A note on average of roots of unity

We consider the problem of characterizing all functions f defined on the set of integers modulo n with the property that an average of some nth roots of unity determined by f is always an algebraic integer. Examples of such functions with this property are linear functions. We show that, when n is a prime number, the converse also holds. That is, any function with this property is representable by a linear polynomial. Finally, we give an application of the main result to the problem of determining self-perfect isometries for the cyclic group of prime order p.