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

Abstract

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

The note was submitted as part of a larger portfolio, which is where the term "improved" comes from, however, this has caused confusion as it seems to present itself as improving the current standing of the problem as a whole