SCHWEITZER 2002 Problem 4

For a given natural number $n$ , consider those sets $A\subseteq \mathbb{Z}_n$ for which the equation $xy=uv$ has no other solution in the residual classes $x,y,u,v\in A$ than the trivial solutions $x=u$ , $y=v$ and $x=v$ , $y=u$ . Let $g(n)$ be the maximum of the size of such sets $A$ . Prove that $$\limsup_{n\to\infty}\frac{g(n)}{\sqrt{n}}=1$$