Square packings and rectifiable doubling measures

TL;DR

This paper constructs doubling measures on Euclidean and metric spaces that are rectifiable in certain dimensions but purely unrectifiable in others, using a novel square packing approach to analyze Lipschitz images.

Contribution

It introduces a new method based on square packing for constructing higher-dimensional Lipschitz images with specific measure and rectifiability properties, extending to general metric spaces.

Findings

Existence of doubling measures that are m-rectifiable and purely (m-1)-unrectifiable.

Construction of Lipschitz images with prescribed Hausdorff dimension and measure properties in Heisenberg group.

Every compact metric space with Assouad dimension less than m is a Lipschitz image of a subset of [0,1]^m.

Abstract

We prove that for all integers $2\leq m\leq d-1$, there exists doubling measures on $\mathbb{R}^d$ with full support that are $m$-rectifiable and purely $(m-1)$-unrectifiable in the sense of Federer (i.e. without assuming $\mu\ll\mathcal{H}^m$). The corresponding result for 1-rectifiable measures is originally due to Garnett, Killip, and Schul (2010). Our construction of higher-dimensional Lipschitz images is informed by a simple observation about square packing in the plane: $N$ axis-parallel squares of side length $s$ pack inside of a square of side length $\lceil N^{1/2}\rceil s$. The approach is robust and when combined with standard metric geometry techniques allows for constructions in complete Ahlfors regular metric spaces. One consequence of the main theorem is that for each $m\in\{2,3,4\}$ and $s<m$, there exist doubling measures $\mu$ on the Heisenberg group $\mathbb{H}^1$ and Lipschitz maps $f:E\subset\mathbb{R}^m\rightarrow\mathbb{H}^1$ such that $\mu\ll\mathcal{H}^{s-\epsilon}$ for all $\epsilon>0$, $f(E)$ has Hausdorff dimension $s$, and $\mu(f(E))>0$. This is striking, because $\mathcal{H}^m(f(E))=0$ for every Lipschitz map $f:E\subset\mathbb{R}^m\rightarrow\mathbb{H}^1$ by a theorem of Ambrosio and Kirchheim (2000). Another application of the square packing construction is that every compact metric space $\mathbb{X}$ of Assouad dimension strictly less than $m$ is a Lipschitz image of a compact set $E\subset[0,1]^m$. Of independent interest, we record the existence of doubling measures on complete Ahlfors regular metric spaces with prescribed lower and upper Hausdorff and packing dimensions.