Preservation of the residual classes numbers by polynomials

Abstract:

Let $K$ be a global field and let $\mathcal {O}_{K,S}$ be the ring of $S$-integers of $K$ for some finite set $S$ of primes of $K$. We prove that whatever the infinite subset $E\subseteq \mathcal {O}_{K,S}$ and the polynomial $f(X)\in K[X]$, the subsets $E$ and $f(E)$ have the same number of residual classes modulo $\mathfrak {m}$ for almost all maximal ideals $\mathfrak {m}$ of $\mathcal {O}_{K,S}$ if and only if $\deg (f)=1$ when the characteristic of $K$ is 0 and $f(X)=g(X^{p^k})$ for some integer $k$ and some polynomial $g$ with $\deg (g)=1$ when the characteristic of $K$ is $p>0$.