Distance problems for planar hypercomplex numbers

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 Szemer\'edi-Trotter theorem.