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 exists 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

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

Let $C=\{(x,y,2y-x)\in A\times A \times A\}$ be the set of $3$-term arithmetic progressions within $A$.

[The original version asked about $x+y \in A$, settled by the example of Anthony Quas.]

I need to prove (or refute) that there exists a lower bound $u(n)$ on $\frac{|C|}{|A|} $ such that

$$\lim_{n\rightarrow\infty}\frac{\log(u(n))}{\log(n)}>0.$$

thanks to the helpers