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 Zₙ double struck upper Z Subscript n . Specifically, we prove that for any modulus n ≥ 1 n ≥ 1 n greater than or equals 1 , if f : Z n → Z n f: Zₙ → Zₙ f colon double struck upper Z Subscript n Baseline right arrow double struck upper Z Subscript n Baseline is a function whose averages ( 1 / n ) ∑ x = 0 n − 1 ω f ( x ) + b x (1/n) ∑ ₓ₌₀ⁿ⁻¹ ω ᶠ⁽ˣ⁾⁺ᵇˣ left parenthesis 1 slash n right parenthesis sigma summation Underscript x equals 0 Overscript n minus 1 Endscripts omega Superscript f left parenthesis x right parenthesis plus b x (where ω = e 2 π i / n ω =e2π i/n omega equals e Superscript 2 pi i divided by n ) are algebraic integers for all b ∈ Z n b ∈ Zₙ</