Tao 2023 infinite partial sumsets primes

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corollary_1_6: Tao and Ziegler's unconditional result that the primes contain half of an infinite sumset: there are infinite increasing sequences of natural numbers (a_i) and (b_j) with a_i + b_j prime whenever i < j.

theorem_1_3: Tao and Ziegler's conditional prime analogue of Erdős's B + B + t question: assuming the Dickson-Hardy-Littlewood conjecture that every admissible tuple is prime-producing, some infinite set B of primes has b + b' + 1 prime for all distinct b, b' in B; Remark 1.4 shows the analogue fails for some subsets of the primes of relative density 1.

theorem_1_5: Tao and Ziegler's unconditional theorem that some infinite increasing sequence of natural numbers has every initial segment (h_1, ..., h_k) prime-producing, meaning infinitely many n make n + h_1, ..., n + h_k all prime.

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Tao, Terence and Ziegler, Tamar, Infinite partial sumsets in the primes. J. Anal. Math. 151 (2023), 375--389, DOI 10.1007/s11854-023-0323-y. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2301.10303), every other right reserved. The copy read for this card is arXiv:2301.10303v4 (27 Jan 2024); labels below follow it. Read status: claims checked; the statements of Theorems 1.3 and 1.5, Corollary 1.6, Conjecture 1.2 and Remark 1.4 were read clause by clause on the printed pages, and no proof is checked step by step.

Corollary 1.6 shows unconditionally that there exist infinite increasing sequences a_1 < a_2 < ... and b_1 < b_2 < ... of natural numbers with a_i + b_j prime whenever 1 <= i < j, so the primes contain half of an infinite sumset. It is deduced from the main Theorem 1.5, which produces an infinite sequence h_1 < h_2 < ... such that every initial k-tuple (h_1, ..., h_k) is prime-producing, meaning infinitely many n make n + h_1, ..., n + h_k simultaneously prime. The paper also records Theorem 1.3, that assuming the Dickson-Hardy-Littlewood conjecture (Conjecture 1.2, that every admissible tuple is prime-producing) there is an infinite set B of primes with b + b' + 1 prime for all distinct b, b' in B -- the prime analog of Erdos's question about B + B + t inside a set of positive upper density, proved for such sets by Kra, Moreira, Richter and Robertson; the paper notes that Theorem 1.3 gives a new proof of Granville's result that, under Conjecture 1.2, the primes contain a sumset A + B of two infinite sets. The unconditional Theorem 1.5 draws instead on an adaptation of Maynard's sieve, whose bounded-gaps result gives that every admissible k-tuple contains a prime-producing l-tuple with l >> log k, together with Bergelson's intersectivity lemma. Remark 1.4 shows the conclusion of Theorem 1.3 can fail when the primes are replaced by a subset of relative density 1, since such a set can have gaps tending to infinity while any such B forces bounded gaps infinitely often; it says a similar remark applies to Theorem 1.5 and Corollary 1.6. For problem 431, which asks whether A + B can equal the primes up to finitely many exceptions, the paper bears only on containment: the primes contain half of an infinite sumset unconditionally, and a full sumset A + B of infinite sets under Dickson-Hardy-Littlewood. It says nothing on whether such a sumset can exhaust the primes.

Source: https://arxiv.org/abs/2301.10303.

Bears on.

• #431: the paper bears only on containment. Corollary 1.6 places half of an infinite sumset inside the primes unconditionally (a_i + b_j prime for i < j), and the paper says Theorem 1.3 reproves Granville's result that, under the Dickson-Hardy-Littlewood conjecture, 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.
• #656: the paper poses the problem's question with A the primes, a set of density zero outside the problem's hypothesis. Theorem 1.3 answers it with t = 1 assuming the Dickson-Hardy-Littlewood conjecture, and Remark 1.4 shows the conclusion fails for some subset of the primes of relative density 1.

Results.

• Theorem 1.5 (p. 2): There is an infinite sequence h_1 < h_2 < ... of natural numbers such that (h_1, ..., h_k) is prime-producing for every k.
• Corollary 1.6 (p. 2): There exist infinite sequences a_1 < a_2 < ... and b_1 < b_2 < ... of natural numbers with a_i + b_j prime whenever 1 <= i < j.
• Theorem 1.3 (p. 2): Assuming Conjecture 1.2 (Dickson-Hardy-Littlewood: every admissible tuple, one avoiding a residue class mod every prime, is prime-producing), there is an infinite set B of primes with b + b' + 1 prime for all distinct b, b' in B; the page also records Remark 1.4 (p. 2), that the conclusion fails for some subset of the primes of relative density 1.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.