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arXiv · 2607.17324

On Erdos-Falconer distance problem in even dimensions

Thang Pham·Chun-Yen Shen·Boqing Xue
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Abstract

Let $q$ be an odd prime power and $\mathbb{F}_q$ be the finite field of order $q$. We prove an extraction theorem for the Erd\H{o}s-Falconer distance conjecture in even dimensions, showing that the conjecture for all even dimensions reduces to the planar case. As consequences, we obtain improved thresholds on the pinned distance problem and the distribution of triangles, achieving new records of $\frac{d}{2}+\frac{1}{4}$ over prime fields and $\frac{d+1}{2}+\frac{1}{10}$ over arbitrary finite fields, respectively.

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Thang Pham, Chun-Yen Shen, Boqing Xue. 2026-07-19. On Erdos-Falconer distance problem in even dimensions. https://arxiv.org/abs/2607.17324

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