Oliver Roche-Newton
Published 2014-07-07Version 1
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.
Comments: 12 pages. This note is not intended for journal publication, since the main result and proof are too similar to the work of arXiv:1106.1148. However, the subtle differences between the proof of the main result here and that of arXiv:1106.1148 mean that it is necessary to record the result carefully, particularly with future applications in mind
Szemerédi-Trotter type results in arbitrary finite fields
If $(A+A)/(A+A)$ is small then the ratio set is large
Ratio sets of random sets
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