Primes with Beatty and Chebotarev conditions

Abstract

We study the prime numbers that lie in Beatty sequences of the form ⌊αn+β⌋ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence with α of finite type and a Chebotarev class of some Galois extension is precisely the product of the densities α−1⋅|C||G|. Moreover, we show that the primes in the intersection of these sets satisfy a Bombieri–Vinogradov type theorem. This allows us to prove the existence of bounded gaps for such primes. As a final application, we prove a common generalization of the aforementioned bounded gaps result and the Green–Tao theorem.