Year: 2020
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: FitzPatrick, David
Institution: Princeton University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3006024682