Infinite partial sumsets within the primes

January 26, 2023

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Tamar Ziegler and I’ve simply uploaded to the arXiv our paper “Infinite partial sumsets within the primes“. This can be a brief paper impressed by a current results of Kra, Moreira, Richter, and Robertson (mentioned for example in this Quanta article from final December) displaying that for any set ${A}$ of pure numbers of constructive higher density, there exists a sequence ${b_1 < b_2 < b_3 < dots}$ of pure numbers and a shift ${t}$ such that ${b_i + b_j + t in A}$ for all ${i<j}$ this solutions a query of ErdÅ‘s). In view of the “transference precept“, it’s then believable to ask whether or not the identical end result holds if ${A}$ is changed by the primes. We are able to present the next outcomes:

Theorem 1

We comment that it was proven by Balog that there (unconditionally) exist arbitrarily lengthy however 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 end result may also be recovered from the later outcomes of Ben Inexperienced, myself, and Tamar Ziegler.) Additionally, it had beforehand been proven by Granville that on the Hardy-Littlewood prime tuples conjecture, there existed growing sequences ${a_1 < a_2 < dots}$ and ${b_1 < b_2 < dots}$ of pure 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 after all in keeping with the previous being conditional and the latter unconditional). The conclusion (ii) additionally implies the well-known theorem of Maynard that for any given ${k}$ , there exist infinitely many ${k}$ -tuples of primes of bounded diameter, and certainly our proof of (ii) makes use of the identical “Maynard sieve” that powers the proof of that theorem (although we use a formulation of that sieve nearer to that in this weblog put up of mine). Certainly, the failure of (iii) principally arises from the failure of Maynard’s theorem for dense subsets of primes, just by eradicating these clusters of primes which might be unusually intently spaced.

Our proof of (i) was initially impressed by the topological dynamics strategies utilized by Kra, Moreira, Richter, and Robertson, however we managed to condense it to a purely elementary argument (taking over solely half a web page) that makes no reference to topological dynamics and builds up the sequence ${b_1 < b_2 < dots}$ recursively by repeated software of the prime tuples conjecture.

The proof of (ii) takes up the vast majority of the paper. It’s best to phrase the argument when it comes to “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 equal to the existence of arbitrarily lengthy prime-producing tuples; our theorem is equal to the stronger assertion that there exist an infinite sequence ${h_1 < h_2 < dots}$ such that each preliminary phase ${(h_1,dots,h_k)}$ is prime-producing. The primary new instrument for attaining that is the next cute measure-theoretic lemma of Bergelson:

Lemma 2 (Bergelson intersectivity lemma) Let ${E_1,E_2,dots}$ be subsets of a chance area ${(X,mu)}$ 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 brief proof, although not a wholly apparent one. Firstly, by deleting a null set from ${X}$ , one can assume that each one finite intersections ${E_{i_1} cap dots cap E_{i_k}}$ are both constructive measure or empty. Secondly, a routine software of Fatou’s lemma reveals that the maximal perform ${limsup_N frac{1}{N} sum_{i=1}^N 1_{E_i}}$ has a constructive integral, therefore should be constructive in some unspecified time in the future ${x_0}$ . Thus there’s a subsequence ${E_{i_1}, E_{i_2}, dots}$ whose finite intersections all comprise ${x_0}$ , thus have constructive measure as desired by the earlier discount.

It seems that one can’t fairly mix the usual Maynard sieve with the intersectivity lemma as a result of the occasions ${E_i}$ that present up (which roughly correspond to the occasion that ${n + h_i}$ is prime for some random quantity ${n}$ (with a well-chosen chance distribution) and a few shift ${h_i}$ ) have their chance going to zero, fairly than being uniformly bounded from beneath. To get round this, we borrow an thought from a paper of Banks, Freiberg, and Maynard, and group the shifts ${h_i}$ into varied clusters ${h_{i,1},dots,h_{i,J_1}}$ , chosen in such a means that the chance that no less than one of ${n+h_{i,1},dots,n+h_{i,J_1}}$ is prime is bounded uniformly from beneath. One then applies the Bergelson intersectivity lemma to these occasions and makes use of many purposes of the pigeonhole precept to conclude.