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

Drmota, Michael; Gutenbrunner, Georg

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 125-150.

Soient $K$ un corps fini et $Q\in K\left[T\right]$ un polynôme de degré au moins égal à 1. Une fonction $f$ sur $K\left[T\right]$ est dite (complètement) $Q$-additive si $f(A+BQ)=f\left(A\right)+f\left(B\right)$ pour tous $A,B\in K\left[T\right]$ tels que $deg\left(A\right)<deg\left(Q\right)$. Nous montrons que les vecteurs $(f_1\left(A\right),...,f_d\left(A\right))$ sont asymptotiquement équirépartis dans l’ensemble image $\{(f_1\left(A\right),...,f_d\left(A\right)):A\in K\left[T\right]\}$ si les $Q_j$ sont premiers entre eux deux à deux et si les $f_j:K\left[T\right]\to K\left[T\right]$ sont $Q_j$-additives. En outre, nous établissons que les vecteurs $(g_1\left(A\right),g_2\left(A\right))$ sont asymptotiquement indépendants et gaussiens si $g_1,g_2:K\left[T\right]\to ℝ$ sont $Q_1$- resp. $Q_2$-additives.

Let $K$ be a finite field and $Q\in K\left[T\right]$ a polynomial of positive degree. A function $f$ on $K\left[T\right]$ is called (completely) $Q$-additive if $f(A+BQ)=f\left(A\right)+f\left(B\right)$, where $A,B\in K\left[T\right]$ and $deg\left(A\right)<deg\left(Q\right)$. We prove that the values $(f_1\left(A\right),...,f_d\left(A\right))$ are asymptotically equidistributed on the (finite) image set $\left\{\right(f_1\left(A\right),...,$ $f_d\left(A\right)):A\in K\left[T\right]\}$ if $Q_j$ are pairwise coprime and $f_j:K\left[T\right]\to K\left[T\right]$ are $Q_j$-additive. Furthermore, it is shown that $(g_1\left(A\right),g_2\left(A\right))$ are asymptotically independent and Gaussian if $g_1,g_2:K\left[T\right]\to ℝ$ are $Q_1$- resp. $Q_2$-additive.

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