The Joint Distribution of Q-additive Functions on Polynomials over Finite Fields

Let K be a finite field and Q ∈ K[T ] a polynomial of positive degree. A function f on K[T ] is called (completely) Q-additive if f(A + BQ) = f(A) + f(B), where A,B ∈ K[T ] and deg(A) < deg(Q). We prove that the values (f1(A), ..., fd(A)) are asymptotically equidistributed on the (finite) image set {(f1(A), ..., fd(A)) : A ∈ K[T ]} if Qj are pairwise coprime and fj : K[T ] → K[T ] are Qj-additive. Furthermore, it is shown that (g1(A), g2(A)) are asymptotically independent and Gaussian if g1, g2 : K[T ]→ R are Q1resp. Q2-additive.