Geometry of measures in real dimensions via H\"older parameterizations

Abstract

We investigate the influence that $s$-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $\mathbb{R}^n$ when $s$ is a real number between $0$ and $n$. This topic in geometric measure theory has been extensively studied when $s$ is an integer. In this paper, we focus on the non-integer case, building upon a series of papers on $s$-sets by Mart\'in and Mattila from 1988 to 2000. When $0<s<1$, we prove that measures with almost everywhere positive lower density and finite upper density are carried by countably many bi-Lipschitz curves. When $1\leq s<n$, we identify conditions on the lower density that ensure the measure is either carried by or singular to $(1/s)$-H\"older curves. The latter results extend part of the recent work of Badger and Schul, which examined the case $s=1$ (Lipschitz curves) in depth. Of further interest, we introduce H\"older and bi-Lipschitz parameterization theorems for Euclidean sets with "small" Assouad dimension.