As is well known, Goldbach conjectured that every even positive integer (greater than 2) is the sum of two primes. While this is a difficult open problem, progress has been made from a number of different directions. Perhaps most notably, Chen has shown that every sufficiently large even positive integer is the sum of a prime and an almost prime (that is an integer that is a product of at most two primes). In another direction, Montgomery and Vaughan have shown that if is the set of positive even integers that cannot be expressed as the sum of two primes then
for some positive constant . This is stronger than the observation (which was made much earlier) that almost every positive integer can be expressed as the sum of two primes. In this post we’ll be interested in sets of integers with the property that most integers can be expressed as the sum of two elements from the set. To be more precise we’ll say that a set of positive integers has the Goldbach property (GP) if the sumset consists of a positive proportion of the integers. From the preceding discussion we have that the set of primes has the GP. (This discussion is closely related to the theory of thin bases.)
A natural first question in investigating such sets would be to ask how thin such a set can be. Simply considering the number of possible distinct sums, the reader can easily verify that a set (of positive integers) with the GP must satisfy
This is to say that a set of positive integers with the GP must satisfy for all large . Recall that the prime number theorem gives us that for the set of primes . Thus the primes are much thicker than a set with the GP needs to be (at least from naive combinatorial considerations).
Considering this, one might ask if there is a subset of the primes with the GP but having significantly lower density in the integers. I recently (re)discovered that the answer to this question is yes. In particular we have that
Theorem 1 There exists a subset of the primes such that has positive density in the integers and