Does there exist a characterization of sets $S$ such that $|S-S|$ is "almost quadratic" in $|S|$? For instance, what are some examples of sets such that $|S-S|$ is on the order of $\frac{{|S|}^2}{\log |S|}$?
As an example of what I mean, we can consider $S = A + R$ where $A$ is an arithmetic progression of size $\log n$ and $R$ is a noise, random subset of size $\frac{n}{\log n}$; $S$ consists of random translations of an arithmetic progression. We expect $|S|$ to be on the order of $n$, and $|S-S|$ to be on the order of $\frac{n^2}{\log n}$.
I was wondering, besides examples resembling the one I gave above, whether there are "constructive" sets with this difference set property. For instance, are there any such results regarding the difference sets of squares or Fibonacci numbers?