Sum-ratio estimates over arbitrary finite fields

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.

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