Sum-ratio estimates over arbitrary finite fields

Oliver Roche-Newton

Sum-ratio estimates over arbitrary finite fields

Combinatorics 2014-07-08 v1 Number Theory

Authors: Oliver Roche-Newton

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.

Keywords

multiplicative functions exponential sums finite fields

Cite

@article{arxiv.1407.1654,
  title  = {Sum-ratio estimates over arbitrary finite fields},
  author = {Oliver Roche-Newton},
  journal= {arXiv preprint arXiv:1407.1654},
  year   = {2014}
}

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

Sum-ratio estimates over arbitrary finite fields

Abstract

Keywords

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