On equivalence of unbounded metric spaces at infinity

TL;DR

This paper studies the asymptotic behavior of unbounded metric spaces at infinity by analyzing pretangent spaces and establishes conditions for their equivalence, with specific results for the real line.

Contribution

It provides necessary and sufficient conditions for two unbounded subspaces to have the same pretangent spaces at infinity and characterizes subspaces of the real line isometric to their pretangent spaces.

Findings

Conditions for equivalence of pretangent spaces are established.

Characterization of subspaces of the real line isometric to their pretangent spaces.

Framework for understanding asymptotic geometry of unbounded metric spaces.

Abstract

Let $(X, d)$ be an unbounded metric space. To investigate the asymptotic behavior of $(X, d)$ at infinity, one can consider a sequence of rescaling metric spaces $(X, \frac{1}{r_n} d)$ generated by given sequence $(r_n)_{n \in \mathbb N}$ of positive reals with $r_n \to \infty$. Metric spaces that are limit points of the sequence $(X, \frac{1}{r_n} d)_{n \in \mathbb N}$ will be called pretangent spaces to $(X, d)$ at infinity. We found the necessary and sufficient conditions under which two given unbounded subspaces of $(X, d)$ have the same pretangent spaces at infinity. In the case when $(X, d)$ is the real line with Euclidean metric, we also describe all unbounded subspaces of $(X, d)$ isometric to their pretangent spaces.