Garaev's Inequality in finite fields not of prime order

TL;DR

This paper extends Garaev's sum-product theorem to finite fields of non-prime order, addressing the complexities introduced by subfields and establishing a new sum-product estimate under specific conditions.

Contribution

It provides the first sum-product result in finite fields of non-prime order with explicit conditions, generalizing previous prime order results.

Findings

Sum-product estimate with exponent 49/48 in non-prime finite fields

Conditions analogous to Hausdorff dimension less than 1/2 are necessary

Addresses challenges posed by subfields in the sum-product problem

Abstract

We prove a version of Garaev's sum product theorem in the set of finite fields with non-prime order. Because of the presence of subfields, this seems to require some hypotheses on the set. We work under a condition analogous to having Hausdorff dimension less than 1/2. Under these conditions, we obtain a sum-product theorem with exponent 49/48.