Integrality of Averages of Roots of Unity and Perfect Isometries

Abstract

We establish a criterion for the integrality of averages of roots of unity and apply it to settle a conjecture regarding the linearity of functions on $\mathbb{Z}_n$. Specifically, we prove that for any modulus $n \ge 1$, if a function $f: \mathbb{Z}_n \to \mathbb{Z}_n$ satisfies that the averages $\frac{1}{n} \sum_{x=0}^{n-1} ω^{f(x)+bx}$ (where $ω=e^{2πi/n}$) are algebraic integers for all $b \in \mathbb{Z}_n$, then $f$ is necessarily linear modulo $n$. This provides a short, elementary proof that works uniformly for all $n$ and avoids the finite-field machinery used in previous partial results. Furthermore, when $n=p^r$, we utilize a local-global integrality argument to show that any normalized sum of $p^r$-th roots of unity that is $p$-adically integral must be either $0$ or a single root of unity. As an application, we completely characterize the perfect isometries of the cyclic group $C_{p^r}$: they are precisely those induced by affine permutations $x \mapsto αx + β$ with $\gcd(α, p^r)=1$.