SCHWEITZER 2012 Problem 2

Call a subset $A$ of the cyclic group $(\mathbb{Z}_n,+)$ rich if for all $x,y \in \mathbb{Z}_n$ there exists $r \in \mathbb{Z}_n$ such that $x-r,x+r,y-r,y+r$ are all in $A$ . For what $\alpha$ is there a constant $C_\alpha>0$ such that for each odd positive integer $n$ , every rich subset $A \subset \mathbb{Z}_n$ has at least $C_\alpha n^\alpha$ elements?