Fuglede's conjecture on cyclic groups of order $p^n q$
Discrete analysis (2017)
Voir la notice de l'article provenant de la source Scholastica
arXiv
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. $\mathbb{Z}_{p^n}$.
Romanos-Diogenes Malikiosis; Mihail N. Kolountzakis. Fuglede's conjecture on cyclic groups of order $p^n q$. Discrete analysis (2017). http://geodesic.mathdoc.fr/item/DAS_2017_a8/
@article{DAS_2017_a8,
author = {Romanos-Diogenes Malikiosis and Mihail N. Kolountzakis},
title = {Fuglede's conjecture on cyclic groups of order $p^n q$},
journal = {Discrete analysis},
year = {2017},
language = {en},
url = {http://geodesic.mathdoc.fr/item/DAS_2017_a8/}
}