On $(\alpha_n)$-regular sets

TL;DR

This paper introduces $(eta_n)$-regular sets in uniformly perfect metric spaces, exploring their properties, measure restrictions, and conditions for being fat or thin, extending known results from the real line to more general spaces.

Contribution

It defines a new class of regular sets in metric spaces, analyzes their measure-theoretic properties, and generalizes previous real-line results to broader metric space contexts.

Findings

Characterization of fat and thin $(eta_n)$-regular sets

Conditions under which doubling measures remain doubling when restricted

Extension of real-line results to uniformly perfect metric spaces

Abstract

We define $(\alpha_n)$ -regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the necessary and sufficient conditions to be fat (or thin). In addition we discuss restrictions of doubling measures to these sets, and in particular give a sufficient condition to retain at least some of the restricted measures doubling on the set. Our main result generalizes and extends analogous results that were previously known to hold in the real-line.