Sum-ratio estimates over arbitrary finite fields

2014/07/07 by Roche-Newton, Oliver
#11B75 #68R05 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1407.1654

Abstract

The aim of this note is to record a proof that the estimate max\|A+A|,|A:A|\≫|A|^12/11 holds for any set A⊂\mathbbF_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 \citeLiORN, with the product set A⋅A in the place of the ratio set A:A. The sum-ratio estimate here beats the sum-product estimate in \citeLiORN 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 \citeLiORN, but a proof was not given.

Related

• An improved sum-product estimate over finite fields
• Sum-product estimates in finite fields
• An improved sum-product estimate for general finite fields
• Sum-product estimates over arbitrary finite fields
• A sum-product estimate in fields of prime order