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Claim. Write $J$ for the set of limit points of $(p_{n+1}-p_n)/\log n$. Theorem 3 of J. Pintz, Polignac numbers, conjectures of Erdős on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture, From Arithmetic to Zeta-Functions, Springer (2016), 367--384, states: "There is an ineffective constant $c>0$ such that $[0,c]\subset J$." Each $C$ in $[0,c]$ is then the limit of $(p_{n_i+1}-p_{n_i})/\log n_i$ along some sequence $n_i$, which answers the question of Problem 5 yes for those $C$. The chapter is described on its library card.

Covers. $C=0$ and every $C$ in $[0,c]$ for an ineffective $c>0$ that the paper does not determine; no positive $C$ is named.

Depends on. Nothing in this wiki.

Standing. Claimed. The result appeared as a chapter of the edited volume From Arithmetic to Zeta-Functions (Springer 2016), with no evidence that the chapter was refereed, so no refereed evidence is listed. Not reviewed: the site's commentary credits $[0,c]\subset S$ to [Pi16], but the site labels the problem OPEN, so that commentary is not acceptance.