I have a question!
Can someone explain how (the intuition, method?) one can try to construct an additive set of cardinality $N$ with a small sum set (around $N$) and a very large difference set (say, around $N^2$)?
I have a question! Can someone explain how (the intuition, method?) one can try to construct an additive set of cardinality $N$ with a small sum set (around $N$) and a very large difference set (say, around $N^2$)? 


A great work on this has been done by Imre Ruzsa; see, for instance, his paper "Many differences, few sums" in Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 51 (2008), 27–38 (2009). As a very brief answer to your question, you cannot have a set $A$ of cardinality $N$ with $2A\sim N$ and $AA\sim N^2$ since if $2A=\alpha N$, then $$ \sqrt\alpha N \le AA \le \alpha^2 N; $$ you will find this inequality in the aforementioned paper by Ruzsa. The only way to construct sets with many differences and few sums I can think of (but perhaps, not the only one known to the mankind) is to use the tensor power trick. Start with you favorite set $A_0$ with $2A_0=\alphaA_0$ and $A_0A_0=\betaA_0$, and consider the cartesian power $A:=A_0^k$ with a large $k$. You have $N=A=A_0^k$, $2A=\alpha^k N$, and $AA=\beta^k N$; hence, if $A_0$ is chosen so that $\alpha<\beta$, then $AA$ is much larger than $2A$ (for $k$ large). 

