# Constructions of small set with big difference set.

Does anyone know any constructions of small set with 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 $n$.

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What do you mean by $\mathbb Z_n$? And how does $A$ relate to $A_n$? –  tomasz Jul 8 '12 at 1:37

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 $\sqrt{n}$.

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user31373's answer doesn't solve the problem. If you extend the sequence to include "200, 300, etc." and then try to find the difference "85", you find it is nowhere in the set. –  user136176 Mar 18 at 5:20
@user136176: 90 - 5 = 85. I am a bit confused as to what you are thinking of. (I've converted your answer to a comment, since it is a comment on an existing answer. You will gain the ability to comment everywhere once you have accrued 50 reputation points on the website.) –  Willie Wong Mar 18 at 9:07
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 $n=q^2+q+1$.