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Stable sets of primes in number fields

We define a new class of sets -- stable sets -- of primes in number fields. For example, Chebotarev sets $P_{M/K}(σ)$, with $M/K$ Galois and $σ\in \Gal(M/K)$, are very often stable. These sets have positive (but arbitrary small) Dirichlet density and generalize sets with density 1 in the sense that arithmetic theorems like certain Hasse principles, the Grunwald-Wang theorem, the Riemann's existence theorem, etc. hold for them. Geometrically this allows to give examples of infinite sets $S$ with arbitrary small positive density such that $\Spec \mathcal{O}_{K,S}$ is algebraic $K(π,1)$ (for all $p$ simultaneous).