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Statement

Setting (pp. 1--2). $p_n$ is the $n$th prime and $\mathbb{L}$ is the set of limit points of the sequence $\{(p_{n+1}-p_n)/\log p_n\}_{n=1}^{\infty}$.

Theorem 1 (p. 2, quoted). "Let $\beta_1\leq\beta_2\leq\beta_3\leq\beta_4$ be any real numbers. Then $\mathbb{L}\cap\{\beta_j-\beta_i:\ 1\leq i<j\leq 4\}\neq\emptyset.$"

The paper notes (p. 2) that Banks, Freiberg and Maynard proved the same statement with nine reals in place of four, and Pintz with five. It derives Corollary 2 and Corollary 3 from Theorem 1 through two general propositions about sets meeting every such difference set (Propositions 4 and 5, pp. 3--4).

Conditional remark (Remark 2, p. 6). The author says that the argument would give Theorem 1 with three reals in place of four, and so $\mu(\mathbb{L}\cap[0,T])\geq T/2$, if the prime-pair bound (1.6) held with a constant $A<3$ in place of $3.99$, and would give $\mathbb{L}=[0,\infty]$ if it held with $A<2$; he adds that by the parity principle this should be as hard as a lower bound for the same sum over prime pairs.

Source. Jori Merikoski, Limit points of normalized prime gaps, J. Lond. Math. Soc. (2) 102 (2020), 99--124, doi:10.1112/jlms.12314; arXiv:1811.03008. Labels and pages here are those of arXiv v3: Theorem 1 on p. 2, the outline of its proof on pp. 5--6 (Section 1.3), the proof in Sections 2--5 (pp. 7--25), with a correction to the proofs of Lemmas 15 and 16 in Section 6 (pp. 26--31). The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement were read clause by clause on the printed pages. The proof was read for its structure only and not checked step by step. Nothing here is independently reviewed.

Proof pointer

Section 5, pp. 24--25, following the argument of Banks, Freiberg and Maynard (their Section 6). A modified Erdős--Rankin construction (Lemma 20, pp. 24--25, quoted from Banks, Freiberg and Maynard) gives, for large $N$, an admissible $K$-tuple $\mathcal{H}$ split into four equal parts $\mathcal{H}_i$ whose elements are $(\beta_i+\epsilon+o(1))\log N$ (after translating so that $\beta_1\geq0$), and a residue class $b$ modulo a smooth $W$ such that every prime in $(n,n+z]$ with $n\equiv b\ (W)$ lies in $n+\mathcal{H}$. The modified Maynard--Tao sieve (Proposition 18, p. 22) then gives $n\in[N,2N]$ in that class with primes in two different parts, hence two consecutive primes whose normalized gap is close to some $\beta_j-\beta_i$. Proposition 18 is deduced on p. 23 from Proposition 19 with $a=100$, so that $\lceil3.99a\rceil+1=4a$, by pigeonhole. Proposition 19 rests on the bound for sums over prime pairs with constant $3.99$ in place of $4$ (Proposition 12, p. 13), proved in Section 3 with Chen's sieve and a modified Bombieri--Vinogradov theorem for primes and almost-primes (Section 2).

Dependencies

Banks, Freiberg and Maynard's modified Erdős--Rankin construction (Lemma 20, pp. 24--25) and their Maynard--Tao sieve estimates (their Lemmas 4.6 and 4.7, as used on pp. 23--24); the paper's Proposition 12 (p. 13), Proposition 18 (p. 22) and Proposition 19 (pp. 22--23).

Bears on

• Problem 5: the problem asks whether every $C\geq0$ is a limit point of $(p_{n+1}-p_n)/\log n$, the same set $\mathbb{L}$ since $\log p_n\sim\log n$. Theorem 1 shows that $\mathbb{L}$ meets the difference set of any four reals; it does not place any given $C>0$ in $\mathbb{L}$, and the paper says (p. 2) that besides $0$ and $\infty$ no real number is known to lie in $\mathbb{L}$.