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Source. Terence Tao and Tamar Ziegler, Infinite partial sumsets in the primes, J. Anal. Math. 151 (2023), 375--389, read in the arXiv version identified on the source card; labels and pages are that version's.

Statement

Setting (Definition 1.1, p. 1). The natural numbers are $\mathbb{N}=\{1,2,3,\dots\}$. A tuple $(h_1,\dots,h_k)$ of natural numbers is admissible if for each prime $p$ it avoids at least one residue class mod $p$, and prime-producing if there are infinitely many $n$ for which $n+h_1,\dots,n+h_k$ are simultaneously prime. Every prime-producing tuple is admissible.

Theorem 1.5 (p. 2, quoted). "There exists an infinite sequence $h_1<h_2<\dots$ of natural numbers, such that the $k$-tuple $(h_1,\dots,h_k)$ is prime-producing for every $k$."

The theorem is unconditional. The paper notes (pp. 2--3) that it is equivalent to Corollary 1.6, and (p. 3) that Corollary 1.6 implies Maynard's result that arbitrarily long prime-producing tuples exist. The proof places the $h_i$ inside any prescribed infinite admissible set, for example the odd squares (p. 3; the general form is Proposition 5.1, p. 11).

Read depth. Claims checked: the definition and the statement were read clause by clause on the printed pages. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Sections 3 and 4, pp. 5--11. Proposition 3.1 (p. 5), proved with a variant of the Maynard sieve in Section 4, gives for an admissible tuple $(h_{i,j})$ with blocks of sizes $J_1,\dots,J_I$ a probability measure on $[N,2N]$ under which each $n+h_{i,j}$ is prime with probability $\gg\theta_i\log J_i/J_i$ while pairs in one block are both prime with probability $\ll(\theta_i\log J_i/J_i)^2$. Taking $J_i=2^{2^i}$, $\theta_i=2^{-i}$ and the $h_{i,j}$ distinct odd squares, a Furstenberg-type limit gives one measure on $2^\Sigma$ in which, by the second moment method, the event that some $n+h_{i,j}$ in block $i$ is prime has measure $\gg1$ for every $i$. Bergelson's intersectivity lemma (Lemma 3.2, p. 7) then gives $i_1<i_2<\dots$ with all finite intersections of positive measure, and the pigeonhole principle picks one index $j_r$ in each chosen block (p. 7).

Dependencies

The Maynard sieve: Proposition 4.1 (p. 9), a slight variant of Lemma 4.5 of Banks, Freiberg and Maynard whose proof is sketched from the estimates of Polymath 8b, and Lemma 4.2 (p. 10), taken from Lemma 4.6 of Banks, Freiberg and Maynard; and Bergelson's intersectivity lemma (Lemma 3.2, p. 7, quoted from Bergelson's Theorem 1.1).

Bears on

• Problem 431: through its equivalent form, Corollary 1.6, the theorem gives infinite sequences $(a_i)$ and $(b_j)$ with $a_i+b_j$ prime for $i<j$, half of an infinite sumset inside the primes. It says nothing on whether a sumset of two infinite sets can agree with the primes up to finitely many exceptions.