Vinogradov’s theorem with Fouvry–Iwaniec primes

Abstract

We show that every sufficiently large $x\equiv 3(4)$ can be written as the sum of three primes, each of which is a sum of a square and a prime square. The main tools are a transference version of the circle method and various sieve related ideas. In particular, a majorant of the set of primes of interest is constructed that overestimates it by a factor of less than $3$ and for which we have good control of the Fourier transform.