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arXiv · 1805.08900

A new bound on Erd\H{o}s distinct distances problem in the plane over prime fields

Alex Iosevich·Doowon Koh·Thang Pham·Chun-Yen Shen·Le Anh Vinh
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Abstract

In this paper we obtain a new lower bound on the Erd\H{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\subset \mathbb{F}_p^2$ with $|A|\le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |\Delta(A)| \gg |A|^{\frac{1}{2}+\frac{149}{4214}}.$$ Our result gives a new lower bound of $|\Delta{(A)}|$ in the range $|A|\le p^{1+\frac{149}{4065}}$. The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower bound on the number of distinct distances between a line and a set in $\mathbb{F}_p^2$. The latter is the new feature that allows us to improve the previous bound due Stevens and de Zeeuw.

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Alex Iosevich, Doowon Koh, Thang Pham, Chun-Yen Shen, Le Anh Vinh. 2018-05-22. A new bound on Erd\H{o}s distinct distances problem in the plane over prime fields. https://arxiv.org/abs/1805.08900

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