On Thin Sets of Primes Expressible as Sumsets

In this paper we will use the following notation. Given a set of positive integers S, we let S(x) denote the number of elements in S that are ≤ x, and we let |S| denote the total number of elements of S. Given two sets of positive integers A and B, we denote the sumset {a + b : a ∈ A, b ∈ B} by A + B; and so, the number of elements in A + B that are ≤ x will be (A + B)(x). For a finite set of integers J , and integers q ≥ 2 and r, we let J(r, q) denote the set of elements of J which are ≡ r (mod q). We will also use Vinogradov’s notation: The statements “f(x) ≪ g(x)” and “g(x) ≫ f(x)” are both equivalent to “f(x) = O(g(x))”; and, we will use “f(x) ≪y g(x)” to indicate that the implied constant in the big-oh depends on a parameter y. Finally, by the statement f(x) ∼ g(x) we mean that