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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 $N$?
#333
:
[ErGr80]
number theory
,
additive basis
Erdős and Newman
[ErNe77]
have proved this is true when $A$ is the set of squares.
See also
[806]
.
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