OPEN

Let $f(N)$ be maximal such that there exists $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=\lfloor N^{1/2}\rfloor$ such that $\lvert (A+A)\cap [1,N]\rvert=f(N)$. Estimate $f(N)$.

Erdős and Freud [ErFr91] proved
\[\left(\frac{3}{8}-o(1)\right)N \leq f(N) \leq \left(\frac{1}{2}+o(1)\right)N,\]
and note that it is closely connected to the size of the largest quasi-Sidon set (see [840]).