Erdős problem 221

Is there a set $A\subset\mathbb{N}$ such that, for all large $N$, $$\lvert A\cap\{1,\ldots,N\}\rvert \ll N/\log N$$ and such that every large integer can be written as $2^k+a$ for some $k\geq 0$ and $a\in A$?

Lorentz [Lo54] proved there is such a set with, for all large $N$, $$\lvert A\cap\{1,\ldots,N\}\rvert \ll \frac{\log\log N}{\log N}N$$ The answer is yes, proved by Ruzsa [Ru72].