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Integrality of Averages of Roots of Unity and Perfect Isometries

This paper establishes a criterion for the integrality of averages of roots of unity to prove that functions on $\mathbb{Z}_n$ satisfying specific algebraic conditions are necessarily linear, thereby providing an elementary proof of a conjecture and fully characterizing the perfect isometries of cyclic groups $C_{p^r}$ as those induced by affine permutations.

Original authors: Chatchawan Panraksa, Pornrat Ruengrot

Published 2026-06-01

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CC BY 4.0

Original authors: Chatchawan Panraksa, Pornrat Ruengrot

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, circular clock with $n$ hours on it. In mathematics, the numbers on this clock are called roots of unity. If you take a step around this clock, you eventually return to the start.

Now, imagine you have a mysterious machine (a function) that takes a number from the clock and spits out another number on the same clock. You don't know how this machine works yet. It could be doing something chaotic, or it could be following a simple, straight-line rule (like "add 3 to every number" or "double every number").

This paper is about figuring out exactly how that machine works, just by looking at the average of its outputs.

The Main Mystery: The "Smoothness" Test

The authors ask a specific question: If you take the machine's output, mix it with some other numbers, and calculate the average, what does that tell you?

In math, there's a special club called algebraic integers. Think of these as the "VIPs" of the number world. Most averages of these clock numbers are messy fractions or complex decimals that aren't VIPs. However, if the average is a VIP (an algebraic integer), it's a huge clue.

The Big Discovery (The Main Theorem):
The authors proved a simple rule: If you test this machine with every possible setting and find that the average is always a VIP (an algebraic integer), then the machine must be a simple, straight-line machine.

In everyday terms:

The Machine: A function $f(x)$ .
The Test: Checking if the average of its outputs is a "VIP number."
The Result: If the test passes every time, the machine isn't chaotic. It's doing something simple like $f(x) = \text{something} \times x + \text{something else}$ .

Before this paper, mathematicians only knew this was true if the clock had a prime number of hours (like 3, 5, or 7). They had to use very heavy, complicated tools (like "finite-field machinery") to prove it. The authors of this paper found a short, simple, and universal way to prove it works for any clock size, big or small, prime or not. They didn't need the heavy tools; they just needed a clever trick with averages.

The Special Case: Clocks with Prime Power Hours

The paper then zooms in on a specific type of clock: one where the number of hours is a power of a prime number (like $2^3=8$ , $3^2=9$ , or $5^3=125$ ).

Here, they used a "local-global" argument. Imagine checking the smoothness of a surface:

Global: Looking at the whole surface from far away.
Local: Zooming in very close to see the tiny details.

They showed that if a sum of these clock numbers is "smooth" (integral) when you zoom in on the specific prime number involved, it forces the whole sum to be either zero or just a single clock number. It can't be a messy mixture of many different numbers.

The Real-World Application: Perfect Symmetries

Why does this matter? The authors apply this math to a concept called Perfect Isometries.

Think of a group of dancers (the cyclic group). Each dancer has a specific "move" or "song" (a character) they can perform. A Perfect Isometry is a way to swap the dancers around so that the group still looks and sounds perfectly balanced, but the individual dancers have changed roles.

The paper answers the question: "What are all the possible ways to swap these dancers to keep the group perfect?"

Using their new "average test," they proved that the only ways to do this are the simplest swaps possible:

You can shift everyone by a fixed amount (like everyone moves 2 spots to the right).
You can multiply everyone's position by a number that doesn't get stuck (like everyone doubles their position, but only if the clock size allows it).

In short, the "perfect" symmetries of these groups are exactly the affine permutations—the simple, straight-line rearrangements. There are no hidden, complex, or chaotic ways to swap them that also work perfectly.

Summary

The Rule: If the average of a function's outputs on a clock is always a "special number," the function must be a simple straight line.
The Improvement: This works for any clock size, not just prime sizes, and the proof is much simpler than before.
The Application: This proves that the only "perfect" ways to rearrange a specific type of mathematical group are the simple, straight-line rearrangements. No complex tricks allowed!

Technical Summary: Integrality of Averages of Roots of Unity and Perfect Isometries

Problem Statement

This paper addresses two interconnected problems in algebraic number theory and character theory:

Linearity of Functions on $\mathbb{Z}_n$ : It investigates the conditions under which a function $f: \mathbb{Z}_n \to \mathbb{Z}_n$ must be linear modulo $n$ . Specifically, it examines the conjecture (posed in [2]) that if the averages of roots of unity defined by $\mu_b(f) = \frac{1}{n} \sum_{x=0}^{n-1} \omega^{f(x)+bx}$ are algebraic integers for all $b \in \mathbb{Z}_n$ (where $\omega = e^{2\pi i/n}$ ), then $f$ is necessarily of the form $f(x) \equiv \alpha x + \beta \pmod n$ . Previous proofs for the case where $n$ is prime relied on the theory of permutation polynomials over finite fields and results by Stothers.
Characterization of Perfect Isometries: It seeks to completely determine the perfect isometries of the cyclic group $C_{p^r}$ (where $p$ is prime and $r \ge 1$ ). While partial results existed, a unified characterization for prime power orders was lacking.

Methodology

The authors employ a combination of elementary algebraic number theory and local-global principles, avoiding the finite-field machinery used in prior work.

Criterion for Integrality: The paper utilizes a known criterion (Lemma 2.1) stating that an average of $n$ roots of unity is an algebraic integer if and only if the sum of the roots is zero or all the roots are identical.
Elementary Argument for Linearity: For the general modulus $n$ , the authors apply the integrality criterion to the multiset of terms $\{\omega^{f(x)+bx}\}$ . By analyzing the conditions under which the sum vanishes versus when terms are constant, they derive a contradiction if $f$ is not linear, thereby proving linearity without invoking permutation polynomials.
Local-Global Integrality: For the specific case $n = p^r$ , the authors utilize the structure of the cyclotomic field $F = \mathbb{Q}(\zeta_{p^r})$ . They leverage the fact that $p$ is totally ramified in $F$ , with a unique prime ideal $\mathfrak{p} = (1-\omega)$ . By establishing a local-global identity for the ring of integers $\mathcal{O}_F$ , they show that if a normalized sum of $p^r$ -th roots of unity is $p$ -adically integral (i.e., integral in the localization $\mathcal{O}_{F, \mathfrak{p}}$ ), it is globally integral.
Application to Character Theory: The results are applied to Broué's definition of perfect isometries. The integrality condition for perfect isometries on $C_{p^r}$ is reduced to the $p$ -adic integrality of specific character sums.

Key Contributions and Results

1. Proof of the Linearity Conjecture (Theorem 3.1)

The paper proves that for any modulus $n \ge 1$ , if the averages $\mu_b(f)$ are algebraic integers for all $b \in \mathbb{Z}_n$ , then $f$ is linear modulo $n$ .

Significance: This settles the conjecture from [2] for all $n$ , not just primes.
Methodological Advantage: The proof is short and elementary. It avoids the detour through permutation polynomials and Stothers' results, providing a uniform argument for all $n$ .

2. Classification of Integral Averages for Prime Powers (Theorem 4.3)

For $n = p^r$ , the authors prove that any normalized sum $S/p^r$ of $p^r$ -th roots of unity (where coefficients are non-negative integers summing to $p^r$ ) that is $p$ -adically integral must be either $0$ or a single root of unity (i.e., $S = p^r \omega^{j_0}$ ).

Mechanism: This relies on Theorem 4.2, which establishes that $p$ -adic integrality at the unique prime above $p$ implies global integrality in the cyclotomic field.

3. Characterization of Perfect Isometries (Theorem 5.3)

The paper completely characterizes the perfect isometries of the cyclic group $C_{p^r}$ .

Result: A bijection $f: \mathbb{Z}_n \to \mathbb{Z}_n$ induces a perfect isometry of $C_{p^r}$ if and only if $f$ is an affine permutation of the form $f(x) \equiv \alpha x + \beta \pmod n$ with $\gcd(\alpha, n) = 1$ .
Derivation: The proof combines the $p$ -adic integrality condition (condition (i) of the definition) with the linearity result from Theorem 3.1. The separation condition (condition (ii)) is then shown to hold automatically for such affine maps.

Significance

The paper claims that these results illustrate how a basic fact regarding the averages of roots of unity, when combined with minimal cyclotomic number theory, yields clean structural theorems.

It provides a unified, elementary proof for the linearity of functions on $\mathbb{Z}_n$ that works uniformly for all moduli, removing the dependency on finite-field techniques.
It offers a complete characterization of perfect isometries for cyclic groups of prime power order, resolving the structure of these isometries in terms of simple affine permutations.
The work demonstrates the utility of local-global arguments in cyclotomic fields for solving problems in character theory.