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Statement

Setting (p. 2). Constants $C_U,\epsilon>0$ are fixed, with $\epsilon$ small, and

$$y=\exp\Bigl((1-\epsilon)\frac{\log x\log_3x}{\log_2x}\Bigr),\qquad z=\frac{x}{\log_2x},\qquad U=C_U\frac{x\log y}{\log_2x} \qquad(2.1);$$

$P_t=\prod_{p\le t}p$, and for even $m$

$$\mathcal R_m=\{z<p\le U/m:\ (mp-1,P_y)=1\}\qquad(2.5),$$

with $p$ prime. Here (2.4) defines $\mathcal R$ as the set of $mp\le U$ with $p>z$, $m$ $y$-smooth and $(mp-1,P_y)=1$. Since $Uz^{-1}=C_U\log y$, every $m$ in Proposition 5 is below $y$ once $x$ is large, and for such $m$ the set $\mathcal R_m$ consists of the primes $p$ with $mp\in\mathcal R$.

Proposition 5 (p. 3). "Fix $\delta>0$. Let $m<Uz^{-1}(\log_2x)^{-2}$ be even and let $\mathcal I_m\subseteq[x/2,x]$ be an interval of length at least $\delta|\mathcal R_m|\log x$.

Then for $x>x_0(\delta,C_U)$, there exists a choice of residue classes $a_q\pmod q$ for each prime $q\in\mathcal I_m$ such that

$$p\in\mathcal R_m\Rightarrow p\equiv a_q\pmod q\text{ for some prime }q\in\mathcal I_m."$$

So every prime of $\mathcal R_m$ lies in at least one chosen class, using one class for each prime of $\mathcal I_m$.

The form proved (p. 4). The paper proves an equivalent form: for any fixed $\epsilon,\delta>0$ and any interval $\mathcal I_m\subseteq[x/2,x]$ of length at least $\delta|\mathcal R_m|\log x$, classes $a_q\bmod q$ for the primes $q\in\mathcal I_m$ can be chosen so that all but $\epsilon|\mathcal R_m|$ elements of $\mathcal R_m$ lie in one of them. Appending an interval of length $2\epsilon|\mathcal R_m|\log x$ and using one of its primes for each remaining element gives the proposition with $2\epsilon+\delta$ in place of $\delta$, which is why the paper calls the two forms equivalent.

Source. J. Maynard, Large gaps between primes, Ann. of Math. (2) 183 (2016), no. 3, 915--933, doi:10.4007/annals.2016.183.3.3, read in the arXiv:1408.5110v2 preprint (28 October 2019) identified on the source card; the setting on p. 2, the statement on p. 3, the equivalent form on p. 4, the proof on pp. 4--17.

Read depth. Claims checked: the setting, the statement and the equivalent form were read clause by clause on the page images. The proof (Sections 3 to 6 and the completion on p. 17) was read for its structure; the sieve estimates of Lemmas 6 and 7 were not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 4--17. Section 3 (pp. 4--5) is a probabilistic argument: for each prime $q\in\mathcal I_m$ a class is drawn independently from a probability measure $\mu_{m,q}$, and if almost every $p\in\mathcal R_m$ has expected number $\sum_q\mu_{m,q}(p)$ of hits at least $t$, the expected number of uncovered primes is at most $e^{-t}|\mathcal R_m|$ by (3.1). Section 4 (pp. 5--6) builds $\mu_{m,q}$ in (4.1) from sieve weights adapted from the author's work on small gaps between primes: a GPY-type factor on the tuple $n+h_1q,\dots,n+h_kq$, with an admissible set whose elements are multiples of $P_w$, times a Selberg-sieve factor on $m(n+h_jq)-1$. Section 5 fixes the weights (5.3) through smooth functions $F_{\ell,j}$ and $G$, independent of $q$. Section 6 evaluates the normalizing constant (Lemma 6, p. 7), bounds $\sum_q\mu_{m,q}(p_0)$ from below for $p_0\in\mathcal R_m$ away from the ends (Lemma 7, p. 11), and recalls the integral estimate $kJ_k^{(1)}(F)J_k^{(2)}(G)/(I_k^{(1)}(F)I_k^{(2)}(G))\gg\log k$ (Lemma 8, p. 16). The completion (p. 17) combines these: all but $o_k(|\mathcal R_m|)$ primes of $\mathcal R_m$ have expected number of hits $\gg\delta\log k$, which exceeds $\log\epsilon^{-1}$ once $k$ is large in terms of $\delta$ and $\epsilon$, giving the form proved above.

Dependencies

Lemma 3 of the same paper (p. 3), on the size of $\mathcal R_m$, and Lemmas 6--8 (pp. 7, 11, 16); the weights of the author's Small gaps between primes, Ann. of Math. (2) 181 (2015), and Dense clusters of primes in subsets (the paper's references [9] and [8]), with Lemma 8 resting on [9, Proposition 4.3 (iii)] and [9, §7]; the analytic method of Polymath's Variants of the Selberg sieve, and bounded intervals containing many primes (reference [11]), adapting its Lemma 4.1.

Bears on

• Problem 4: the proposition is the step that lets the constant $C_U$ in $U$ be arbitrarily large, and so is what makes Theorem 1 answer the problem yes.