Integrality of Averages of Roots of Unity and Perfect Isometries
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arXiv
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| Format: | Preprint |
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2026
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| author | Panraksa, Chatchawan Ruengrot, Pornrat |
| author_facet | Panraksa, Chatchawan Ruengrot, Pornrat |
| contents | 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 $\gcd(α, p^r)=1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_31161 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Integrality of Averages of Roots of Unity and Perfect Isometries Panraksa, Chatchawan Ruengrot, Pornrat Number Theory Representation Theory Primary 11R18, Secondary 20C20 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 $\gcd(α, p^r)=1$. |
| title | Integrality of Averages of Roots of Unity and Perfect Isometries |
| topic | Number Theory Representation Theory Primary 11R18, Secondary 20C20 |
| url | https://arxiv.org/abs/2605.31161 |