A HYBRID OF THEOREMS OF VINOGRADOV AND PIATETSKI-SHAPIRO

It was proved by Vinogradov that every sufficiently large odd integer can be written as the sum of three primes. We show that this remains the case when the primes so utilized are restricted to an explicit thin set. One may take, for example, the "Piatetski-Shapiro primes" p = [n1/gamma] with any gamma > 20/21 . By a similar argument it would follow that, for arbitrary theta, 0 < theta < 1 , and suitable lambda = lambda(theta) > 0, one may take the set of primes for which {p(theta)} < p(-lambda).