Sum-product phenomenon in finite fields not of prime order (Q633930)

Let \(F=\mathbb F_{p^n}\) be a finite field, and let \(A\) be a subset of \(F\) so that for any \(A'\subset A\) with \(|A'|\geq |A|^{15/16}\) and for any subfield \(G\subset F\) and for any elements \(c,d\in F\) if \[ A'\subset cG+d, \] then \[ |A'|\leq |G|^{1/2}. \] Then it must be that \[ \max(|A+A|, |f(A,A)|) \gtrapprox |A|^{17/16} \] where \(f: \mathbb F_p\times \mathbb F_p \rightarrow \mathbb F_p\) is a function defined by \(f(x,y)=x(g(x)+cy)\), where \(c\in \mathbb F_p^*\) and \(g: \mathbb F_p \rightarrow \mathbb F_p\) is any function.