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arXiv · 1805.08910

Sum-product estimates over arbitrary finite fields

Doowon Koh·Sujin Lee·Thang Pham·Chun-Yen Shen
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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 $A\subset \mathbb{F}_q$ we have \[|(A-A)^2+(A-A)^2|\gg |A|^{1+\frac{1}{21}}.\] This can be viewed as the Erd\H{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that \[\max\{|A+A|, |A^2+A^2|\}\gg |A|^{1+\frac{1}{42}}, ~|A+A^2|\gg |A|^{1+\frac{1}{84}}.\]

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Doowon Koh, Sujin Lee, Thang Pham, Chun-Yen Shen. 2018-05-23. Sum-product estimates over arbitrary finite fields. https://arxiv.org/abs/1805.08910

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