An improved sum-product estimate over finite fields

TL;DR

This paper improves the sum-product estimate for subsets of finite fields with non-prime order, nearly matching the best known bounds over prime fields, and provides a new bound involving set sizes and logarithmic factors.

Contribution

It introduces an enhanced sum-product estimate for finite fields of non-prime order, extending prior results primarily known for prime fields.

Findings

New sum-product estimate involving |A+A| and |A·A|

Estimate matches the best known prime field bounds up to a logarithmic factor

Applicable under certain conditions for subsets of finite fields

Abstract

This paper gives an improved sum-product estimate for subsets of a finite field whose order is not prime. It is shown, under certain conditions, that $$\max\{|A+A|,|A\cdot{A}|\}\gg{\frac{|A|^{12/11}}{(\log_2|A|)^{5/11}}}.$$ This new estimate matches, up to a logarithmic factor, the current best known bound obtained over prime fields by Rudnev (\cite{mishaSP}).