Thin sets of primes and the Goldbach property
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
This last inequality shows that is as thin as such a set can be. In fact, one can show that a generic subset of the primes of this density will work. This result, in an more quantitative form, has been observed by Granville and earlier by Wirsing. Here I’ll present the somewhat different argument I found which is based on a theorem of Bourgain.
Some properties of sets
In this section we define sets and record some of their properties. Let . For we say that is a set (with constant ) if for every function such that (here denotes the -th Fourier coefficient of ) we have
There is a rich theory to sets (being an instance of the Restriction phenomenon in Fourier analysis), however we will only need a few key properties. The following theorem was proved in this paper of Rudin (which is also where sets were first defined).
This tells us, for example, that a set must satisfy for large . These sets also have a number of nice combinatorial properties. Here we record a basic lemma (which can be found as Lemma 4.30 in Tao and Vu’s Additive Combinatorics) which is easily deduced from Parseval’s identity.
We now state a theorem of Bourgain.
Theorem 4 For any there exists a set, consisting of only of primes such that
The proof of this theorem is fairly involved and relies on number theoretic as well as probabilistic arguments. We will only need this result in the case when . In this case (since is an even integer and norms can be expanded) both parts of the argument can be significantly simplified. In particular the only number theoretic information that is required are upper bounds on the number of ways an integer can be represented as the sum of two primes (see, for example, this paper of Green and Tao). Information of this sort can be readily obtained by standard sieve methods.
Let us now fit these pieces together to prove the theorem.
Proof of Theorem 1
Let be the set given by Bourgain’s theorem. By virtue of the fact that is a set we have that
On the other hand, for sufficiently large we also have that . Applying Lemma 3, we have that . We can conclude that
and the theorem follows.
The argument here (via the proof of Bourgain’s theorem) constructs the set in a probabilistic manner. This is also the case in the arguments of Granville and Wirsing. In fact all of these arguments proceed by showing that a generic (in an appropriate sense) has the property that one is seeking. It would be very interesting to construct an explicit example of such a set.
As we remarked earlier we are only using Bourgain’s theorem when is an even integer, in which case his argument can be simplified considerably. It would be interesting to find an analogous combinatorial interpretation of sets for . Presumably this would lead to a deeper result about the primes.
The proof of Bourgain’s theorem uses very little about the primes. Similar ideas can be applied to other number theoretic sets, such as polynomial sequences. In fact, his paper gives a general criterion for a set to containing a subsets of maximal density.