Constructions of small set with big difference set

Question

Does anyone know any constructions of a small set with a big difference set? Mathematically speaking:

Let $A\subseteq \mathbb{Z}$, such that $A-A=\mathbb{Z}_n$. Please give a sequence $(A_n)_{n\in \mathbb{N}}$ such that $|A_n|$ is small in terms of $n$.

Answer 1

Here is an example which you can easily generalize: 0,1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,100. It gives the best possible order $\sqrt{n}$.

Answer 2

Do you know about perfect cyclic difference sets? If $q$ is a prime power, than there is a set $A$ of $q+1$ integers such that $A-A$ covers ${\bf Z}/n{\bf Z}$ with $n=q^2+q+1$.