We give the following characterization of rectifiable metric spaces. A metric space with positive lower Hausdorff density is rectifiable if and only if, for any subset F and f:F→ Y, a Lipschitz map into a metric space with positive measure image (of the same dimension), there exists a positive measure subset A⊂ F so that f is biLipschitz on A. We also give a characterization in terms of a full biLipschitz decomposition. These characterizations are new even for subsets of Euclidean space. One of our tools is Alberti representations. On the way we give a method for constructing independent Alberti representations, which may be of independent interest. We use this to characterize unrectifiable metric spaces as those spaces for which there exist a positive measure subset S and a Lipschitz map ϕ into a lower dimensional Euclidean space so that S is ^1-null with respect to all curve fragments that are quantitatively transversal to ϕ.

Alberti, Csörnyei and Preiss introduced a notion of a "pointwise (weak) tangent field" for a subset of Euclidean space – a field that contains almost every tangent line of every curve passing through the set – and showed that all area-zero sets in the plane admit one-dimensional tangent fields. We extend their results in two distinct directions. First, a special case of our pointwise result shows that each doubling subset of Hilbert space admits a pointwise tangent field in this sense, with dimension bounded by the Nagata (or Assouad) dimension of the set. Second, inspired by the Analyst's Traveling Salesman Theorem of Jones, we introduce new, "coarse" notions of tangent field for subsets of Hilbert space, which take into account both large and small scale structure. We show that doubling subsets of Hilbert space admit such coarse tangent fields, again with dimension bounded by the Nagata (or Assouad) dimension of the set. For porous sets in the plane, this result can be viewed as a quantitative version of the Alberti–Csörnyei–Preiss result, though our results hold in all (even infinite) dimensions.

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 $s0$, $f(E)$ 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.

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 $2\leq m0$, 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.

A well-known open problem asks whether every bi-Lipschitz homeomorphism of ℝ^d factors as a composition of mappings of small distortion. We show that every bi-Lipschitz embedding of the unit cube [0,1]^d into ℝ^d factors into finitely many global bi-Lipschitz mappings of small distortion, outside of an exceptional set of arbitrarily small Lebesgue measure, which cannot in general be removed. Our main tool is a corona-type decomposition theorem for bi-Lipschitz mappings. As corollaries, we obtain a related factorization result for bi-Lipschitz homeomorphisms of the d-sphere, and we show that bi-Lipschitz embeddings of the unit d-cube in ℝ^d can be approximated by global piecewise affine homeomorphisms outside of a small set.

We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. We prove that it is always possible to decompose the domain of such a mapping into pieces on which the mapping “behaves like a projection mapping” along with a “garbage set” that is arbitrarily small in an appropriate sense. Moreover, our control is quantitative, i.e., independent of both the particular mapping and the metric space it maps into. This improves a theorem of Azzam-Schul from the paper “Hard Sard”, and answers a question left open in that paper. The proof uses ideas of quantitative differentiation, as well as a detailed study of how to supplement Lipschitz mappings by additional coordinates to form bi-Lipschitz mappings.

In their 1991 and 1993 foundational monographs, David and Semmes characterized uniform rectifiability for subsets of Euclidean space in a multitude of geometric and analytic ways. The fundamental geometric conditions can be naturally stated in any metric space and it has long been a question of how these concepts are related in this general setting. In this paper we prove their equivalence. Namely, we show the equivalence of Big Pieces of Lipschitz Images, Bi-lateral Weak Geometric Lemma and Corona Decomposition in any Ahlfors regular metric space. Loosely speaking, this gives a quantitative equivalence between having Lipschitz charts and approximations by nicer spaces. En route, we also study Reifenberg parameterizations.

We show that if an Ahlfors–David regular set E $E$ of dimension k $k$ has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by BP ( BP ( LG ) ) $\mathop \mathrm{BP}(\mathop \mathrm{BP}(\mathop \mathrm{LG}))$ ), then E ⊂ E ∼ $E\subset \tilde{E}$ where E ∼ $\tilde{E}$ is Ahlfors–David regular of dimension k $k$ and has Big Pieces of Lipschitz Graphs (denoted usually by BP ( LG ) ) $\mathop \mathrm{BP}(\mathop \mathrm{LG}))$ . Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors–David regular sets F ${\mathcal {F}}$ replacing LG $\mathop \mathrm{LG}$ . A simple corollary is the stability of the BP operator after two iterations. This was previously only known in the Euclidean setting for the case F = LG ${\mathcal {F}}= \mathop \mathrm{LG}$ with substantially more complicated proofs.

We give a simple quantitative condition, involving the "mapping content" of Azzam–Schul, that implies that a Lipschitz map from a Euclidean space to a metric space must be close to factoring through a tree. Using results of Azzam–Schul and the present authors, this gives simple checkable conditions for a Lipschitz map to have a large piece of its domain on which it behaves like an orthogonal projection. The proof involves new lower bounds and continuity statements for mapping content, and relies on a "qualitative" version of the main theorem recently proven by Esmayli–Hajłasz.

We show that if an Ahlfors–David regular set E $E$ of dimension k $k$ has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by BP ( BP ( LG ) ) $\mathop \mathrm{BP}(\mathop \mathrm{BP}(\mathop \mathrm{LG}))$ ), then E ⊂ E ∼ $E\subset \tilde{E}$ where E ∼ $\tilde{E}$ is Ahlfors–David regular of dimension k $k$ and has Big Pieces of Lipschitz Graphs (denoted usually by BP ( LG ) ) $\mathop \mathrm{BP}(\mathop \mathrm{LG}))$ . Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors–David regular sets F ${\mathcal {F}}$ replacing LG $\mathop \mathrm{LG}$ . A simple corollary is the stability of the BP operator after two iterations. This was previously only known in the Euclidean setting for the case F = LG ${\mathcal {F}}= \mathop \mathrm{LG}$ with substantially more complicated proofs.

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $\beta$-numbers, numbers measuring flatness in a given scale and location. This work was generalized to R^n by Okikiolu, to Hilbert space by the second author, and has many variants in a variety of metric settings. Notably, in 2005, Hahlomaa gave a sufficient condition for a subset of a metric space to be contained in a rectifiable curve. We prove the sharpest possible converse to Hahlomaa's theorem for doubling curves, and then deduce some corollaries for subsets of metric and Banach spaces, as well as the Heisenberg group.}

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $\beta$-numbers. These $\beta$-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones' result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones' $\beta$ numbers. In this paper we give a version of P. Jones' theorem for sets of arbitrary (integer) dimension lying in Euclidean space. We estimate the $d$-dimensional Hausdorff measure of a set in terms of an analogous sum of $\beta$-type numbers. There is no assumption of Ahlfors regularity, but rather, only of a lower bound on the Hausdorff content. We adapt David and Semmes' version of Jones' $\beta$-numbers by redefining them using a Choquet integral. A key tool in the proof is G. David and T. Toro's parametrization of Reifenberg flat sets (with holes).

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between μ and 1-dimensional Hausdorff measure H1. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L2 variant of P. Jones’ traveling salesman construction, which is of independent interest.

We prove a version of Peter Jones' analyst's traveling salesman theorem in a class of highly non-Euclidean metric spaces introduced by Laakso and generalized by Cheeger Kleiner. These spaces are constructed as inverse limits of metric graphs, and include examples which are doubling and have a Poincare inequality. We show that a set in one of these spaces is contained in a rectifiable curve if and only if it is quantitatively "flat" at most locations and scales, where flatness is measured with respect to so-called monotone geodesics. This provides a first examination of quantitative rectifiability within these spaces.

We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot-Carathéodory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones's geometric lemma. Our estimates improve on those of \cite{FFP}, by replacing the power $2$ of the Jones-$\beta$-number with any power $r<4$. This complements (in an open ended way) our work \cite{Li-Schul-beta-leq-length}, where we showed that such an estimate was necessary, but with $r=4$.

We show that if a subset K in the Heisenberg group (endowed with the Carnot-Caratheodory metric) is contained in a rectifiable curve, then it satisfies a modified analogue of Peter Jones's geometric lemma. This is a quantitative version of the statement that a finite length curve has a tangent at almost every point. This condition complements that of a work by Ferrari, Franchi, and Pajot (2007) except a power 2 is changed to a power 4. Two key tools that we use in the proof are a geometric martingale argument like that of Schul (2007) as well as a new curvature inequality in the Heisenberg group.

We identify two sufficient conditions for locally finite Borel measures on $\mathbb{R}^n$ to give full mass to a countable family of Lipschitz images of $\mathbb{R}^m$. The first condition, extending a prior result of Pajot, is a sufficient test in terms of $L^p$ affine approximability for a locally finite Borel measure $\mu$ on $\mathbb{R}^n$ satisfying the global regularity hypothesis $$\limsup_{r\downarrow 0} \mu(B(x,r))/r^m <\infty\quad \text{at $\mu$-a.e. $x\in\mathbb{R}^n$}$$ to be $m$-rectifiable in the sense above. The second condition is an assumption on the growth rate of the 1-density that ensures a locally finite Borel measure $\mu$ on $\mathbb{R}^n$ with $$\lim_{r\downarrow 0} \mu(B(x,r))/r=\infty\quad\text{at $\mu$-a.e. $x\in\mathbb{R}^n$}$$ is 1-rectifiable.

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in \(\mathbb {R}^n\), \(n\ge 2\). To each locally finite Borel measure \(\mu \), we associate a function \(\widetilde{J}_2(\mu ,x)\) which uses a weighted sum to record how closely the mass of \(\mu \) is concentrated near a line in the triples of dyadic cubes containing \(x\). We show that \(\widetilde{J}_2(\mu ,\cdot )<\infty \ \mu \)-a.e. is a necessary condition for \(\mu \) to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result is that no assumption is made on the upper Hausdorff density of the measure. Thus we are able to analyze general 1-rectifiable measures, including measures which are singular with respect to 1-dimensional Hausdorff measure.

The purpose of this note is to point out a simple consequence of some earlier work of the authors, Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps. For $f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $f$ is approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994. Our result is in the form of a Carleson-type estimate.

For a given connected set G in d-dimensional Euclidean space, we construct a connected set (Gamma) over tilde. G such that the two sets have comparable Hausdorff length, and the set (Gamma) over tilde has the property that it is quasiconvex, that is, any two points x and y in (Gamma) over tilde can be connected via a path, all of which is in (Gamma) over tilde, which has length bounded by a fixed constant multiple of the Euclidean distance between x and y. Thus, for any set K in d-dimensional Euclidean space, we have a set (Gamma) over tilde as above such that (Gamma) over tilde has comparable Hausdorff length to a shortest connected set containing K. Constants appearing here depend only on the ambient dimension d. In the case where G is Reifenberg flat, our constants are also independent of the dimension d, and in this case, our theorem holds for G in an infinite-dimensional Hilbert space. This work is closely related to k-spanners, which appears in computer science.