kolountzakis

doi:10.19086/da.2071

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 $n\geq1$. 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. $\mathbb{Z}_{p^n}$.

Highlights

Let Ω be a measurable subset of Rn of positive Lebesgue measure
Ω is called spectral, if it accepts an orthogonal basis of exponentials, namely eiλ·x, where λ ranges through Λ ⊆ Rn; the set Λ is called the spectrum of Ω
The novel contribution of our present work is the proof of Spectral⇒Tile direction for cyclic groups of order N = pnq, establishing the veracity of Fuglede’s conjecture in this setting

Summary

Introduction

Let Ω be a measurable subset of Rn of positive Lebesgue measure. Ω is called spectral, if it accepts an orthogonal basis of exponentials, namely eiλ·x, where λ ranges through Λ ⊆ Rn; the set Λ is called the spectrum of Ω. Let A ⊆ ZN be a set that tiles ZN by translations, where N = pnqm for distinct primes p, q. The novel contribution of our present work is the proof of Spectral⇒Tile direction for cyclic groups of order N = pnq, establishing the veracity of Fuglede’s conjecture in this setting.

Results

Conclusion