Attaining the exponent $5/4$ for the sum-product problem in finite fields

TL;DR

This paper improves the exponent in the finite field sum-product problem from 11/9 to 5/4, showing that for certain sets in finite fields, either the sum set or the product set is significantly larger.

Contribution

The paper establishes a new lower bound exponent of 5/4 for the sum-product problem in finite fields, surpassing previous bounds of 11/9.

Findings

The maximum of sum or product sets is at least proportional to |A|^{5/4}.

Results hold for sets with size up to roughly p^{1/2} in finite fields.

Improves upon previous sum-product bounds by Rudnev, Shakan, and Shkredov.

Abstract

We improve the exponent in the finite field sum-product problem from $11/9$ to $5/4$, improving the results of Rudnev, Shakan and Shkredov. That is, we show that if $A\subset \mathbb{F}_p$ has cardinality $|A|\ll p^{1/2}$ then \[ \max\{|A\pm A|,|AA|\} \gtrsim |A|^\frac54 \] and \[ \max\{|A\pm A|,|A/A|\}\gtrsim |A|^\frac54\,. \]