let $A\subset\mathbb{Z}/n\mathbb{Z}$ such that: $|A|>n^{d}$ ($0< d <1$)

$D=\{(x,y)\,|\, x+y\in A\}$

and $C=(A\times A)\cap D$
i need to prove (or refute) that there exist a lower bound u(n) such that: $lim_{n\rightarrow\infty}\frac{log(u(n))}{log(n)}>0$
and: $|C|\geq u(n)|A|$

thanks to the helpers