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arXiv 2002.06631math.COmath.MG
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平面超复数的距离问题

Distance problems for planar hypercomplex numbers

David FitzPatrick

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20 pages
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摘要：

本文研究了平面超复数的单位距离和相异距离问题：对偶数$\mathbb{D}$和对偶数$\mathbb{S}$ .我们表明，不同的距离问题在$\mathbb{S}^2$的行为类似于原来的问题在$\mathbb{R}^2$。其他三个问题的表现与它们的真实类似物相当不同。我们通过引入点集的多重性的各种概念来研究这三个问题。我们的分析是基于研究几何的对偶平面和双平面。我们还依赖于离散几何的经典结果，例如Szemerédi-Trotter定理。

英文摘要：

We study the unit distance and distinct distances problems over the planar hypercomplex numbers: the dual numbers $\mathbb{D}$ and the double numbers $\mathbb{S}$. We show that the distinct distances problem in $\mathbb{S}^2$ behaves similarly to the original problem in $\mathbb{R}^2$. The other three problems behave rather differently from their real analogs. We study those three problems by introducing various notions of multiplicity of a point set. Our analysis is based on studying the geometry of the dual plane and of the double plane. We also rely on classical results from discrete geometry, such as the Szemerédi-Trotter theorem.

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