On the Sum Product Estimates and Two Variables Expanders

Abstract

Let ${\mathbb F}_p$ be the finite field of a prime order~$p$. Let $F \colon {\mathbb F}_p \times {\mathbb F}_p\rightarrow {\mathbb F}_p$ be a function defined by $F(x,y)=x(f(x)+by)$, where $b \in {\mathbb F}_p^*$ and $f\colon {\mathbb F}_p \rightarrow {\mathbb F}_p$ is any function. We prove that if $A \subset {\mathbb F}_p$ and $|A|<p^{1/2}$ then

$|A+A|+|F(A,A)| \gtrapprox |A|^{\frac{13}{12}}.$

Taking $f=0$ and $b=1$, we get the well-known sum-product theorem by Bourgain, Katz and Tao, and Bourgain, Glibichuk and Konyagin, and also improve the previous known exponent from $\frac{14}{13}$ to $\frac{13}{12}$.