Lewko's blog

Thin sets of primes and the Goldbach property

Posted in Fourier Analysis, math.NT, Uncategorized by Mark Lewko on November 19, 2009

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 {E} is the set of positive even integers that cannot be expressed as the sum of two primes then

 \displaystyle |[0,N]\cap E| \leq |N|^{1-\delta}

for some positive constant {\delta>0}. 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 {S} has the Goldbach property (GP) if the sumset {S+S} 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 {S} (of positive integers) with the GP must satisfy

 \displaystyle \liminf_{N\rightarrow\infty} \frac{[0,N]\cap S}{\sqrt{N}} > 0.

This is to say that a set of positive integers with the GP must satisfy {[0,N]\cap S \gg \sqrt{N}} for all large {N}. Recall that the prime number theorem gives us that {|[0,N]\cap\mathcal{P}| \approx N/\ln(N)} for the set of primes {\mathcal{P}}. 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 {\mathcal{Q}} such that {\mathcal{Q}+\mathcal{Q}} has positive density in the integers and

\displaystyle \limsup_{N\rightarrow\infty} \frac{[0,N]\cap \mathcal{Q}}{\sqrt{N}} < \infty.

 This last inequality shows that {\mathcal{Q}} 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 {\Lambda(p)} sets

In this section we define {\Lambda(p)} sets and record some of their properties. Let {S \subset {\mathbb Z}}. For {p>2} we say that {S} is a {\Lambda(p)} set (with constant {c}) if for every function {f \in L^2(\mathbb{T})} such that {\text{supp}(\hat{f}) \subseteq S} (here {\hat{f}(n)} denotes the {n}-th Fourier coefficient of {f}) we have

\displaystyle ||f||_{L^p} \leq c ||f||_{2}. \ \ \ \ \ (1)

There is a rich theory to {\Lambda(p)} 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 {\Lambda(p)} sets were first defined).

Theorem 2 If {S} is a {\Lambda(p)} set for {p>2}, and {A} is an arithmetic progression then \displaystyle |S \cap A| \ll_{p} |A|^{2/p}..  More specifically, we have that

\displaystyle \limsup_{N\rightarrow\infty} \frac{|S\cap[-N,N]|}{N^{2/p}}<\infty .

This tells us, for example, that a {\Lambda(4)} set {S} must satisfy {S\cap[-N,N] \leq c \sqrt{N}} for large {N}. 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.

Lemma 3 Let {S\subset {\mathbb Z}} be a {\Lambda(4)} set with constant {||S||_{\Lambda(4)}} and let {S'} be a finite subset of {S}. We then have

\displaystyle |S'+S'| \geq \frac{|S|^2}{||S||_{\Lambda(4)}^{4}}.

 We now state a theorem of Bourgain.

 Theorem 4 For any {p>2} there exists a {\Lambda(p)} set, {\mathcal{Q}} consisting of only of primes such that

\displaystyle \liminf_{n\rightarrow\infty} \frac{[0,N]\cap \mathcal{Q}}{N^{2/p}}>0.

 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 {p=4}. In this case (since {4} 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 {\mathcal{Q}} be the {\Lambda(4)} set given by Bourgain’s theorem. By virtue of the fact that {\mathcal{Q}} is a {\Lambda(4)} set we have that 

\displaystyle \limsup_{N\rightarrow\infty}\frac{|\mathcal{Q}\cap[0,N]|}{N^{1/2}} < \infty.

On the other hand, for sufficiently large {N} we also have that {|S\cap [0,N]| \gg N^{1/2}}. Applying Lemma 3, we have that {|(\mathcal{Q}+\mathcal{Q})\cap [0,2N]|\gg N}. We can conclude that 

\displaystyle \liminf_{N\rightarrow\infty} \frac{|(\mathcal{Q}+\mathcal{Q})\cap [0,2N]|}{N} >0

and the theorem follows.

Some Remarks

The argument here (via the proof of Bourgain’s theorem) constructs the set {\mathcal{Q}} 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 {p} is an even integer, in which case his argument can be simplified considerably. It would be interesting to find an analogous combinatorial interpretation of {\Lambda(p)} sets for {2<p<4}. 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 {\Lambda(p)} subsets of maximal density.

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4 Responses

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  1. binbinzhou said, on January 6, 2010 at 10:41 am

    we should put more on the Mobius function!

  2. binbinzhou said, on January 6, 2010 at 11:16 am

    It is a good approach to GC, however i do not think this method finally resolve GC!

  3. binbin zhou said, on January 7, 2010 at 1:21 am

    We should think more on Mobius function!

  4. Marcelo Fernandes said, on May 30, 2012 at 10:13 pm

    Reblogged this on Marcelo Fernandes de Almeida.

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