Erdős problem 43

If Sidon sets $A, B \subseteq \{1, \dots, N\}$ satisfy $(A-A) \cap (B-B) = \{0\}$, must $\binom{|A|}{2} + \binom{|B|}{2} \le \binom{f(N)}{2} + O(1)$, where $f(N)$ is the largest Sidon-set size in $[N]$ - and can the bound be improved by a fixed proportion when $|A| = |B|$?