Singular Sets of Uniformly Asymptotically Doubling Measures

TL;DR

This paper establishes a dimension bound on the singular set of Radon measures that are uniformly asymptotically doubling, showing it has dimension at most n-3 under certain convergence conditions.

Contribution

It introduces a new bound on the singular set dimension for uniformly asymptotically doubling measures, advancing understanding of measure regularity.

Findings

Singular set dimension is at most n-3 for such measures

Proves convergence of doubling ratios implies dimension bounds

Provides a framework for analyzing measure singularities

Abstract

In the following paper, we prove a dimension bound on the singular set of a Radon measure assuming its doubling ratio converges uniformly on compact sets. More precisely, we prove that if a Radon measure is $n$-Uniformly Asymptotically Doubling, then $\dim(\mathcal{S}_{\mu}) \leq n-3$, where $\mathcal{S}_{\mu}$ is the singular set of the measure.