Sum-product estimates via directed expanders

TL;DR

This paper characterizes polynomials over finite fields that exhibit a sum-product type phenomenon, showing that small sum sets imply large polynomial image sets using directed expander constructions.

Contribution

It provides a complete characterization of polynomials over finite fields for which small sum sets lead to large polynomial image sets, extending sum-product estimates.

Findings

Identifies all polynomials with sum-product type behavior

Uses directed expander graphs to analyze polynomial images

Generalizes the sum-product problem to polynomial functions

Abstract

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, 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 $|A+A|$ is small, then $|P(A)|$ is large.} \vskip2mm The case $P=x_1x_2$ corresponds to the well-known sum-product problem.