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Statement

Setting (p. 4). Write $d_n=p_{n+1}-p_n$. The paper recalls Erdős's 1948 theorem (2.12), $\liminf d_{n+1}/d_n<1<\limsup d_{n+1}/d_n$, and quotes his 1955 remark that "One would of course conjecture that" $\liminf d_{n+1}/d_n=0$ and $\limsup d_{n+1}/d_n=\infty$ (2.13), "but these conjectures seem very difficult to prove."

Theorem 5 (p. 4). As $n\to\infty$,

$$\liminf_{n\to\infty}\frac{d_{n+1}/d_n}{(\log n)^{-1}}<\infty \qquad\text{and}\qquad \limsup_{n\to\infty}\frac{d_{n+1}/d_n}{\log n}>0 .$$

The first inequality says $d_{n+1}/d_n\ll1/\log n$ for infinitely many $n$, and the second that $d_{n+1}/d_n\gg\log n$ for infinitely many $n$; together they give both parts of (2.13).

Proof pointer

Page 11; the paper proves the second inequality and says the first is analogous. From an admissible $k$-tuple, $k\ge3.5\cdot10^6$, the Main Theorem, together with Selberg's upper-bound sieve (Lemma 3, p. 7), gives positions $i<j$ and a sequence $N_\nu\to\infty$ along which at least $(c_1(k,\mathcal H)+o(1))N/\log^kN$ integers $n\le N$ make $n+h_i$, $n+h_j$ consecutive primes (6.6), so the gap $d=h_j-h_i$ is bounded. If the next gap were at most $\varepsilon\log N$ times it for all of these, Lemma 3 and the singular-series average (Lemma 4, p. 8) would bound their number by $O_{k,c_1}(\mathfrak S(\mathcal H)CN\varepsilon/\log^kN)$ (6.11) with $C=h_k-h_1$, too few once $\varepsilon$ is small.

Read depth

Claims checked: the statement was read clause by clause on the printed pages of arXiv:1305.6289v1, and the proof on p. 11 was followed. The first inequality's proof is not written out in the paper. Nothing here is independently reviewed.

Dependencies

The Main Theorem (p. 6) and Lemmas 3 and 4 (pp. 7--8) of this paper, the lemmas taken from earlier works cited there.

Source. János Pintz, Polignac numbers, conjectures of Erdős on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture, arXiv:1305.6289v1 (2013); published in From Arithmetic to Zeta-Functions, Springer (2016), 367--384, doi:10.1007/978-3-319-28203-9_22. Labels and pages here are those of arXiv v1. The edition read is named on the source card.

Bears on

None recorded. Problem 218 also compares consecutive gaps, but asks about the density of $n$ with $d_{n+1}\ge d_n$ and about infinitely many $n$ with $d_{n+1}=d_n$; Theorem 5 answers neither.