Square packings and rectifiable doubling measures

TL;DR

This work constructs doubling measures on complete metric spaces that are $m$-rectifiable yet purely $(m-1)$-unrectifiable for $2\le m\le d-1$, without assuming $\mu\ll\mathcal{H}^m$. The authors develop a robust square-packing method to build higher-dimensional Lipschitz images from trees of sets, and combine it with generalized $b$-adic cubes (KRS-cubes) and quasi-Bernoulli measures to achieve precise control over both Hausdorff and packing dimensions. They prove the existence of such measures on complete Ahlfors regular spaces; in particular, for $\mathbb{X}=\mathbb{H}^1$ and $m\in\{2,3,4\}$, there are doubling measures charging Lipschitz images of $\mathbb{R}^m$ with almost maximal Hausdorff dimension, while $\mathcal{H}^m$-mass remains zero under Lipschitz maps to $\mathbb{H}^1$. An additional consequence is that any compact metric space with Assouad dimension less than $m$ is a Lipschitz image of a subset of $[0,1]^m$, and one can prescribe bounds on both lower/upper Hausdorff and packing dimensions for doubling measures. Overall, the paper advances the rectifiability-program by enabling higher-codimension Lipschitz images to carry doubling measures with finely tuned dimensional profiles, and introduces robust tools that extend to complete metric spaces beyond Euclidean settings.

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 $μ\ll\mathcal{H}^m$). 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 $μ$ on the Heisenberg group $\mathbb{H}^1$ and Lipschitz maps $f:E\subset\mathbb{R}^m\rightarrow\mathbb{H}^1$ such that $μ\ll\mathcal{H}^{s-ε}$ for all $ε>0$, $f(E)$ has Hausdorff dimension $s$, and $μ(f(E))>0$. 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.

Figures (5)

• Figure 1.1: The $3\times 3$ grid in the proof of Lemma \ref{['root-packing']} when $m=2$, $k=3$, and $s_0>s_1=s_2=s_3>s_4=s_5=s_6=s_7=s_8$.
• Figure 1.2: Left: squares with side lengths $s_0>s_1=s_2=s_3$. Right: squares with side lengths $s_0>s_1>s_2'>s_3'$. Both sets of squares fit inside of a square of side length $s=s_0+s_1$. Adjusting the side lengths so that $s_1\approx s_0$ and $s_2',s_3'\ll s_0$, the ratio of the total area of the four squares on the left to the total area of the corresponding squares on the right can be made arbitrarily close to 2. Thus, scaling the picture on the right, there are lists of squares with the same total area, but different optimal packings.
• Figure 2.1: Illustration of the domain of the Lipschitz map $f$ in the "square packing construction" with dimension $m=2$, tree depth $l=2$, and the maximal number of children of sets in $\mathcal{T}_0$ and $\mathcal{T}_1$ given by $N_0=16$ and $N_1=25$, respectively. The side length of each of yellow square is $D_1$ and the side length of each "block" of yellow squares is $(\lceil N_1^{1/2}\rceil -1)D_1$. The side length of each blue square is equal to the side length of a yellow block plus $D_0$. All together, the domain sits inside a square of side length $(\lceil N_0^{1/2}\rceil-1)D_0+\lceil N_0^{1/2}\rceil(\lceil N_1^{1/2}\rceil-1) D_1$.
• Figure 2.2: Modified construction of the Lipschitz map $f$ in Lemma \ref{['l:pack']} with dimension $m=2$, tree depth $l=2$, and maximal number of children of sets in $\mathcal{T}_0$ and $\mathcal{T}_1$ given by $N_0=N_1=5$. The side length of each of yellow square is $D_1$ and the side length of each "block" of yellow squares is $(2+\frac{1}{\sqrt{2}})D_1$. The side length of each blue square is equal to the side length of a yellow block plus $D_0$. All together, the domain sits inside a square of side length $(2+\frac{1}{\sqrt{2}})D_0+(2+\frac{1}{\sqrt{2}})^2 D_1$.
• Figure 5.1: On each metric cube $Q\in\Delta_k$ with $k\geq 0$, we try to force the quasi-Bernoulli measure $\mu_{\mathbf{s}}$ to look locally $s_{k+1}$-dimensional by distributing the mass of the cube to its children so that the central child receives the most mass of any child. The number of children of a cube can fluctuate throughout the space. To get a doubling measure, we choose the weight $\alpha$ of children in $\mathsf{Outer}(Q)$ independently of $Q$. The weight $\beta_Q$ of children in $\mathsf{Inner}(Q)\setminus\{Q^\downarrow\}$ and weight $\gamma_Q$ of the central child $Q^\downarrow$ depend on $\alpha$, $q$, $s_{k+1}$, $L_Q=\#\mathsf{Outer}(Q)$, and $M_Q=\#\mathsf{Inner}(Q)$. In the graphic, we illustrate mass distributions for two cubes $P,Q\in\Delta_k$ with $\mu_{\mathbf{s}}(P)=\mu_{\mathbf{s}}(Q)=1$. On the left $L_P=9$ and $M_P=7$, while on the right $L_Q=13$ and $M_Q=8$.

Theorems & Definitions (69)

• Theorem 1.1
• Example 1.2
• Conjecture 1.3
• Lemma 1.4
• proof
• Corollary 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Remark 1.9