Singular Sets of Uniformly Asymptotically Doubling Measures

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.