Fuglede's conjecture holds in ${\mathbb{Z}}_p\times {\mathbb{Z}}_{p^n}$

Abstract

Fuglede's conjecture states that for a subset $\Omega$ of a locally compact abelian group $G$ with positive and finite Haar measure, there exists a subset of the dual group of $G$ which is an orthogonal basis of $L^2(\Omega )$ if and only if it tiles the group by translation. In this paper, we prove a divisibility property for a set in ${\mathbb{Z}}_p\times {\mathbb{Z}}_{p^n}$. Then using the divisibility property and equi-distributed property, we prove that Fuglede's conjecture holds in the group ${\mathbb{Z}}_p\times {\mathbb{Z}}_{p^n}$.

arXiv comment26 pages. Corrected some typos

References in corpus (6)

• Complex Hadamard matrices and the Spectral Set Conjecture
• Fuglede's spectral set conjecture for convex polytopes
• Fuglede's conjecture fails in 4 dimensions over odd prime fields
• Equi-distributed property and spectral set conjecture on ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_p$
• On the structure of spectral and tiling subsets of cyclic groups
• Fuglede's conjecture holds for cyclic groups of order $pqrs$