Sum-product estimates over arbitrary finite fields

2018/05/23 by Koh, Doowon, Lee, Sujin, Pham, Thang +1
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1805.08910

Abstract

In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets A⊂ \mathbbF_q we have |(A-A)²+(A-A)²|≫ |A|^1+(1)/(21). This can be viewed as the Erdős distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that max\|A+A|, |A²+A²|\≫ |A|^1+(1)/(42), ~|A+A²|≫ |A|^1+(1)/(84).

Related

• An improved sum-product estimate over finite fields
• Sum-product estimates in finite fields
• An improved sum-product estimate for general finite fields
• An improved bound on the sum-product estimate in \mathbbF_p
• Sum-ratio estimates over arbitrary finite fields