Thin and fat sets for doubling measures in metric spaces

TL;DR

This paper investigates the properties of thin and fat sets in uniformly perfect metric spaces, providing conditions under which certain sets are null or positive for all doubling measures.

Contribution

It introduces sufficient conditions for cut-out sets to be classified as thin or fat in the context of doubling measures in metric spaces.

Findings

Characterization of thin sets as null for all doubling measures

Identification of fat sets with positive measure for all doubling measures

Conditions under which cut-out sets are thin or fat

Abstract

We consider sets in uniformly perfect metric spaces which are null for every doubling measure of the space or which have positive measure for all doubling measures. These sets are called thin and fat, respectively. In our main results, we give sufficient conditions for certain cut-out sets being thin or fat.