Sum-Product Type Estimates over Finite Fields

Abstract

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 $$|AA+A|,|A(A+A)|\gg \min \left\{q,\frac{|A{|}^2}{q^{\frac{1}{2}}}\right\},$$ so that if $A\ge q^{\frac{3}{4}}$, then $|AA+A|,|A(A+A)|\gg q$.

Lemma 1.2 is written in a general form. I would like to thank Oliver Roche-Newton for pointing out that the main result of the paper can be improved by using point-line incidence theorem of Le Anh Vinh over finite fields