OPEN
Suppose $A\subseteq \mathbb{N}$ is a Sidon set. How large can
\[\limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}\]
be?
Erdős proved that $1/2$ is possible and Krückeberg
[Kr61] proved $1/\sqrt{2}$ is possible. Erdős and Turán
[ErTu41] have proved this $\limsup$ is always $\leq 1$.
The fact that $1$ is possible would follow if any finite Sidon set is a subset of a perfect difference set (see [707]).
