Sum-product estimates via directed expanders

Let ${\text{F}}_q$ be a finite field of order $q$ and $P$ be a polynomial in ${\text{F}}_q[x_1,x_2]$. For a set $A\subset {\text{F}}_q$, define $P(A):={P(x_1, x_2) | x_i \in A }$. Using certain constructions of expanders, we characterize all polynomials $P$ for which the following holds \vskip2mm \centerline{\it If $\mathrm{\mid }A+A\mathrm{\mid }$ is small, then $\mathrm{\mid }P(A)\mathrm{\mid }$ is large.} \vskip2mm The case $P=x_1{x}_2$ corresponds to the well-known sum-product problem.