Open, the site's label (OPEN; page last edited 23 January 2026), which attaches to the three questions together. The first two are open: Erdős [Er54] constructed a set $A$ with $\lvert A\cap\{1,\ldots,N\}\rvert\ll(\log N)^2$ such that every large integer is $p+a$, improving Lorentz's $(\log N)^3$ [Lo54], but no source reaches $o((\log N)^2)$, and the $O(\log N)$ question is open even for the almost-all variant, where Wolke [Wo96] reached $(\log N)^{1+o(1)}$, Kolountzakis [Ko96] $(\log N)\log\log N$ and Ruzsa [Ru98c] $O(\omega(N)\log N)$ for any $\omega\to\infty$, with $O(\log N)$ known only when the sumset need have lower density at least $1-\varepsilon$ (Ruzsa, Theorem 1). The third question is answered yes: Theorem 2 of Ruzsa [Ru98c] gives $\liminf\lvert A\cap\{1,\ldots,N\}\rvert/\log N\ge e^{\gamma}\approx1.781$ for every such $A$, recorded as the accepted partial claim on Ruzsa's claim page (1998), settling the part liminf on the refereed publication. The parts existence and log are unsettled, so the problem stays open. Erdős offered a prize for the second question, as Guy [Gu04] reports it.