Sum-ratio estimates over arbitrary finite fields

TL;DR

This paper proves a sum-ratio estimate for sets in finite fields that improves upon previous sum-product estimates by a logarithmic factor, under conditions excluding near-subfield sets.

Contribution

It provides a new, more intuitive proof of a sum-ratio estimate over finite fields, extending prior work and improving conditions and bounds.

Findings

Sum-ratio estimate holds for sets not close to subfields.

The estimate surpasses previous sum-product bounds by a logarithmic factor.

The proof is more intuitive and slightly improves conditions for the set A.

Abstract

The aim of this note is to record a proof that the estimate $$\max{\{|A+A|,|A:A|\}}\gg{|A|^{12/11}}$$ holds for any set $A\subset{\mathbb{F}_q}$, provided that $A$ satisfies certain conditions which state that it is not too close to being a subfield. An analogous result was established in \cite{LiORN}, with the product set $A\cdot{A}$ in the place of the ratio set $A:A$. The sum-ratio estimate here beats the sum-product estimate in \cite{LiORN} by a logarithmic factor, with slightly improved conditions for the set $A$, and the proof is arguably a little more intuitive. The sum-ratio estimate was mentioned in \cite{LiORN}, but a proof was not given.