Tamar Ziegler and I have just uploaded to the arXiv our paper “Infinite partial sumsets in the primes“. This is a short paper inspired by a recent result of Kra, Moreira, Richter, and Robertson (discussed for instance in this Quanta article from last December) showing that for any set ${A}$ of natural numbers of positive upper density, there exists a sequence ${b_1 < b_2 < b_3 < \dots}$ of natural numbers and a shift ${t}$ such that ${b_i + b_j + t \in A}$ for all ${i this answers a question of Erdős). In view of the “transference principle“, it is then plausible to ask whether the same result holds if ${A}$ is replaced by the primes. We can show the following results:

Theorem 1
• (i) If the Hardy-Littlewood prime tuples conjecture (or the weaker conjecture of Dickson) is true, then there exists an increasing sequence ${b_1 < b_2 < b_3 < \dots}$ of primes such that ${b_i + b_j + 1}$ is prime for all ${i < j}$.
• (ii) Unconditionally, there exist increasing sequences ${a_1 < a_2 < \dots}$ and ${b_1 < b_2 < \dots}$ of natural numbers such that ${a_i + b_j}$ is prime for all ${i.
• (iii) These conclusions fail if “prime” is replaced by “positive (relative) density subset of the primes” (even if the density is equal to 1).

We remark that it was shown by Balog that there (unconditionally) exist arbitrarily long but finite sequences ${b_1 < \dots < b_k}$ of primes such that ${b_i + b_j + 1}$ is prime for all ${i < j \leq k}$. (This result can also be recovered from the later results of Ben Green, myself, and Tamar Ziegler.) Also, it had previously been shown by Granville that on the Hardy-Littlewood prime tuples conjecture, there existed increasing sequences ${a_1 < a_2 < \dots}$ and ${b_1 < b_2 < \dots}$ of natural numbers such that ${a_i+b_j}$ is prime for all ${i,j}$.

The conclusion of (i) is stronger than that of (ii) (which is of course consistent with the former being conditional and the latter unconditional). The conclusion (ii) also implies the well-known theorem of Maynard that for any given ${k}$, there exist infinitely many ${k}$-tuples of primes of bounded diameter, and indeed our proof of (ii) uses the same “Maynard sieve” that powers the proof of that theorem (though we use a formulation of that sieve closer to that in this blog post of mine). Indeed, the failure of (iii) basically arises from the failure of Maynard’s theorem for dense subsets of primes, simply by removing those clusters of primes that are unusually closely spaced.

Our proof of (i) was initially inspired by the topological dynamics methods used by Kra, Moreira, Richter, and Robertson, but we managed to condense it to a purely elementary argument (taking up only half a page) that makes no reference to topological dynamics and builds up the sequence ${b_1 < b_2 < \dots}$ recursively by repeated application of the prime tuples conjecture.

The proof of (ii) takes up the majority of the paper. It is easiest to phrase the argument in terms of “prime-producing tuples” – tuples ${(h_1,\dots,h_k)}$ for which there are infinitely many ${n}$ with ${n+h_1,\dots,n+h_k}$ all prime. Maynard’s theorem is equivalent to the existence of arbitrarily long prime-producing tuples; our theorem is equivalent to the stronger assertion that there exist an infinite sequence ${h_1 < h_2 < \dots}$ such that every initial segment ${(h_1,\dots,h_k)}$ is prime-producing. The main new tool for achieving this is the following cute measure-theoretic lemma of Bergelson:

Lemma 2 (Bergelson intersectivity lemma) Let ${E_1,E_2,\dots}$ be subsets of a probability space ${(X,\mu)}$ of measure uniformly bounded away from zero, thus ${\inf_i \mu(E_i) > 0}$. Then there exists a subsequence ${E_{i_1}, E_{i_2}, \dots}$ such that

$\displaystyle \mu(E_{i_1} \cap \dots \cap E_{i_k} ) > 0$

for all ${k}$.

This lemma has a short proof, though not an entirely obvious one. Firstly, by deleting a null set from ${X}$, one can assume that all finite intersections ${E_{i_1} \cap \dots \cap E_{i_k}}$ are either positive measure or empty. Secondly, a routine application of Fatou’s lemma shows that the maximal function ${\limsup_N \frac{1}{N} \sum_{i=1}^N 1_{E_i}}$ has a positive integral, hence must be positive at some point ${x_0}$. Thus there is a subsequence ${E_{i_1}, E_{i_2}, \dots}$ whose finite intersections all contain ${x_0}$, thus have positive measure as desired by the previous reduction.

It turns out that one cannot quite combine the standard Maynard sieve with the intersectivity lemma because the events ${E_i}$ that show up (which roughly correspond to the event that ${n + h_i}$ is prime for some random number ${n}$ (with a well-chosen probability distribution) and some shift ${h_i}$) have their probability going to zero, rather than being uniformly bounded from below. To get around this, we borrow an idea from a paper of Banks, Freiberg, and Maynard, and group the shifts ${h_i}$ into various clusters ${h_{i,1},\dots,h_{i,J_1}}$, chosen in such a way that the probability that at least one of ${n+h_{i,1},\dots,n+h_{i,J_1}}$ is prime is bounded uniformly from below. One then applies the Bergelson intersectivity lemma to those events and uses many applications of the pigeonhole principle to conclude.