Let $A\subseteq \mathbb{N}$ be a set such that $|A\cap \{1,\dots ,N\}|\gg \log N$ for all large $N$. Let $f(n)$ count the number of solutions to $n=p+a$ for $p$ prime and $a\in A$. Is it true that $\mathrm{limsup}f(n)=\infty$?