论文标题
平面超复合数字的距离问题
Distance problems for planar hypercomplex numbers
论文作者
论文摘要
我们研究了单位距离和平面超复杂数字上的不同距离问题：双数$ \ mathbb {d} $和双数字$ \ mathbb {s} $。我们表明，$ \ mathbb {s}^2 $中的独特距离问题的行为与$ \ mathbb {r}^2 $中的原始问题相似。其他三个问题的行为与他们的真实类似物有所不同。我们通过引入点集的多种概念来研究这三个问题。我们的分析基于研究双平面和双面的几何形状。我们还依靠离散几何形状的经典结果，例如szemerédi-trotter定理。
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.
