Square packings and rectifiable doubling measures

Square packings and rectifiable doubling measures, Discrete Analysis 2025:3, 40 pp.

A measure $\mu$ on a metric space is $m$-rectifiable if it assigns full measure to a countable collection of Lipschitz images of bounded subsets of $\mathbb{R}^m$. Measures that are 1-rectifiable are a major object of study in geometric measure theory, and their structure is now well understood. Much less is known when $m > 1$. For example, a simple metric characterization of Lipschitz curves has been known since the 1920s, but no such result is available for higher-dimensional Lipschitz images.

This article undertakes a study of the $m>1$ case, with interesting and intriguing results. First, the authors obtain several sufficient conditions for a measure to be $m$-rectifiable for general $m$. Of particular interest is the criterion given in Theorem 2.5, with a general and robust construction of a Lipschitz map based on a geometric observation concerning the packing of axis-parallel cubes in $\mathbb{R}^n$.

The authors then use that criterion to prove their main result: given two integers $m,n$ such that $2\leq m<n$, they construct doubling measures on $\mathbb{R}^n$ with full support that are $m$-rectifiable but purely $(m-1)$-unrectifiable (Theorem 1.1). The doubling condition says that there exists a constant $D>0$, called a doubling constant, such that for all $r>0$ and for all $x$ in the support of $\mu$,
$$0<\mu(B(x,2r))\leq D\mu(B(x,r))<\infty,$$
where $B(x,r)$ is the ball of radius $r$ centred at $x$.

For $m=1$, this was already known. Specifically, Garnett, Killip, and Schul (2010) proved that for any $n\geq 2$ there exists a doubling measure $\mu$ on $\mathbb{R}^n$ and a rectifiable curve $\Gamma\subset\mathbb{R}^n$ such that $\mu(\Gamma)>0$. This is already a highly counterintuitive result: for example, it would be false if $\Gamma$ were required to be even slightly smoother, and the restriction of $\mu$ to $\Gamma$ must be singular with respect to the 1-dimensional Hausdorff measure on $\Gamma$.

The $m\geq 2$ case is much more difficult, requiring the new methods introduced here. Additionally, the authors are able to impose additional conditions on the dimensionality of $\mu$. For example, their result implies that there exist doubling measures $\mu$ on $\mathbb{R}^3$ supported on sets of Hausdorff dimension 0.0001 and
packing dimension 1.9999 that are 2-rectifiable and purely 1-unrectifiable. Both the results and the methods developed here mark significant progress in our understanding of $m$-rectifiable measures for general $m$, and are likely to inspire further work.