An improved bound on the sum-product estimate in $\mathbb{F}_{p}$

TL;DR

This paper improves the lower bounds on the sum-product estimate in finite fields, showing that for a non-empty subset of _{p}, either the sum set or product set must be significantly large, refining previous bounds.

Contribution

The paper presents a new, tighter lower bound on the sum-product estimate in _{p}, advancing the understanding of additive and multiplicative growth in finite fields.

Findings

Established a new lower bound involving |^{15/14} and |^{11/12} p^{1/12} terms.

Demonstrated that either the sum set or product set of A is at least proportional to |A|^{15/14} divided by a logarithmic factor.

Improved upon previous sum-product bounds in finite fields, narrowing the gap between known upper and lower estimates.

Abstract

We give an improved bound on the famed sum-product estimate in a field of residue class modulo $p$ ($\mathbb{F}_{p}$) by Erd\H{o}s and Szemeredi, and a non-empty set $A \subset \mathbb{F}_{p}$ such that: $$ \max \{|A+A|,|A A|\} \gg \min \left\{\frac{|A|^{15 / 14} \max \left\{1,|A|^{1 / 7} p^{-1 / 14}\right\}}{(\log |A|)^{2 / 7}}, \frac{|A|^{11 / 12} p^{1 / 12}}{(\log |A|)^{1 / 3}}\right\}, $$ and more importantly: $$\max \{|A+A|,|A A|\} \gg \frac{|A|^{15 / 14}}{(\log |A|)^{2 / 7}}.$$