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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). 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.

Conjecture 1.2 (Dickson--Hardy--Littlewood conjecture, p. 1, quoted). "Every admissible tuple $(h_1,\dots,h_k)$ is prime-producing."

Theorem 1.3 (p. 2, quoted). "Assume Conjecture 1.2. Then there exists an infinite set $B$ of primes such that $b+b'+1$ is prime for every distinct $b,b'\in B$."

The paper presents this as the analogue, for the primes, of Erdős's question whether every set of natural numbers of positive upper density contains an infinite $B$ and a natural number $t$ with $b+b'+t$ in the set for all distinct $b,b'\in B$, which it reports proved by Kra, Moreira, Richter and Robertson (p. 1). It notes (p. 2) that Granville had shown that Conjecture 1.2 implies that the primes contain a sumset $A+B$ of two infinite sets, and that Theorem 1.3 gives a new proof of this; and that, by results of Balog (see also Green and Tao), the theorem holds unconditionally when "infinite set" is replaced by "arbitrarily large finite sets".

Remark 1.4 (p. 2). The theorem does not carry over to every subset of the primes of positive relative density. If $A$ is a set of primes containing $b+b'+1$ for all distinct $b,b'$ in some infinite $B$, then $A$ has bounded gaps infinitely often, since two elements $b_1,b_2$ of $B$ give infinitely many pairs $n+b_1,n+b_2$ in $A$. But for any $h:\mathbb{R}^+\to\mathbb{R}^+$ with $h(x)/\log x\to0$ and $h(x)\to\infty$, the primes $p\ge100$ with no prime in $[p+1,p+h(p)]$ form a set of relative density $1$ in the primes (by a theorem of Gallagher) whose consecutive gaps tend to infinity. The remark says a similar remark applies to Theorem 1.5 and Corollary 1.6.

Read depth. Claims checked: the definition, the conjecture, the theorem and the remark 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

Section 2, pp. 4--5. Call an increasing tuple of primes $3<p_1<\dots<p_k$ good if $p_i+p_j+1$ is prime for $i<j$, consecutive entries differ by more than $2$, and $p_i$ does not divide $p_j+2$ for $i<j$. Proposition 2.1 (p. 4) extends a good $k$-tuple to a good $(k+1)$-tuple, by applying Conjecture 1.2 to the admissible tuple $(0,2,p_1+1,\dots,p_k+1)$; iterating from $p_1=5$ gives the set $B$. Remark 2.2 (pp. 4--5) recasts the construction in the dynamical framework of Kra, Moreira, Richter and Robertson.

Dependencies

Conjecture 1.2, assumed.

Bears on

• Problem 656: the problem's hypothesis is positive upper density, which the primes lack. Theorem 1.3 proves the problem's conclusion for $A$ the primes, with $t=1$, assuming Conjecture 1.2; Remark 1.4 shows the conclusion fails for some subset of the primes of relative density $1$.
• Problem 431: assuming Conjecture 1.2, the paper says Theorem 1.3 reproves Granville's result that the primes contain a sumset $A+B$ of two infinite sets. It says nothing on whether such a sumset can agree with the primes up to finitely many exceptions.