On a generalization of the Neukirch–Uchida theorem

In this article we generalize a part of Neukirch–Uchida theorem for number fields from the birational case to the case of curves $\operatorname{Spec}\mathcal O_{K,S}$, where $S$ a stable set of primes of a number field $K$. Such sets have positive but arbitrarily small Dirichlet density, which must be uniformly bounded from below by some $\epsilon>0$ in the tower $K_S/K$.