The joint distribution of Q-additive functions on polynomials over finite fields
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 125-150.

Soient K un corps fini et QK[T] un polynôme de degré au moins égal à 1. Une fonction f sur K[T] est dite (complètement) Q-additive si f(A+BQ)=f(A)+f(B) pour tous A,BK[T] tels que deg(A)<deg(Q). Nous montrons que les vecteurs (f 1 (A),...,f d (A)) sont asymptotiquement équirépartis dans l’ensemble image {(f 1 (A),...,f d (A)):AK[T]} si les Q j sont premiers entre eux deux à deux et si les f j :K[T]K[T] sont Q j -additives. En outre, nous établissons que les vecteurs (g 1 (A),g 2 (A)) sont asymptotiquement indépendants et gaussiens si g 1 ,g 2 :K[T] sont Q 1 - resp. Q 2 -additives.

Let K be a finite field and QK[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,BK[T] and deg(A)<deg(Q). We prove that the values (f 1 (A),...,f d (A)) are asymptotically equidistributed on the (finite) image set {(f 1 (A),..., f d (A)):AK[T]} if Q j are pairwise coprime and f j :K[T]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] are Q 1 - resp. Q 2 -additive.

@article{JTNB_2005__17_1_125_0,
     author = {Michael Drmota and Georg Gutenbrunner},
     title = {The joint distribution of $Q$-additive functions on polynomials over finite fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {125--150},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     doi = {10.5802/jtnb.481},
     zbl = {1129.11040},
     mrnumber = {2152215},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.481/}
}
Michael Drmota; Georg Gutenbrunner. The joint distribution of $Q$-additive functions on polynomials over finite fields. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 125-150. doi : 10.5802/jtnb.481. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.481/

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