Sum-Product Type Estimates over Finite Fields

Let ${\mathbb{F}}_q$ denote the finite field with $q$ elements where $q=p^l$ is a prime power. Using Fourier analytic tools with a third moment method, we obtain sum-product type estimates for subsets of ${\mathbb{F}}_q$. In particular, we prove that if $A\subset {\mathbb{F}}_q$, then $$\mathrm{\mid }AA+A\mathrm{\mid },\mathrm{\mid }A(A+A)\mathrm{\mid }\gg \min \{q,\frac{\mathrm{\mid }A{\mathrm{\mid }}^2}{q^{\frac{1}{2}}}\},$$ so that if $A\ge q^{\frac{3}{4}}$, then $\mathrm{\mid }AA+A\mathrm{\mid },\mathrm{\mid }A(A+A)\mathrm{\mid }\gg q$.