Asymptotically Scattered Spaces

TL;DR

This paper introduces and characterizes asymptotically scattered metric spaces, classifies thin spaces up to coarse equivalence, and extends the concept to group actions, providing a comprehensive framework for their analysis.

Contribution

It defines asymptotically scattered spaces, characterizes them via forbidden subspaces, classifies thin spaces, and extends the concept to G-spaces and groups.

Findings

Characterization of asymptotically scattered spaces via forbidden subspaces

Classification of thin metric spaces up to coarse equivalence

Extension of asymptotically scattered spaces to G-spaces and groups

Abstract

We define thin and asymptotically scattered metric spaces as asymptotic counterparts of discrete and scattered metric spaces respectively. We characterize asymptotically scattered spaces in terms of prohibited subspaces, and classify thin metric spaces up to coarse equivalence. We introduce the types of asymptotically scattered spaces and construct the spaces of distinct types. We transfer the notion of asymptotically scattered spaces to $G$-spaces and characterize asymptotically scattered spaces of groups.