The joint distribution of Q-additive functions on polynomials over finite fields

Abstract

Let $K$ be a finite field and $Q\in K[T]$ 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 $A,B\in K[T]$ and $\deg(A)<\deg(Q)$.<br />We prove that the values $(f_1(A),\ldots,f_d(A))$ are asymptotically equidistributed on the (finite) image set $\{(f_1(A),\ldots,f_d(A)) :<br />A\in K[T]\}$ if $Q_j$ are pairwise coprime and $f_j : K[T] o K[T]$ are $Q_j$-additive. Furthermore, it is shown that $(g_1(A),g_2(A))$ are asymptotically independent and Gaussian if $g_1,g_2: K[T] o \R$ are $Q_1$- resp. $Q_2$-additive.