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Statement

Setting (p. 4). Write $d_n=p_{n+1}-p_n$. Definition 3 (p. 4): $\mathcal F$ is the class of functions $f:\mathbb Z^+\to\mathbb R^+$ of slow oscillation, meaning that for every $\varepsilon>0$ there is $N(\varepsilon)>0$ with

$$(1-\varepsilon)f(N)\le f(n)\le(1+\varepsilon)f(N)\quad\text{for }N\le n\le2N,\ N>N(\varepsilon).$$

Theorem 4 (p. 4). For every $f\in\mathcal F$ with $f(n)\le\log n$ and $\lim_{n\to\infty}f(n)=\infty$ there is an ineffective constant $c_f>0$ such that

$$[0,c_f]\subset J_f,$$

where $J_f$ is the set of limit points of $d_n/f(n)$.

The paper says the question was asked by Kálmán Győry (p. 4). With $f(n)=\log n$ it gives Theorem 3 (p. 10).

Proof pointer

Page 10. By contradiction, along the lines of the proof of Theorem 2: if the theorem fails, there are, for a small $c^*>0$, intervals $J_\nu=[c_\nu,c_\nu+\delta_\nu]$ with $c_\nu>4\delta_\nu>20c_{\nu+1}$ and $c_1<c^*$ that, for $K$ large, contain no value $d_n/f(n)$ with $n\ge N(K)$ (5.1)--(5.2). The paper builds an admissible $k$-tuple, $3.5\cdot10^6\le k\le K$, with $h_\nu$ in a slightly shrunk copy of $[(c_\nu+\delta_\nu/2)f(N),(c_\nu+\delta_\nu)f(N)]$ (5.5); slow oscillation of $f$ keeps each difference $h_\mu-h_\nu$, $\mu<\nu$, inside $[c_\mu f(n),(c_\mu+\delta_\mu)f(n)]$ for all $n\in[N,2N)$ (5.6)--(5.7). The Main Theorem, whose tuples may have diameter up to $\varepsilon\log N$, which the hypothesis $f(n)\le\log n$ allows, makes some difference equal to $d_n$ for an $n\in[N,2N)$, a contradiction.

Read depth

Claims checked: the definition and the statement were read clause by clause on the printed pages of arXiv:1305.6289v1, and the proof on p. 10 was followed. Nothing here is independently reviewed.

Dependencies

The Main Theorem (p. 6) of this paper.

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

• Problem 5: through its case $f(n)=\log n$, which is Theorem 3; for other $f$ it concerns a different normalization and does not bear on the problem.