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Statement

Level of distribution (p. 1, the paper's definition (1.3)): the primes have level of distribution $\theta>0$ if for every $A>0$

$$\sum_{q\le x^\theta}\max_{(a,q)=1}\Bigl|\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)}\Bigr|\ll_A\frac{x}{(\log x)^A}.$$

Bombieri--Vinogradov gives this for every $\theta<1/2$; the Elliott--Halberstam conjecture asserts it for every $\theta<1$.

Theorem 1.4 (p. 3). Assume that the primes have level of distribution $\theta$ for every $\theta<1$. Then

$$\liminf_{n}\,(p_{n+1}-p_n)\le12,\qquad\liminf_{n}\,(p_{n+2}-p_n)\le600 .$$

The paper remarks (p. 3) that the constant 12 appears optimal for its method in its current form.

Source. J. Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015), no. 1, 383--413, doi:10.4007/annals.2015.181.1.7, read in the arXiv:1311.4600v3 preprint (28 October 2019) identified on the source card; the pages cited are the preprint's printed pages, not the journal's. The definition on p. 1, Theorem 1.4 on p. 3, the proof on p. 6.

Read depth. Claims checked: the statement and the deduction on p. 6 were read clause by clause. The numerical bounds $M_5>2$ and $M_{105}>4$ (Proposition 4.3 (1) and (2), Section 8, pp. 21--24) were read for their structure and not recomputed. Nothing here is independently reviewed.

Proof pointer

P. 6. Take $\theta=1-\epsilon$. With $k=105$ and the admissible set of diameter 600 used for Theorem 1.3, $M_{105}>4$ gives $\theta M_{105}/2>2$, so Proposition 4.2 yields three primes among the $n+h_i$ infinitely often and the second bound. With $k=5$ and $\mathcal H=\{0,2,6,8,12\}$, $M_5>2$ gives $\theta M_5/2>1$ and the first bound.

Dependencies

Proposition 4.2 and Proposition 4.3 (1) and (2) of the same paper; the hypothesis on the level of distribution is assumed, not proved.