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 $\frac{1}{n}{\sum }_{x=0}^{n-1}{\omega }^{f(x)+bx}$ (where $\omega =e^{2\pi i/n}$) 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 $x\mapsto \alpha x+\beta$ with $\gcd (\alpha ,p^r)=1$.

6 pages; accepted for publication in Bulletin of the Australian Mathematical Society