INTEGRALITY OF AVERAGES OF ROOTS OF UNITY AND PERFECT ISOMETRIES

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 Z_n. Specifically, we prove that for any modulus n ≥ 1, if f : Z_n → Z_n is a function whose averages (1/n) Σⁿ_x=⁻₀¹ ω^f(x)+^bx (where ω = e²πi/ⁿ) are algebraic integers for all b ∈ 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 use a local–global integrality argument to show that any normalised sum of p^rth roots of unity that is p-adically integral must be either 0 or a single root of unity. As an application, we completely characterise the perfect isometries of the cyclic group C_pr; they are precisely those induced by affine permutations x → αx + β with gcd(α, p^r) = 1.