A Note on Average of Roots of Unity

TL;DR

This paper characterizes functions on integers modulo n whose averages of nth roots of unity are always algebraic integers, proving linearity for prime n and applying results to self perfect isometries of cyclic groups.

Contribution

It proves that for prime n, functions with the algebraic integer average property are necessarily linear, extending understanding of roots of unity and their applications.

Findings

Linear functions have the property for all n.

For prime n, the property characterizes linear functions.

Application to self perfect isometries of cyclic groups.

Abstract

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 $n$th 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 $p$.