Erdős Problem #237 — Unbounded prime-plus-set representation counts

Let $A\subseteq\mathbb N$ satisfy $|A\cap\{1,\ldots,N\}|\gg\log N$ for all sufficiently large $N$, and let $f(n)$ count representations $n=p+a$ with $p$ prime and $a\in A$. Must $\limsup_{n\to\infty}f(n)=\infty$?