On thin sets of primes expressible as sumsets

Whether or not there exist sets of integers A and B, each with at least two elements, such that A + B coincides with the set of primes for suffi-ciently large elements, remains an open problem. There has been recent progress however, showing that the counting functions A(x) and B(x) must both have size x^1/2+o(1). We show in this paper that further progress can be expected from the structure theory of sumsets. As a first step towards this, we examine sumsets of three sets A, B, C, where each has at least two elements, and A+B+C consists entirely of primes. First we show that, assuming the Hardy-Littlewood conjecture, there exist sets of integers A, B, C, each having at least two elements, with A + B + C consisting entirely of primes, and (A + B + C)(x) ≫ x/ log³ x, and where A + B contains at most 3 elements. Thus, there exist "not very thin" sets of primes that can be expressed as a sumset of three sets. The main result in the paper is a certain "inverse theorem": We show that if A, B, C each have at least two elements A+ B + C consists entirely of primes with (A+ B + C) ≫ x/ log ^κ x, and if A, B C are what we call a "regular triple", then either A + B, B + C or A + C must have at most κ elements. We use many different methods to prove this, including sieve methods, the probabilistic method, and a variety of other combinatorial methods.