Equi‐distributed property and spectral set conjecture on Zp2×Zp

Abstract

In this paper, we show an equi-distributed property in 2-dimensional finite abelian groups Z p n × Z p m , where p is a prime number. By using this equi-distributed property, we prove that Fuglede's spectral set conjecture holds on groups Z p 2 × Z p , namely, a set in Z p 2 × Z p is a spectral set if and only if it is a translational tile.