Mathematics > Number Theory
[Submitted on 23 May 2018 (v1), last revised 16 Jul 2018 (this version, v3)]
Title:Sum-product estimates over arbitrary finite fields
View PDFAbstract:In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $A\subset \mathbb{F}_q$ we have \[|(A-A)^2+(A-A)^2|\gg |A|^{1+\frac{1}{21}}.\] This can be viewed as the Erdős distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that \[\max\{|A+A|, |A^2+A^2|\}\gg |A|^{1+\frac{1}{42}}, ~|A+A^2|\gg |A|^{1+\frac{1}{84}}.\]
Submission history
From: Thang Pham [view email][v1] Wed, 23 May 2018 00:03:42 UTC (9 KB)
[v2] Tue, 29 May 2018 21:59:31 UTC (9 KB)
[v3] Mon, 16 Jul 2018 03:32:01 UTC (10 KB)
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