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

## TL;DR

This paper proves that in the finite abelian group _{p^2} imes \u007_{p}, a set is spectral if and only if it tiles, by establishing an equidistributed property and confirming Fuglede's conjecture for this group.

## Contribution

The paper introduces an equidistributed property in rac{p^2}{p} groups and uses it to prove Fuglede's spectral set conjecture in rac{p^2}{p} groups.

## Key findings

- Fuglede's spectral set conjecture holds on rac{p^2}{p} groups.
- Established an equidistributed property in rac{p^2}{p} groups.
- Spectral sets in rac{p^2}{p} groups are exactly the tiles.

## 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.