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arXiv · 2101.10959

On the two-parameter Erd\H{o}s-Falconer distance problem over finite fields

Clément Francois·Hossein Nassajian Mojarrad·Duc Hiep Pham·Chun-Yen Shen
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Abstract

Given $E \subseteq \mathbb{F}_q^d \times \mathbb{F}_q^d$, with the finite field $\mathbb{F}_q$ of order $q$ and the integer $d \ge 2$, we define the two-parameter distance set as $\Delta_{d, d}(E)=\left\{\left(\|x_1-y_1\|, \|x_2-y_2\|\right) : (x_1,x_2), (y_1,y_2) \in E \right\}$. Birklbauer and Iosevich (2017) proved that if $|E| \gg q^{\frac{3d+1}{2}}$, then $ |\Delta_{d, d}(E)| = q^2$. For the case of $d=2$, they showed that if $|E| \gg q^{\frac{10}{3}}$, then $ |\Delta_{2, 2}(E)| \gg q^2$. In this paper, we present extensions and improvements of these results.

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Clément Francois, Hossein Nassajian Mojarrad, Duc Hiep Pham, Chun-Yen Shen. 2021-01-26. On the two-parameter Erd\H{o}s-Falconer distance problem over finite fields. https://arxiv.org/abs/2101.10959

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