Integrality of Averages of Roots of Unity and Perfect Isometries

Source: arXiv:2605.31161 · Published 2026-05-29 · By Chatchawan Panraksa, Pornrat Ruengrot

TL;DR

This paper addresses the problem of characterizing functions f: Z_n → Z_n for which the averages of roots of unity expressions µ_b(f) = (1/n) ∑{x=0}^{n-1} ω^{f(x)+bx} (with ω=e^{2πi/n}) are algebraic integers for all b ∈ Z_n. The authors prove a new, short, and elementary uniform criterion that if these averages are algebraic integers for every b, then f must be a linear polynomial modulo n, i.e. f(x) ≡ αx + β (mod n). This settles a previous conjecture from the literature that was known only for prime n using finite field techniques. Furthermore, specializing to prime power modulus n = p^r, they use a local-global integrality argument in the cyclotomic field and p-adic valuation to show any normalized sum of p^r-th roots of unity that is p-adically integral must be zero or a single root of unity. As an application, they fully characterize the perfect isometries of the cyclic group C as precisely those induced by affine permutations f(x)= αx + β with gcd(α, p^r) = 1. The results unify and simplify prior approaches by avoiding finite-field machinery and show how a basic algebraic integer criterion yields strong structural insights in number theory and character theory.

Key findings

• For any modulus n ≥ 1, if f : Z_n → Z_n satisfies that µ_b(f) = (1/n) ∑_x ω^{f(x)+bx} is an algebraic integer for all b ∈ Z_n, then f is linear modulo n (Theorem 3.1).
• This linearity characterization holds uniformly for all n, not just primes, avoiding prior use of finite-field machinery.
• When n = p^r is a prime power, any normalized sum S of p^r-th roots of unity with positive integer coefficients summing to p^r, whose average S/p^r is p-adically integral, must be either 0 or a single root of unity (Theorem 4.3).
• The rational prime p is totally ramified in Q(ζ_{p^r}), with a unique prime ideal (1 - ω) dividing p, enabling a local-global integrality argument in the cyclotomic ring OF = Z[ω].
• The perfect isometries of the cyclic group C_{p^r} correspond exactly to affine permutations f(x) ≡ αx + β with gcd(α, p^r) = 1 (Theorem 5.3), characterized via the p-adic integrality of µ_I(g^a, g^b).
• The paper generalizes and simplifies the previously known prime modulus case which used the theory of permutation polynomials and Stothers' result.
• The algebraic integer criterion for averages of roots of unity states they are integral iff the sum is zero or all roots coincide (Lemma 2.1), a key tool for proofs.

Methodology — deep read

The authors start with the problem setting: fix a primitive nth root of unity ω = e^{2πi/n} and consider functions f: Z_n → Z_n. For each b ∈ Z_n, define the average µ_b(f) = (1/n) Σ_{x=0}^{n-1} ω^{f(x)+bx}. The main assumption is that all µ_b(f) are algebraic integers in the cyclotomic field Q(ω).

1. Threat Model & Assumptions: The problem is purely mathematical; no adversarial setting is considered. The question is which functions f yield integral averages for all b without prior knowledge of f.

2. Data: The input is the function f with domain and codomain Z_n, no external dataset. The integers n and prime powers p^r form parameters. No preprocessing.

3. Architecture / Algorithm: The core is proof-based. They leverage a classical criterion (Lemma 2.1) that an average of n roots of unity is an algebraic integer iff either the roots are all equal or they sum to zero. Applying this to the multisets {ω^{f(x)+bx} : x ∈ Z_n} for each b, they argue that either f(x)+bx is constant mod n (implying linearity) or all sums vanish. Summing the vanishing cases over b yields a contradiction unless some b collapses the expression to a constant, yielding f linear.

For n = p^r, the authors work in cyclotomic rings OF = Z[ω] with ω = ζ_{p^r}. Using valuations v_p and the fact p totally ramifies, they apply a local-global integrality approach: if a normalized sum μ = S/p^r ∈ F (cyclotomic field) is p-adically integral in the localization OF_p, then it is globally integral in OF (Theorem 4.2). Combining this with the criterion from Lemma 2.1 on sums of roots of unity with positive integer multiplicities leads to the result that μ can be 0 or a single root of unity.

For perfect isometries of C_{p^r}, defined via character bijections I: Irr(G) → Irr(G) and associated functions μ_I(g^a, g^b), the integrality condition translates directly into the previously studied algebraic integer condition, characterizing isometries via affine permutations.

4. Training regime: Not applicable; pure math proofs.

5. Evaluation protocol: The proofs proceed by contradiction, algebraic number theory, and applying classical lemmas. Results settle conjectures and match prior known prime cases.

6. Reproducibility: The work is theoretical. All proofs are constructive and explicit with self-contained lemmas. No code or datasets are involved.

Example end-to-end: Suppose n = 6 and f: Z_6 → Z_6 satisfies that for every b ∈ Z_6, the average μ_b(f) = (1/6) Σ_x e^{2πi(f(x)+bx)/6} is integral over the algebraic integers. By Lemma 2.1 applied to the multisets {e^{2πi(f(x)+bx)/6}}, either the sum of roots is zero or all roots are equal. The proof shows for all b if the sums vanish, then aggregate sum leads to contradiction. Hence there is b_0 such that f(x) + b_0 x is constant mod 6, so f is affine mod 6.

Similarly for n = p^r, p-adic valuation arguments confirm the sum structure. This yields a complete classification of perfect isometries of cyclic p-groups.

All arguments avoid finite field theory permutation polynomial machinery used in prior work, leading to a short and uniform proof for all n.

Technical innovations

• A uniform, elementary proof that algebraic integrality of the averages µ_b(f) for all b forces linearity of f modulo n, valid for all n without finite field machinery.
• Use of a local-global integrality criterion exploiting the total ramification of p in Q(ζ_{p^r}) to show p-adically integral averages must be trivial sums (zero or a single root of unity).
• Complete characterization of perfect isometries of cyclic p-groups C_{p^r} via affine permutations, connecting representation theory and cyclotomic integrality.
• Avoidance of previous dependency on permutation polynomial theory over finite fields, simplifying and extending prior partial results.

Baselines vs proposed

• Previous result [2] for prime n: linearity proved using permutation polynomial theory; here extended to all n without that machinery.
• Characterization of perfect isometries matches Broué's framework and refines previous partial results for cyclic p-groups.

Limitations

• Results are proven purely for cyclic groups and functions on Z_n; extensions to non-cyclic or more general groups are not addressed.
• The p-adic integrality results rely on total ramification of p in cyclotomic fields; behavior for composite moduli with multiple prime factors beyond prime powers is unclear.
• No computational or algorithmic implementation of perfect isometry testing is provided — the paper is purely theoretical.
• The integrality condition tested is quite strong; the behavior when only some averages are integral or approximate integrality holds is not studied.
• The approach does not analyze or discuss adversarial settings or noise robustness, limiting direct applicability to bot-defense or CAPTCHA domains.

Open questions / follow-ons

• Can the methods and integrality criteria extend to characterize functions or isometries for more general finite groups beyond cyclic groups of prime power order?
• What structural characterizations arise for functions when the algebraic integrality condition is relaxed or only holds for a subset of parameters b?
• Is there an efficient algorithmic procedure to test perfect isometry property or to find the affine parameters α, β given a function f computationally?
• How might these integral-average characterizations connect to other problems in number theory or representation theory using roots of unity?

Why it matters for bot defense

Although this paper's focus is strictly theoretical in algebraic number theory and character theory, the core criterion that averages of roots of unity are integral only under highly structured conditions (linear functions modulo n) could inspire bot-defense researchers analyzing similar root-of-unity based features. In CAPTCHA or bot-defense contexts where one designs challenge-response puzzles involving group structures or complex exponential sums, understanding when aggregate exponential averages force linearity or triviality may help construct robust puzzles or identify invariants difficult for bots to fake. The local-global integrality argument at prime powers also exemplifies how p-adic valuations can strongly constrain sums of complex exponentials, which may be exploited for secure protocol design.

Overall, while not directly addressing CAPTCHA or bot detection, the methods show how seemingly complicated sums simplify dramatically under integrality constraints. Insights into perfect isometries could analogously inform symmetric transformations or character-based invariants utilized in advanced bot-defense schemes leveraging algebraic structures.

Cite

@article{arxiv2605_31161,
  title={ Integrality of Averages of Roots of Unity and Perfect Isometries },
  author={ Chatchawan Panraksa and Pornrat Ruengrot },
  journal={arXiv preprint arXiv:2605.31161},
  year={ 2026 },
  url={https://arxiv.org/abs/2605.31161}
}

Read the full paper

• arXiv:2605.31161 (abstract)
• arXiv:2605.31161 (PDF)