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arXiv · 1905.04179

A new perspective on the distance problem over prime fields

Alex Iosevich·Doowon Koh·Thang Pham
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Abstract

Let $\mathbb{F}_p$ be a prime field, and ${\mathcal E}$ a set in $\mathbb{F}_p^2$. Let $Δ({\mathcal E})=\{||x-y||: x,y \in {\mathcal E} \}$, the distance set of ${\mathcal E}$. In this paper, we provide a quantitative connection between the distance set $Δ({\mathcal E})$ and the set of rectangles determined by points in ${\mathcal E}$. As a consequence, we obtain a new lower bound on the size of $Δ({\mathcal E})$ when ${\mathcal E}$ is not too large, improving a previous estimate due to Lund and Petridis and establishing an approach that should lead to significant further improvements.

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Alex Iosevich, Doowon Koh, Thang Pham. 2019-05-10. A new perspective on the distance problem over prime fields. https://arxiv.org/abs/1905.04179

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