Lower bounds for maximal Sidon sets in an interval

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Let $A\subset[1,N]$ be a maximal strong Sidon set and write $k=|A|$.

Blocking criterion and counting

For $x\notin A$, adjoining $x$ violates the Sidon property if and only if

$$x=a+b-c\quad\text{or}\quad 2x=a+b$$

for some $a,b,c\in A$. A collision between two new sums $x+a,x+b$ is trivial. The other possible collisions give exactly the two displayed forms.

For a triple blocker outside $A$, the minus element differs from both plus elements. Counting unordered plus pairs gives at most

$$\binom{k}{2}(k-2)+k(k-1)=\frac{k^2(k-1)}2$$

such expressions. There are at most $\binom{k}{2}$ integer midpoint blockers. Including the occupied points yields the bound

$$\boxed{N\le\frac{k^3+k}{2}.}$$