The general Goldbach problem with Beatty primes

In this paper, we are concerned with the representation of large positive integers as the sum of a fixed number of prime numbers taken from Beatty sequences. We are able to avoid any conditions on the Beatty sequences that they be of finite type which have been required hitherto. We use a form of the Hardy–Littlewood–Vinogradov method, but the proofs are quite delicate.