OPEN
Let $A\subseteq \mathbb{N}$ be a set of density zero. Does there exist a basis $B$ such that $A\subseteq B+B$ and
\[\lvert B\cap \{1,\ldots,N\}\rvert =o(N^{1/2})\]
for all large $N$?
Erdős and Newman
[ErNe77] have proved this is true when $A$ is the set of squares.
See also [806].
