# Sum-Product Type Estimates over Finite Fields

## TL;DR

This paper establishes sum-product estimates over finite fields using Fourier analysis and third moment methods, showing that large subsets have product-plus-sum sets nearly as large as the entire field.

## Contribution

It introduces a novel combination of Fourier analytic tools and third moment techniques to derive new sum-product estimates over finite fields.

## Key findings

- For subsets A of finite fields, |AA+A| and |A(A+A)| are large, at least proportional to min{q, |A|^2/q^{1/2}}.
- If |A| ≥ q^{3/4}, then |AA+A| and |A(A+A)| are at least proportional to q.
- The methods improve understanding of sum-product phenomena in 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 $|AA+A|,|A(A+A)|\gg q$.