Equi-distributed property and spectral set conjecture on ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_p$

Abstract

In this paper, we show an equi-disctributed property in $2$-dimensional finite abelian groups ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_p$ where $p$ is a prime number. By using this equi-disctributed property, we prove that Fuglede's spectral set conjecture holds on groups ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_p$, namely, a set in ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_p$ is a spectral set if and only if it is a tile.

19 pages