We study the prime numbers that lie in Beatty sequences of the form $\lfloor \alpha n + \beta \rfloor$ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence and a Chebotarev class of some Galois extension is precisely the product of the densities $\alpha^{-1}\cdot\frac{|C|}{|G|}$. 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.

中文翻译：

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具有 Beatty 和 Chebotarev 条件的素数

我们研究位于 $\lfloor \alpha n + \beta \rfloor$ 形式的 Beatty 序列中的素数，并规定了代数分裂条件。我们证明了固定比蒂序列和某个伽罗瓦扩展的 Chebotarev 类中素数的密度恰好是密度的乘积 $\alpha^{-1}\cdot\frac{|C|}{|G|} $. 此外，我们证明这些集合的交集中的素数满足 Bombieri--Vinogradov 型定理。这使我们能够证明此类素数存在有界间隙。作为最终应用，我们证明了上述有界间隙结果和 Green-Tao 定理的共同推广。