Problem #329: Suppose $A\subseteq \mathbb{N}$ is a Sidon set

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Progress

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 [44] and [707]).

This question can also be asked for $B_2[g]$ sequences (i.e. where the number of solutions to $n=a_1+a_2$ with $a_1\leq a_2$ is at most $g$ for all $n$, so that a $B_2[1]$ set is a Sidon set). Kolountzakis [Ko96] constructed a $B_2[2]$ sequence where the $\limsup$ is $1$, and for larger $g$ constructions were provided by Cilleruelo and Trujillo [CiTr01].