Multiscale analysis of 1-rectifiable measures II: characterizations

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 $\mu$ in $n$-dimensional Euclidean space for all $n\geq 2$ in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an $L^2$ gauge the extent to which $\mu$ 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 $\mu$ and 1-dimensional Hausdorff measure. 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 $L^2$ variant of P. Jones' traveling salesman construction, which is of independent interest.


Introduction
A fundamental concept in geometric measure theory is a general notion of rectifiability of a set or a measure, which generalizes the classical notion of rectifiability of a curve. For sets, this notion of rectifiability is due to A.S. Besicovitch [Bes28]. For measures, this notion of rectifiability is due to A.P. Morse and J. Randolph [MR44] and H. Federer [Fed47,Fed69] (see Definition 1.1 below). Rectifiability has been extensively studied for sets and also for measures µ that satisfy an additional regularity assumption, which is often expressed in terms of finiteness µ-almost everywhere of the upper Hausdorff density D m (µ, ·) of the measure. This assumption is equivalent to an a priori relationship between the null sets of the measure µ and null sets of the m-dimensional Hausdorff measure H m , more specifically that µ vanishes on every set of H m measure zero. One reason that this regularity assumption is often imposed it that it allows one to replace the class of Lipschitz images of bounded subsets of R m appearing in Federer's definition of rectifiability of a measure with bi-Lipschitz images or Lipschitz graphs or C 1 graphs without changing the class of rectifiable measures. For arbitrary (or doubling) Radon measures, however, it is known by an example of Garnett, Killip, and Schul [GKS10] that the class of measures that are rectifiable with respect to Lipschitz images is strictly larger than the class of measures that are rectifiable with respect to bi-Lipschitz images. While (countable) rectifiability of a set or a measure is an inherently qualitative property, a quantitative counterpart of the theory of rectifiability was developed in the early 1990s by P. Jones [Jon90] and by G. David and S. Semmes [DS91,DS93]. One goal of theses investigations was to study the connection between rectifiability and singular integral operators. In David and Semmes' theory of uniformly rectifiable sets and measures, it is essential that the measures involved are Ahlfors regular, a strong form of regularity of a measure. The work in this paper addresses studying Federer's definition of rectifiability without imposing the standing regularity hypotheses of past investigations. We repurpose tools from the Jones-David-Semmes theory of quantitative rectifiability to characterize Federer 1-rectifiable measures using snapshots of a measure (beta numbers) at multiple scales. Moreover, we identify the 1-rectifiable and purely 1-unrectifiable parts of an arbitrary Radon measure µ in R n in terms of the pointwise behavior of the lower Hausdorff density D 1 (µ, x) and a weighted geometric square function J * p (µ, x), which records in an L p gauge the extent to which µ admits approximate tangent lines or has rapidly growing density ratios along its support. For the precise statement of these main results, see §2. The central reason we restrict ourselves to m-rectifiability with m = 1 is the special role that connectedness has for one-dimensional sets: every closed, connected set in R n of finite H 1 measure is a Lipschitz image of a closed interval. The current lack of a Lipschitz parameterization theorem for surfaces is the key obstruction to understanding Federer m-rectifiability when m ≥ 2. The innovation of this paper is to provide the first full treatment of rectifiability of arbitrary Radon measures, including measures which have infinite Hausdorff density or which are mutually singular with respect to Hausdorff measure.
The abstract proof of Proposition 1.2 given above does not provide a concrete method for identifying µ m rect and µ m pu for a given Radon measure µ. A fundamental problem in geometric measure theory is to provide (geometric, measure-theoretic) characterizations of µ m rect and µ m pu . When n = 2 and m = 1, this problem was first formulated and investigated by Besicovitch [Bes28,Bes38] for positive and finite measures µ of the form µ = H 1 E, E ⊆ R 2 , and later by Morse and Randolph [MR44] (resp. Moore [Moo50]) for Radon measures µ on R 2 (resp. R n , n ≥ 3) such that µ H 1 . Here and below H m denotes the m-dimensional Hausdorff measure on R n (see e.g. [Fed69] or [Mat95]), normalized to agree with the Lebesgue measure in R m . The condition µ H 1 , called absolute continuity, means that µ(E) = 0 for every Borel set E such that H 1 (E) = 0. In this paper, we provide characterizations of the 1-rectifiable part µ 1 rect and the purely 1-unrectifiable part µ 1 pu of an arbitrary Radon measure µ on R n (see §2). We emphasize that in contrast with previous works, our main result does not require an a priori relationship between µ and the 1-dimensional Hausdorff measure H 1 . A remarkable feature of the proof of our characterization is that we adapt techniques originating from the theory of quantitative rectifiability to study the qualitative rectifiability of measures. In fact, our identification of the rectifiable part of the measure µ is constructive in nature.
The proofs of Theorems 1.3 and 1.4 in arbitrary dimensions 1 ≤ m ≤ n − 1 have a distinguished place in the history of geometric measure theory. When n = 2 and m = 1, Theorem 1.3 is originally due to Besicovitch [Bes28] (also see [Bes38]) and Theorem 1.4 is originally due to Morse and Randolph [MR44]. An extension of the later result to n ≥ 3 was given by Moore [Moo50]. In the aforementioned works, Besicovitch also characterized rectifiability of H 1 E, E ⊆ R 2 , by existence H 1 -a.e. of approximate tangent lines to E; and also in terms of the H 1 measure of orthogonal projections (of subsets) of E onto lines. Characterizations of rectifiability of H m E, E ⊆ R n , in terms of approximate tangents and projections were extended to all dimensions 1 ≤ m ≤ n − 1 by Federer [Fed47]. The next case of Theorem 1.3 was settled by Marstrand [Mar61], who proved the density characterization of rectifiability when n = 3 and m = 2. A few years later, in [Mar64], Marstrand proved that if there exists a Radon measure µ on R n and a real number s > 0 such that lim r↓0 r −s µ(B(x, r)) exists and is positive and finite on a set of positive µ measure, then s is an integer. Mattila's proof of the general case of Theorem 1.3, which is based on Marstrand's approach, was published in [Mat75] nearly 50 years after the pioneering paper by Besicovitch.
To prove the general case of Theorem 1.4, Preiss [Pre87] had to give a careful analysis of the geometry of m-uniform measures µ on R n , i.e. Radon measures with the property that µ(B(x, r)) = ω m r m for all r > 0, for all x ∈ R n such that µ(B(x, r)) > 0 for all r > 0. Although it is obvious that the restriction H m L of m-dimensional Hausdorff measure to an m-plane L in R n is an example of an m-uniform measure, it is less clear if these are the only examples. Surprisingly, starting with m ≥ 3, there exist "non-flat" uniform measures such as H 3 {(x, t) ∈ R 3 × R : |x| 2 = t 2 }; see [Pre87] (also [KP87]). Preiss introduced several original ideas to study the geometry of non-flat uniform measures, including the notion of a tangent measure to a Radon measure. For an in-depth introduction, we refer the reader to the exposition of Preiss' theorem by De Lellis [DL08]. The classification of m-uniform measures in R n is as of yet incomplete, but some progress has recently been made by Tolsa [Tol15b] and Nimer [Nim15,Nim16].
The support spt µ of a Borel measure µ on a metric space X is the set of all x ∈ X such that µ(B(x, r)) > 0 for all r > 0. In [AM16], Azzam and Mourgoglou proved that positive lower density is a sufficient condition for a locally finite Borel measure to be 1-rectifiable under additional global assumptions on the measure and its support. A locally finite Borel measure µ on X is doubling if there is a constant C > 1 such that µ(B(x, 2r)) ≤ Cµ(B(x, r)) for all x ∈ spt µ and for all r > 0. Theorem 1.5 (Azzam and Mourgoglou [AM16]). Assume that µ is a doubling measure whose support is a connected metric space, and let E ⊆ spt µ be compact. Then µ E is 1-rectifiable if and only if D 1 (µ, x) > 0 for µ-a.e. x ∈ E.
It is important to emphasize that in Theorem 1.5, no assumption is made on the upper density of the measure. Thus, Theorem 1.5 characterizes a class of 1-rectifiable measures, which includes measures that are not absolutely continuous with respect to H 1 . Examples of 1-rectifiable doubling measures µ on R n with spt µ = R n and µ ⊥ H 1 , which satisfy the hypothesis of Theorem 1.5, were constructed by Garnett, Killip, and Schul [GKS10]. Interestingly, such examples give measure zero to every bi-Lipschitz image of R m in R n (in particular, they give measure zero to every Lipschitz graph), but nevertheless give full measure to a countable family of Lipschitz images of R. Following [BS15], it is also known that such measures have density D 1 (µ, x) = ∞ µ-a.e., with µ(B(x, r))/r → ∞ at a rapid rate as r → 0.
1.3. L p Jones beta numbers and rectifiability. Let 1 ≤ p < ∞ and let µ be a Radon measure on R n . For every bounded Borel set E ⊆ R n of positive diameter (typically, either a ball B(x, r) or a cube Q) and straight line in R n , we define β p (µ, E, ) ∈ [0, 1] by where we interpret β p (µ, E, ) = 0 if µ(E) = 0. We then define β p (µ, E) ∈ [0, 1] by where the infimum runs over all straight lines in R n . Note that β p (µ, E, ) and β p (µ, E) are increasing in the exponent p for all µ, E, and . Higher dimensional beta numbers β (m) p (µ, E) may be defined by letting range over all m-dimensional affine planes in R n instead of over all lines in R n .
In [Jon90], Peter Jones characterized subsets of finite length curves in the plane in terms of an 2 sum of a sup-norm variant of (1.6); see Definition 3.3 and Theorem 3.4 below. One goal was to bring these quantities into the study of singular integrals [Jon89]. Guy David and Stephen Semmes did this, and more, for curves and surfaces in their investigation of uniformly rectifiable sets and measures (e.g. see [DS91] and [DS93]). Much work has been done in connecting beta numbers and rectifiability (highlights include [Paj97], [Lég99], [Ler03], [Hah08], [LW11], [DT12], [BS15], [ADT16], [BS16], [NV15]), but we single out two recent papers, [Tol15a] and [AT15], and state one theorem, which is a combination of their main results. Theorem 1.6 (Tolsa [Tol15a], Azzam and Tolsa [AT15]). Let 1 ≤ m ≤ n − 1 and let µ be a Radon measure on R n . Then µ is m-rectifiable and µ H m if and only if The factor µ(B(x, r))/r m appearing in (1.8) translates between different conventions in the definition of beta numbers. In [Tol15a] and [AT15], the authors use the convention that integration in the definition of β (m) 2 (µ, B(x, r)) is against the measure r −m µ, whereas our convention is that integration is against the measure µ(B(x, r)) −1 µ. Note that A good way to access a more comprehensive survey about uniform rectifiability, its connection to singular integrals, and its connection to analytic capacity, is to read David and Semmes [DS93], Pajot [Paj02], and Tolsa [Tol14]. We also make some remarks in [BS15, §4].

Conventions.
We may write a b (or b a) to denote that a ≤ Cb for some absolute constant 0 < C < ∞ and write a ∼ b if a b and b a. Likewise we may write a t b (or b t a) to denote that a ≤ Cb for some constant 0 < C < ∞ that may depend on a list of parameters t and write a ∼ t b if a t b and b t a.
Below we use several grids of dyadic cubes. Unless stated otherwise, we take all dyadic cubes in R n to be half open, say of the form Q = j 1 2 k , j 1 + 1 2 k × · · · × j n 2 k , j n + 1 2 k , k, j 1 , . . . , j n ∈ Z.
The side length of Q, which we denote by side Q, is 2 −k ; the diameter of Q, which we denote by diam Q, is 2 −k √ n. Let ∆(R n ) denote the collection of all dyadic cubes in R n and let ∆ 1 (R n ) denote the collection of all dyadic cubes in R n of side length at most 1. For any cube Q and λ > 0, we let λQ denote the unique cube in R n that is obtained by dilating Q by a factor of λ with respect to the center of Q. Note that side λQ = λ side Q and diam λQ = λ diam Q for all cubes Q and for all λ > 0.

Main results and organization of the paper
In our main result, Theorem A, we characterize the 1-dimensional rectifiable and purely unrectifiable parts of arbitrary Radon measures in terms of the pointwise behavior of the lower density and a geometric square function to be defined below. Also, see Theorem E, where we characterize the rectifiable and purely unrectifiable parts of a pointwise doubling measure in terms of the pointwise behavior of a simpler geometric square function alone.
For any dyadic cube Q in R n , define the set ∆ * (Q) of nearby cubes to be the set of all dyadic cubes R such that The constant 1600 √ n in the definition of ∆ * (Q) is chosen to be large enough to invoke Proposition 3.6 in the proof of Lemma 5.3, but has not been optimized.
Let 1 ≤ p < ∞ and let µ be a Radon measure on R n . For all Q ∈ ∆(R n ), we define β * p (µ, Q) ∈ [0, 1] by where as usual the infimum runs over all straight lines in R n . Note that β * p (µ, Q) = 0 whenever µ(1600 √ n Q) = 0, and β * p (µ, Q) is increasing in p for all µ and Q. When p = 2, dµ(x). Define a density-normalized Jones function J * p (µ, x) associated to the numbers β * p (µ, Q) as follows. For every n ≥ 2, 1 ≤ p < ∞, and Radon measure µ on R n , define (In the definition, we use the convention 0/0 = 0 and 1/0 = ∞.) This is a variant of the density-normalized Jones function J 2 (µ, x) used in [BS15], which was associated to the beta numbers β 2 (µ, 3Q). Peter Jones conjectured circa 2000 that these types of weighted Jones functions could be used to characterize rectifiabilty of a measure (personal communication). In this paper's main result, Theorem A, we verify Jones' conjecture by using J * p (µ, x) to identify the 1-rectifiable and purely 1-unrectifiable parts of an arbitrary Radon measure.
Theorem A (characterization of the 1-rectifiable / purely 1-unrectifiable decomposition). Let n ≥ 2 and let 1 ≤ p ≤ 2. If µ is a Radon measure on R n , then the decomposition µ = µ 1 rect + µ 1 pu in (1.3) is given by Corollary B (characterization of 1-rectifiable measures). Let n ≥ 2 and let 1 ≤ p ≤ 2. If µ is a Radon measure on R n , then µ is 1-rectifiable if and only if D 1 (µ, x) > 0 and J * p (µ, x) < ∞ at µ-a.e. x ∈ R n . Corollary C (characterization of purely 1-unrectifiable measures). Let n ≥ 2 and let 1 ≤ p ≤ 2. If µ is a Radon measure on R n , then µ is purely 1-unrectifiable if and only if D 1 (µ, x) = 0 or J * p (µ, x) = ∞ at µ-a.e. x ∈ R n . The proof of Theorem A and its corollaries takes up § §4-6 below. A description of each of these sections appears at the end of this section. The restriction to exponents p ≥ 1 in Theorem A appears in the proof of Lemma 5.2; the restriction to p ≤ 2 appears in the proof of Proposition 4.4. It is an open problem to determine if the conclusion of the theorem holds in the range p > 2. The restriction to half open cubes (in the characteristic function χ Q ) in the definition of J * p (µ, x) is imposed so that in the proof of Theorem 5.1, we may use Lemma 5.6 (also see Remark 5.7).
The methods that we develop to prove Theorem A also yield a characterization of rectifiability of a measure with respect to a single rectifiable curve. Let 1 ≤ p < ∞ and let µ be a Radon measure on R n . For all Q ∈ ∆(R n ), we define β * * p (µ, Q) ∈ [0, 1] by (2.8) β * * p (µ, Q) = inf max {β p (µ, 3R, ) : R ∈ ∆ * (Q)} , where the infimum runs over all straight lines in R n .
Theorem D (Traveling salesman theorem for measures). Let n ≥ 2 and let 1 ≤ p < ∞. Let µ be a finite Borel measure on R n with bounded support. If Γ ⊆ R n is a rectifiable curve such that µ(R n \ Γ) = 0, then Conversely, if S * * p (µ) < ∞, then there is a rectifiable curve Γ such that µ(R n \ Γ) = 0 and (2.10) H 1 (Γ) n diam spt µ + S * * p (µ). Theorem D may be viewed as an extension of the Analyst's traveling salesman theorem (see §3), which characterizes subsets of rectifiable curves. A characterization of measures that are supported on a rectifiable curve was already known for Ahlfors regular measures (see [Ler03, Theorem 5.1]), but in this generality is new even for absolutely continuous measures of the form µ = H 1 E. For the proof of Theorem D, see §6.
For measures satisfying an additional weak regularity property, we also obtain simpler characterizations of the 1-rectifiable and purely 1-unrectifiable parts. Let µ be a Radon measure on R n and let 1 ≤ p < ∞. The density-normalized Jones function J p (µ, x) is defined by A Radon measure µ on R n is called pointwise doubling if The class of pointwise doubling measures includes the class of Radon measures µ on R n with 0 < D 1 (µ, x) ≤ D 1 (µ, x) < ∞ for µ-a.e. x ∈ R n , but is strictly larger.
Theorem E (characterization of the 1-rectifiable / purely 1-unrectifiable decomposition for pointwise doubling measures). Let n ≥ 2 and let 1 ≤ p ≤ 2. If µ is a pointwise doubling measure on R n , then the decomposition µ = µ 1 rect + µ 1 pu in (1.3) is given by See §7 for the proof of Theorem E. It is an open problem to decide if the conclusion of Theorem E holds for arbitrary Radon measures.
Remark 2.1 (Added in May 2016). Shortly after a second draft of this paper appeared on the arXiv in April 2016, the problem following Theorem E was answered in the negative by Martikainen and Orponen [MO16]. For all ε > 0, Martikainen and Orponen construct an example of a probability measure µ supported in the unit square in the plane for which (1) J 2 (µ, x) ≤ ε for all x ∈ spt µ, and (2) D 1 (µ, x) = 0 at µ-a.e. x ∈ R 2 .
(We caution the interested reader that [MO16] uses different notation for J 2 (µ, x).) Thus, the measure µ is purely 1-unrectifiable by Corollary C (or Lemma 4.2) despite having J 2 (µ, ·) uniformly bounded. This shows that 1-rectifiable or purely 1-unrectifiable Radon measures cannot be characterized in terms of pointwise control of the Jones function J 2 (µ, ·) alone. Moreover, let us note that since µ is a finite measure with bounded support, (1). This shows that in Theorem D, the numbers β * * 2 (µ, Q), which take into account how µ looks in cubes R nearby Q, cannot be replaced with the simpler numbers β 2 (µ, 3Q). For further discussion in this direction, see Remarks 4.6 and 5.8.

2.1.
Organization. In Section 3, we recall a metric characterization of rectifiable curves in R n as well as the Analyst's traveling salesman theorem, which characterizes subsets of rectifiable curves in R n in terms of a quadratic sum of Jones' beta numbers. Both are indispensable tools in the theory of 1-rectifiable sets and measures. At the end of the section, we state Proposition 3.6, which is a flexible extension of Jones' original traveling salesman construction that we use to draw rectifiable curves capturing positive measure in § §5 and 7.
The proofs of Theorems A and D are developed over § §4-6. In Section 4, we focus on proving necessary conditions for a Radon measure to be 1-rectifiable, or equivalently, sufficient conditions for a Radon measure to be purely 1-unrectifiable. In particular, we prove that if µ is a Radon measure and Γ is a rectifiable curve in R n , then J * 2 (µ, x) < ∞ at µ-a.e. x ∈ Γ (see Theorem 4.3). This result is some generalization and extension of the main result of the predecessor [BS15] of the current paper. In Section 5, we establish sufficient conditions, which guarantee that a Radon measure is 1-rectifiable. In fact, we introduce beta numbers β * ,c p (µ, Q), which are adapted to cubes R ∈ ∆ * (Q) such that µ(3R) ≥ c diam 3R, and prove that for every Radon measure µ in R n , is a density-normalized Jones function that is associated with the beta numbers β * ,c p (µ, Q) (see Theorem 5.1). The proof of our main result, Theorem A, as well as the proofs of Corollary B, Corollary C, and Theorem D are recorded in Section 6, using the results of Sections 4 and 5.
In Section 7, we show how to modify proofs in Section 5 in order to prove that for every Radon measure in R n , . Theorem E is then proved by combining this result with the main result of [BS15].
In the last two sections, § §8 and 9, we give a self-contained proof of Proposition 3.6, which is modeled on Jones' traveling salesman construction. The proof of the proposition gives an algorithm for drawing a rectifiable curve Γ through the leaves V = lim k→∞ V k of a "tree-like" sequence of 2 −k -separated sets V k . For example, the sets V k could be 2 −k -nets of points in a bounded set E ⊆ R n (as in the proof of the Analyst's traveling salesman theorem) or the sets V k could be µ centers of mass (of the triples) of dyadic cubes of side length 2 −k (as in the proof of Lemma 5. 3). An important technical difference between Jones' original construction and Proposition 3.6 is that the latter does not require V k+1 ⊇ V k . The added flexibility provided by Proposition 3.6 is crucial for the proofs of the sufficient conditions for 1-rectifiable measures, which we present in § §5 and 7.

The Analyst's traveling salesman theorem, again
As Lipschitz maps are continuous and do not increase Hausdorff measure by more than a constant multiple, every rectifiable curve Γ is a closed, connected set such that H 1 (Γ) < ∞. It is a remarkable fact-and an essential fact for the theory of 1-rectifiable sets and measures-that the converse of this observation is also true. For a proof of this fact that is valid in Hilbert space, see [Sch07, Lemma 3.7].
Corollary 3.2. If Γ 1 , Γ 2 , · · · ⊆ R n is a sequence of uniformly bounded, closed, connected sets, then there exists a compact, connected set Γ ⊆ R n and a subsequence ( It is known that the constant 32 in Corollary 3.2 may be replaced with 1 (for example, see [Fal86, Theorem 3.18]), but knowledge of the optimal constant will not be important for the development below. The constant 32 in Lemma 3.1 is not optimal and likely may be replaced with 2. However, once again, knowledge of the optimal constant is not crucial for the applications to follow.
Next, we recall the Analyst's traveling salesman theorem, which characterizes subsets of rectifiable curves in R n . The theorem was first conceived and proved by P. Jones [Jon90] for sets in the plane and then extended by Okikiolu [Oki92] for sets in R n , for all n ≥ 3. For a formulation of the theorem in infinite-dimensional Hilbert space, see Schul [Sch07].
Partial information is also known in the Heisenberg group; see Li and Schul [LS16,LS14] (as well as the previous work by Ferrari, Franchi, and Pajot [FFP07] and Juillet [Jui10]). For traveling salesman type theorems in graph inverse limit spaces, see G.C. David and Schul [DS16].
where ranges over all lines in R n . By convention, we set The cube dilation factor 3 appearing in Theorem 3.4 is somewhat arbitrary and may be replaced with any value strictly greater than 1. In particular, in §4, we need the "necessary" half of the Analyst's traveling salesman theorem with a dilation factor strictly greater than 3. For a derivation of Corollary 3.5 from Theorem 3.4, see [BS15,§2].
Corollary 3.5. For all n ≥ 2 and 3 < a < ∞, there is a constant C = C (n, a) ∈ (1, ∞) such that if E ⊆ R n is bounded and Γ is a connected set containing E, then The following proposition is modeled on and is some extension of a lemma from [JLS] (currently in preparation by P. Jones, G. Lerman, and the second author of this paper) and has roots in P. Jones' proof of the Analyst's traveling salesman theorem from [Jon90]. The variant in [JLS] is a criterion for constructing Lipschitz graphs, whereas Proposition 3.6 is a criterion for constructing rectifiable curves. For a related criterion for constructing bi-Lipschitz surfaces, see [DT12, Theorem 2.5]. One technical difference between Jones' original construction and Proposition 3.6 is that in the latter we do not assume V k+1 ⊇ V k . This added flexibility is crucial for our applications in sections 5 and 7 below.
k=0 be a sequence of nonempty finite subsets of B(x 0 , C r 0 ) such that Suppose that for all k ≥ 1 and for all v ∈ V k we are given a straight line k,v in R n and a number α k,v ≥ 0 such that Then the sets V k converge in the Hausdorff metric to a compact set V ⊆ B(x 0 , C r 0 ) and there exists a compact, connected set Γ ⊆ B(x 0 , C r 0 ) such that Γ ⊇ V and Remark 3.7. The "sufficient" half of the Analyst's traveling salesman theorem is an application of Proposition 3.6. To see this, suppose that E ⊆ R n is a bounded set Each cube Q ∈ ∆(R n ) can be associated to the pair (k, v) in this way for at most C(n) values of k ≥ 1 and v ∈ V k by (V I ). Thus, by Proposition 3.6, there exists a compact, connected set Γ ⊆ R n containing V := lim k→∞ V k = E with The proof of Proposition 3.6 is deferred to § §8 and 9, which are independent of § §4-7.    Our goal in the remainder of the section is to prove the following theorem. Theorem 4.3. Let n ≥ 2. If µ is a Radon measure on R n and Γ is a rectifiable curve, then the function J * 2 (µ, ·) ∈ L 1 (µ Γ) and J * 2 (µ, x) < ∞ at µ-a.e. x ∈ Γ.
At the core of Theorem 4.3 is the following quantitative statement, which is some extension and generalization of [BS15, Proposition 3.1]. In particular, let us stress that the lower Ahlfors regularity condition on E ⊆ Γ has been removed.
Proposition 4.4. Let n ≥ 2. If ν is a finite Borel measure on R n and Γ is a rectifiable curve, then Proof. The proof that we present is an adaptation of the proof of Proposition 3.1 in [BS15], the forerunner to this paper by the same name. For clarity, we develop the part of the proof that needs to be altered. Fix constants ε > 0 and a > 3 to be specified later, ultimately depending on only the ambient dimension n. Define two families ∆ Γ and ∆ 2 of dyadic cubes in R n , as follows.
Note that ∆ Γ and ∆ 2 consist of the cubes appearing in (4.1) for which either β Γ (aQ) or εβ * 2 (ν, Q) is the dominant quantity, respectively. It follows that (4.2) We shall estimate the terms I and II separately. The former will be controlled by H 1 (Γ) and the latter will be controlled by ν(R n \ Γ).
To estimate I, we note that by Jones' (when n = 2) and by Okikiolu's (when n ≥ 3) traveling salesman theorems (in the form of Corollary 3.5), where C is a finite constant determined by n and a.
In order to estimate II, first decompose R n \ Γ into a family T of Whitney cubes with the following specifications.
• The union over all sets in T is R n \ Γ.
• Each set in T is a half open cube in R n of the form [a 1 , b 1 ) × · · · × [a n , b n ).
(To obtain this decomposition, one can modify the standard Whitney decomposition in Stein's book [Ste70] by replacing each closed cube with the corresponding half open cube.) Here dist(T, Γ) = inf x∈T inf y∈Γ |x − y|. For each k ∈ Z, we define First we will estimate β * 2 (ν, Q) 2 diam Q for each Q ∈ ∆ 2 and then we will estimate II. Fix Q ∈ ∆ 2 , say with side Q = 2 −k 0 , and pick any line in R n such that We will estimate β * 2 (ν, Q) 2 from above using : By the triangle inequality and (4.4), we have Using this and the inequality (p + q) 2 ≤ 2p 2 + 2q 2 , it follows that Now, letting R range over ∆ * (Q) and declaring that ε be chosen so that (4/3)a 2 ε 2 = 1/3, we conclude that Note that if x ∈ F R for some R ∈ ∆ * (Q), then x ∈ Γ by (4.4) and 3R ⊆ aQ. Thus, we may employ the Whitney decomposition T of R n \ Γ to estimate the right hand side of (4.5): Because 3R ⊆ aQ for all R ∈ ∆ * (Q) by our choice of a above, it follows that Recall that side Q = 2 −k 0 . If T ∈ T k (aQ), then by bounding the distance between a point in T ∩ aQ and a point in Γ ∩ 1600 √ n Q, we observe that This estimate is valid for every cube Q ∈ ∆ 2 . We emphasize that equation II n ν(R n \ Γ).
Remark 4.5. The proof of Proposition 4.4 is robust in the sense that it does not overly rely on the specific geometry or combinatorics of sets in ∆ * (Q). For example, a version of the proposition holds if the triples 3R of cubes R ∈ ∆ * (Q) appearing the definition of β * 2 (µ, Q) are replaced with a family of balls that are nearby Q and whose diameters are comparable to the diameter of Q, provided that all relevant constants are chosen uniformly across Q ∈ ∆(R n ). We leave details to the interested reader.
Proof of Theorem 4.3. Let µ be a Radon measures on R n and let Γ be a rectifiable curve. Then Let K be the closure of the union of cubes which is compact since Γ is bounded. Then the restriction ν = µ K is finite and by Proposition 4.4. Chaining together inequalities in (4.8) and (4.9), we conclude that J * 2 (µ, ·) ∈ L 1 (µ Γ). Therefore, J * 2 (µ, x) < ∞ at µ-a.e. x ∈ Γ, as well. Remark 4.6. Recall from (2.8) that if µ is a Radon measure on R n and Q ∈ ∆(R n ), then where the infimum runs over all straight lines in R n . Define the density-normalized Jones function J * * 2 (µ, x) associated to the numbers β * * 2 (µ, Q) by In Theorem 4.3, we showed that if µ is a Radon measure and Γ is a rectifiable curve, then J * 2 (µ, x) < ∞ at µ-a.e. x ∈ Γ. However, it is currently an open problem to decide whether µ is a Radon measure and Γ is a rectifiable curve imply that J * * 2 (µ, x) < ∞ at µ-a.e. x ∈ Γ. For the motivation for this problem, see Remark 5.8. 5. Sufficiency: D 1 (µ, x) > 0 and J * p (µ, x) < ∞ µ-a.e. implies µ is 1-rectifiable Our goal in this section is to show that D 1 (µ, ·) > 0 almost everywhere and J * p (µ, ·) < ∞ almost everywhere are together a sufficient condition for a Radon measure µ on R n to be 1-rectifiable. As an intermediate step, we first introduce and work with beta numbers and weighted Jones functions that are adapted to cubes with uniformly large (coarse) density.
Lemma 5.2. Let n ≥ 2 and let 1 ≤ p < ∞. Let µ be a Radon measure on R n , let E be a Borel set of positive diameter such that 0 < µ(E) < ∞, and let denote the center of mass of E with respect to µ. For every straight line in R n , Proof. For every affine subspace in R n , the function dist(·, ) p is convex provided that 1 ≤ p < ∞. Thus, by Jensen's inequality.
Let us define a tree of dyadic cubes to be a set T of dyadic cubes with unique maximal element Q 0 (ordered by inclusion) such that if R ∈ T , then Q ∈ T for all dyadic cubes R ⊆ Q ⊆ Q 0 . Denote Q 0 by Top(T ). An infinite branch of T is defined to be a chain Q 0 ⊇ Q 1 ⊇ Q 2 ⊇ . . . of cubes in T such that side Q l = 2 −l side Q 0 for all l ≥ 0. We define the set Leaves(T ) of leaves of T to be The following lemma is the heart of Theorem 5.1.

Lemma 5.3 (drawing rectifiable curves through the leaves of lower Ahlfors regular trees).
Let n ≥ 2, let 1 ≤ p < ∞, and let c > 0. Let µ be a Radon measure on R n . If T is a tree of dyadic cubes such that then there exists a rectifiable curve Γ in R n such that Γ ⊇ Leaves(T ) and T ). Proof. Applying a dilation and a translation, we may assume without loss of generality that Top(T ) = [0, 1) n . By deleting irrelevant cubes from T , we may also assume without loss of generality that every cube Q ∈ T belongs to an infinite branch of T . To proceed, we will aim to use Proposition 3.6. Set parameters Below we will freely use the fact that diam 3Q = r 0 side Q for all Q ∈ T . For each Q ∈ T , let z 3Q denote the µ center of mass of 3Q, i.e.
For each k ≥ 0, let Z k = {z 3Q : Q ∈ T and side Q = 2 −k } and choose V k to be any maximal 2 −k r 0 -separated subset of Z k . Pick any x 0 ∈ 3Q 0 . Then Also note that V k satisfies condition (V I ) of Proposition 3.6 by the definition of V k .
To check condition (V II ), let k ≥ 0 and let v ∈ V k , say v = z 3Q for some Q ∈ T with side Q = 2 −k . Recall that by assumption every cube in T belongs to an infinite branch of T . Hence there exists R ∈ T such that R ⊆ Q and side R = 1 2 side Q. By maximality, there exists v = z 3P ∈ V k+1 for some P ∈ T such that side P = side R and Thus, condition (V II ) is satisfied.
To check condition (V III ), let k ≥ 1 and let v ∈ V k , say v = z 3Q for some Q ∈ T with side Q = 2 −k . Let R denote the parent of Q, which necessarily belongs to T . By maximality, there exists v = z 3P ∈ V k−1 for some P ∈ T such that side P = side R and Thus, condition (V III ) is satisfied. Now, for each k ≥ 0 and v ∈ V k , let Q k,v ∈ T denote a dyadic cube of side length 2 −k such that v = z 3Q k,v . Next, we will choose lines k,v in R n and numbers α k,v ≥ 0 to use with Proposition 3.6. Let k ≥ 1 and v ∈ V k . By definition of β * ,c p (µ, Q k,v ), we can choose a line k,v such that max{β p (µ, 3R, k,v ) : R ∈ ∆ * (Q) and µ(3R) ≥ c diam 3R} and 3Q j,x ⊆ 1600 √ nQ k,v (with room to spare). By Lemma 5.2 and (5.6), it follows that for any Therefore, the lines k,v and numbers α k,v satisfy (3.1). Furthermore, By Proposition 3.6, there exists a connected, compact set Γ ⊆ R n such that and Γ ⊇ V = lim k→∞ V k . By Lemma 3.1, Γ is a rectifiable curve. It remains to check that Γ ⊇ Leaves(T ). Let y ∈ Leaves(T ), say y = lim k→∞ y k for some sequence of points y k ∈ Q k , for some infinite branch Q 0 ⊇ Q 1 ⊇ Q 2 ⊇ . . . of T . Let z k = z 3Q k denote the center of mass of 3Q k and let v k ∈ V k be any point which minimizes the distance to z k . On one hand, |y k − z k | ≤ 2 −k r 0 = diam 3Q k , since y k , z k ∈ 3Q k . On the other hand, |z k − v k | ≤ 2 −k r 0 by maximality of V k in Z k . Thus, by the triangle inequality, whence y = lim k→∞ v k ∈ lim k→∞ V k ⊆ Γ. Therefore, since y ∈ Leaves(T ) was arbitrary, Γ ⊇ Leaves(T ).
The specialization to trees of lower Ahlfors regular cubes in the previous lemma can be avoided by making an assumption on the behavior of the measure in all nearby cubes. Recall from the introduction that β * * p (µ, Q) = inf max R∈∆ * (Q) β p (µ, 3R, ). Lemma 5.4 (drawing rectifiable curves through the leaves of a tree). Let n ≥ 2, let 1 ≤ p < ∞, and let µ be a Radon measure on R n . If T is a tree of dyadic cubes such that then there exists a rectifiable curve Γ in R n such that Γ ⊇ Leaves(T ) and Proof. The proof follows the same pattern as the proof of Lemma 5.3. However, instead of choosing k,v according to (5.6), one uses the definition of β * * p (µ, Q) to choose a line k,v such that (5.9) max{β p (µ, 3R, k,v ) : R ∈ ∆ * (Q)} ≤ 2 max β * * p (µ, Q k,v ). We leave the details to the reader.
Next, we state and prove a localization lemma for measure-normalized sums over trees of dyadic cubes, which is modeled on [BS16, Lemma 3.2]. Let T be a tree of dyadic cubes and let b : T → [0, ∞). For each Radon measure µ on R n , define the µ-normalized sum function S T ,b (µ, x) by where we interpret 0/0 = 0 and 1/0 = ∞.  Proof. Suppose that T , µ, b, N , ε, A, and A are given as above. If µ(A) = 0, then we may declare every dyadic cube Q ∈ T to be a bad cube and the conclusion of the lemma holds trivially. Thus, suppose that µ(A) > 0. Declare that a dyadic cube Q ∈ T is a bad cube if there exists a dyadic cube R ∈ T such that Q ⊆ R and µ(A ∩ R) ≤ εµ(A)µ(R). We call a dyadic cube Q ∈ T a good cube if Q is not a bad cube. Properties (1) and (2) are immediate. To check property (3), observe that where the last inequality follows because the maximal bad cubes are pairwise disjoint. Let us emphasize that this uses our assumption that T is composed of half open cubes. It follows that This verifies property (4).
In fact, note that x ∈ Top(T x ) ∩ Leaves(T x ) and the tree T x satisfies condition (5.3) of Lemma 5.3. Because each tree T x is determined by Q x and ∆ 1 (R n ) is countable, the collection {T x : D 1 (µ, x) > (3/2) √ n · c} of trees is enumerable, say it suffices to prove that the measure µ A y,N is 1-rectifiable for all y ∈ R n such that D 1 (µ, y) > (3/2) √ n · c and for all N < ∞, where A y,N := {x ∈ Top(T y ) : J * ,c p (µ, x) ≤ N }. Fix y ∈ R n such that D 1 (µ, y) > (3/2) √ n · c and fix N < ∞. Set η y := µ(Top(T y )). Given 0 < ε < η y , let T y,N,ε := Good(T y , N, ε) ⊆ T y denote the tree given by Lemma 5.6 applied with T = T y and b(Q) ≡ β * p (µ, Q) 2 diam Q (see Remark 5.7). Then T y,N,ε inherits property (5.3) from T y and, by Lemma 5.6, S * ,c p (µ, T y,N,ε ) < N ε and µ(A y,N ∩ Leaves(T y,N,ε )) ≥ (1 − εη y )µ(A y,N ). Thus, by Lemma 5.3, there exists a rectifiable curve Γ y,N,ε in R n such that Γ y,N,ε ⊇ Leaves(T y,N,ε ). In particular, Γ y,N,ε captures a large portion of the mass of A y,N : (Of course, Lemma 5.3 also gives a quantitative bound on the length of Γ y,N,ε depending only on n, c, N , ε, and diam Top(T y ), but we do not need it here.) To finish, choose 0 < ε k < η y for all k ≥ 1 so that lim k→∞ ε k = 0. Then Therefore, µ A y,N is 1-rectifiable. As noted above, this completes the proof.
Remark 5.8. By substituting Lemma 5.3 in the proof of Theorem 5.1 with Lemma 5.4, one can verify that if µ is a Radon measure on R n and 1 ≤ p < ∞, then the measure is the density-normalized Jones function associated with the numbers β * * p (µ, x). However, see Remark 4.6.
By uniqueness of the decomposition µ = µ 1 rect + µ 1 pu in Proposition 1.2, to show that µ 1 rect = µ R and µ 1 pu = µ P it suffices to prove that the measure µ R is 1-rectifiable and the measure µ P is purely 1-unrectifiable.
On one hand, since J * ,c p (µ, x) ≤ J * p (µ, x) for all x ∈ R n and for all c > 0, Therefore, µ R is 1-rectifiable by (6.1) and Theorem 5.1. On the other hand, because J * p (µ, x) is increasing in p and p ≤ 2, Therefore, µ P is purely 1-unrectifiable by Lemma 4.2 and Theorem 4.3.

Proof of Corollaries B and C.
A measure µ is 1-rectifiable if and only if µ 1 pu = 0, and a measure µ is purely 1-unrectifiable if and only if µ 1 rect = 0. Therefore, Corollary B and Corollary C follow immediately from Theorem A.
Proof of Theorem D. Let n ≥ 2 and let 1 ≤ p < ∞. Let µ be a finite Borel measure with bounded support. To prove the first statement, suppose that there exists a rectifiable curve Γ such that µ(R n \ Γ) = 0. Then spt µ ⊆ Γ, since Γ is closed. For every Q ∈ ∆(R n ), let Q be any line such that Then for every dyadic cube Q ∈ ∆(R n ) and nearby cube R ∈ ∆ * (Q), Hence, since 3R ⊆ 1600 √ nQ for all R ∈ ∆ * (Q), Therefore, by Corollary 3.5. The second statement is given by Corollary 5.5.

Variations for pointwise doubling measures
In the forerunner [BS15] to this paper, the authors gave a necessary condition for a Radon measure on R n to be 1-rectifiable using the L 2 density-normalized Jones function J 2 (µ, x) (see (2.11)).
Examining the proof of Theorem 7.1 in [BS15], one deduces that E J 2 (µ, x) dµ(x) < ∞ for every rectifiable curve Γ ⊆ R n and for every Borel set E ⊆ R n of the form E = {x ∈ Γ : µ(B(x, r)) ≥ cr for all 0 < r ≤ r 0 } for some c > 0 and r 0 > 0.
Thus, the proof of Theorem 7.1 yields the following stronger formulation of the theorem.
We now give a second application of Proposition 3.6 and the tools of §5, which in combination with Lemma 4.2 and Theorem 7.2, provides characterizations in terms of J p (µ, x) of 1-rectifiable and purely 1-unrectifiable pointwise doubling measures.
Let Q ↑ ∈ ∆(R n ) denote the parent of Q ∈ ∆(R n ). That is, let Q ↑ denote the unique dyadic cube such that Q ↑ ⊇ Q and side Q ↑ = 2 side Q.

Lemma 7.3 (drawing rectifiable curves through the leaves of uniformly doubling trees).
Let n ≥ 2, let 1 ≤ p < ∞, and let 0 < D < ∞. Let µ be a Radon measure on R n . If T is a tree of dyadic cubes such that then there exists a rectifiable curve Γ in R n such that Γ ⊇ Leaves(T ) and Proof. As in the proof of Lemma 5.3, we may assume without loss of generality that Top(T ) = [0, 1) n and every cube Q ∈ T belongs to an infinite branch of T . Set parameters C = 4 and r 0 = 3 √ n. For each Q ∈ T , let z 3Q = µ(3Q) −1 3Q z dµ(z) denote the center of mass of 3Q. For each k ≥ 0, let Z k = {z 3Q : Q ∈ T and side Q = 2 −k } and choose V k to be any maximal 2 −k r 0 -separated subset of Z k . Pick any x 0 ∈ 3Q 0 . Then The set V k satisfies condition (V I ) of Proposition 3.6 by definition. The set V k also satisfies conditions (V II ) and (V III ) of Proposition 3.6 (see the proof of Lemma 5.3). For each k ≥ 0 and v ∈ V k , choose a dyadic cube Q k,v ∈ T such that v = z 3Q k,v . Then, for each k ≥ 1 and v ∈ V k , choose a minimal dyadic cube Q k,v ∈ T such that Q k,v ⊇ Q k,v and such that 3 Q k,v contains 3Q j,v for every j ∈ {k − 1, k} and v ∈ V j ∩ B(v, 65C 2 −k r 0 ). Since 65C 2 −k r 0 = 780 √ n2 −k , we obtain (the overestimate) for all j ∈ {k − 1, k} and v ∈ V j ∩ B(v, 65C 2 −k r 0 ). Thus, by the doubling condition, We are ready to pick lines k,v and numbers α k,v ≥ 0 to use in Proposition 3.6. Let k ≥ 1 and let v ∈ V k . Choose k,v to be any straight line in R n such that Then, by (7.5) and (7.6), for all j ∈ {k − 1, k} and v ∈ V j ∩ B(v, 65C 2 −k r 0 ). Hence condition (3.1) of Proposition 3.6 holds by Lemma 5.2 and (7.7). Next, note that each cube Q ∈ T appears as Q k,v for at most C(n) pairs (k, v) by (7.5). Thus, condition (3.2) of Proposition 3.6 holds by (7.2). By Proposition 3.6, there exists a connected, compact set Γ ⊆ R n such that and Γ ⊇ V = lim k→∞ V k . By Lemma 3.1, Γ is a rectifiable curve. Finally, as in the proof of Lemma 5.3, Γ ⊇ Leaves(T ).
Theorem 7.4. Let µ be a Radon measure on R n and let 1 ≤ p < ∞. Then the measure Proof. Let µ be a Radon measure on R n and let 1 ≤ p < ∞. Write For every x ∈ spt µ such that Double(µ, x) < ∞, there exists an integer 1 ≤ D x < ∞ and r x > 0 such that µ(B(x, 2r)) ≤ 2 Dx µ(B(x, r)) for all 0 < r ≤ r x . Hence for every dyadic cube Q ∈ ∆(R n ) containing x such that 2 log 2 6 √ n −1 side Q ≤ r x . Thus, if Double(µ, x) < ∞, then x belongs to the leaves of the tree where Q x is defined to be the maximal dyadic cube containing x with side Q x ≤ min{r x /2 log 2 6 √ n −1 , 1}.
In fact, note that x ∈ Top(T x ) ∩ Leaves(T x ) and the tree T x satisfies condition (7.1) of Lemma 7.3. Because each tree T x is determined by Q x and D x , and ∆ 1 (R n ) and N are countable, the collection {T x : Double(µ, x) < ∞} of trees is enumerable, say for some sequence of points such that Double(µ, x i ) < ∞ for all i ≥ 1. Therefore, since it suffices to prove that the measure µ A y,N is 1-rectifiable for all y ∈ R n such that Double(µ, y) < ∞ and for all N < ∞, where A y,N := {x ∈ Top(T y ) : J p (µ, x) ≤ N }.
We now have all the necessary components to prove Theorem E.

Drawing rectifiable curves I: description of the curves and connectedness
The goal of this and the next section is to prove Proposition 3.6, which for the reader's convenience we now restate.
Proposition 8.1. Let n ≥ 2, let C > 1, let x 0 ∈ R n , and let r 0 > 0. Let (V k ) ∞ k=0 be a sequence of nonempty finite subsets of B(x 0 , C r 0 ) such that Suppose that for all k ≥ 1 and for all v ∈ V k we are given a straight line k,v in R n and a number α k,v ≥ 0 such that Then the sets V k converge in the Hausdorff metric to a compact set V ⊆ B(x 0 , C r 0 ) and there exists a compact, connected set Γ ⊆ B(x 0 , C r 0 ) such that Γ ⊇ V and By viewing ∞ k=0 V k as vertices of an abstract tree T , where each vertex v ∈ V k+1 is connected by an edge to a nearest vertex in V k , one may view Proposition 8.1 as a criterion for being able to draw a rectifiable curve Γ (i.e. a connected, compact set Γ with H 1 (Γ) < ∞) through the leaves V = lim k→∞ V k of T . Convergence of the sets V k in the Hausdorff metric is guaranteed by Lemma 8.2, whose proof we defer to Appendix A.
Lemma 8.2. Let B ⊆ R n be a bounded set and let V 0 , V 1 , . . . be a sequence of nonempty finite subsets of B. If the sequence satisfies (V III ) for some C > 0 and r 0 > 0, then V k converges in the Hausdorff metric to a compact set V ⊆ B.
The power of 2 in the quantity α 2 k,v in Proposition 8.1 is a consequence of the following application of the Pythagorean formula. For a proof of Lemma 8.3, see Appendix A.
Lemma 8.3. Suppose that V ⊆ R n is a 1-separated set with #V ≥ 2 and there exist lines 1 and 2 and a number 0 ≤ α ≤ 1/16 such that dist(v, i ) ≤ α for all v ∈ V and i = 1, 2.
Let π i denote the orthogonal projection onto i . There exist compatible identifications of If v 1 and v 2 are consecutive points in V relative to the ordering of π 1 (V ), then Moreover, In § §8.1-8.3 and §9, we prove Proposition 8.1 assuming Lemmas 8.2 and 8.3. To begin, in §8.1, we make some reductions and give a high level overview of the proof of the proposition. Next, in §8.2, we give a self-contained description of rectifiable curves Γ k that contain V k and converge in the Hausdorff metric to the curve Γ in the statement of the proposition. By construction, the sets Γ k are evidently closed. In §8.3, we verify that the sets Γ k are connected. In § §9.1-9.5 of the next section, we make detailed estimates on the length of Γ k , which yield the estimate (8.3) on the length of Γ. Finally, to complete the proof of Proposition 8.1, we supply proofs of Lemmas 8.2 and 8.3 in Appendix A.
8.1. Overview of the proof of Proposition 8.1. By scale invariance, it suffices to prove the proposition with r 0 = 1. Let n ≥ 2 and C > 1 be given, let x 0 ∈ R n , let r 0 = 1, and assume that V 0 , V 1 , V 2 , . . . is a sequence of nonempty finite sets in B(x 0 , C ) satisfying (V I ), (V II ), (V III ). By Lemma 8.2, there exists a compact set V ⊆ B(x 0 , C ) such that V k converges to V in the Hausdorff metric as k → ∞. Suppose that for all k ≥ 1 and v ∈ V k we are given a straight line k,v in R n and a number α k,v ≥ 0 satisfying (8.1) and (8.2). If #V k = 1 for infinitely many k, then V is a singleton and the conclusion is trivial. Thus, we shall assume that #V k ≥ 2 for all sufficiently large k. Let k 0 ≥ 0 be the least index such that #V k ≥ 2 for all k ≥ k 0 .
To complete the proof, we will construct a sequence Γ k 0 , Γ k 0 +1 , Γ k 0 +2 , . . . of closed, connected subsets of B(x 0 , C ) such that Γ k ⊇ V k and where C > 1 depends only on n and C . By Corollary 3.2, there exists a compact, connected set Γ and a subsequence (Γ k j ) ∞ j=1 of (Γ k ) ∞ k=k 0 such that Γ k j → Γ in the Hausdorff metric as j → ∞ and Γ satisfies (8.3) with r 0 = 1 and implicit constant 32C. We note In the argument that follows, the points in ∞ k=k 0 V k are called vertices. A vertex x ∈ V k is said to belong to generation k. Property (V I ) states that vertices of the same generation are uniformly separated. Property (V II ) ensures that every vertex is relatively close to some vertex of the next generation. And property (V III ) guarantees that every vertex of generation k ≥ k 0 + 1 is relatively close to some vertex of the previous generation. By associating each vertex to a nearest vertex of the previous generation, the set of all vertices may be viewed as a tree with #V k 0 roots. 8.2. Description of the curves. Each curve Γ k will be defined to be the union of finitely many closed line segments [v , v ] ("edges") between vertices v , v ∈ V k and closed sets B[j, w , w ] ("bridges") that connect vertices w , w ∈ V j for some k 0 ≤ j ≤ k and pass through vertices of generation j nearby w and w for every j > j. Bridges will be frozen in the sense that once a bridge appears in some Γ k , the bridge remains in Γ k for all k ≥ k.
The precise construction depends on a few auxiliary choices. First, choose a small parameter 0 < ε ≤ 1/32 so that the conclusions of Lemma 8.3 hold for α = 2ε. Second, for each generation k ≥ k 0 and vertex v ∈ V k , define an extension E[k, v] to vertices in future generations as follows: Given any v ∈ V k , pick a sequence of vertices v 1 , v 2 , . . . inductively so that v 1 is a vertex in V k+1 that is closest to v 0 = v, v 2 is a vertex in V k+2 that is closest to v 1 , and so on. Then define Once extensions have been chosen, for each generation k ≥ k 0 and for each pair of vertices v , v ∈ Γ k , we define the bridge B[k, v , v ] by We remark that in the special case To define the initial curve Γ k 0 , consider each pair of vertices Suppose that Γ k 0 , . . . , Γ k−1 have been defined for some k ≥ k 0 + 1. In order to define the next set Γ k , we first describe Γ k,v , the "new part" of Γ k nearby v, for every v ∈ V k . Then we declare Γ k to be the union of new parts and old bridges. That is, Let v be an arbitrary vertex in V k . The definition of Γ k,v splits into two cases.
Case I: Suppose that α k,v ≥ ε. To define Γ k,v , we mimic the construction of the initial curve Γ k 0 . Consider every pair of vertices This ends the description of Γ k,v in Case I.
Case II: Suppose that α k,v < ε. Identify the straight line k,v with R (in particular, pick directions "left" and "right") and let π k,v denote the orthogonal projection onto k,v . By Lemma 8.3 and , arranged from left to right according to the relative order of π k,v (v i ) in k,v (identified with R), where l, m ≥ 0. We start by describing the "right half" ). There will be three subcases. Starting from v 0 and working to the right, include each closed . Let t ≥ 0 denote the number of edges that were included in Γ R k,v .
Case II-NT: If t ≥ 1 (that is, at least one edge was included), then we say that the vertex v is not terminal to the right and are done describing Γ R k,v . Case II-T1 and Case II-T2: If t = 0 (that is, no edges were included), then we say that the vertex v is terminal to the right and continue our description of Γ R k,v , splitting into subcases depending on how Γ k−1 looks nearby v. Let w v be a vertex in V k−1 that is closest to v. Enumerate the vertices in V k−1 ∩ B(v, 65C 2 −k ) starting from w v and moving right (with respect to the identification of k,v with R) by ). There are two alternatives: In this case, we set Γ R k,v = B[k, v, v 1 ]. The first alternative defines Case II-T1. The second alternative defines Case II-T2. This concludes the description of Γ R k,v . Next, define the "left half" Also, define the terminology v is not terminal to the left and v is terminal to the left by analogy with the corresponding terminology to the right. Having separately defined both the "left half" Γ L k,v and the "right half" This concludes the description of Γ k,v in Case II.
8.3. Connectedness. By construction, for all k ≥ k 0 , every point x ∈ Γ k is connected to V k inside Γ k , because x belongs to an edge [v , v ] between vertices v , v ∈ V k or x belongs to a bridge B[j, u , u ] between vertices u , u ∈ V j for some k 0 ≤ j ≤ k. Thus, to prove that Γ k is connected, it suffices to prove that every pair of points in V k can be connected inside Γ k . We argue by double induction.
The set V k 0 is connected in Γ k 0 , because Γ k 0 contains [v , v ] or B[k 0 , v , v ] for every pair of vertices v , v ∈ V k 0 . In subsequent generations, if v , v ∈ V k and |v − v | < 30C 2 −k , then v and v are connected in Γ k . This can be seen by inspection of the various cases in the definition of Γ k,v . Suppose for induction that V k−1 is connected in Γ k−1 for some k ≥ k 0 + 1. Let x and y be arbitrary vertices in V k and let w x , w y ∈ V k−1 denote vertices that are closest to x and y, respectively. Because V k−1 is connected in Γ k−1 , w x and w y can be joined in Γ k−1 by a tour of p + 1 vertices in V k−1 , say where each pair w i , w i+1 of consecutive vertices is connected in Γ k−1 by an edge [w i , w i+1 ] or by a bridge B[j, u , u ] for some k 0 ≤ j ≤ k − 1 and u , u ∈ V j with the property that w i ∈ E[j, u ] and w i+1 ∈ E[j, u ].
Set v 0 = x, which satisfies |v 0 − w 0 | = |x − w x | < C 2 −k by (V III ). Suppose for induction that 0 ≤ t ≤ p − 1 and there exists a vertex v t ∈ V k such that |v t − w t | < C 2 −(k−1) and v 0 and v t are connected in Γ k . If t ≤ p − 2, choose v t+1 to be any vertex in V k such that |v t+1 − w t+1 | < C 2 −(k−1) , which exists by (V II ). Otherwise, if t = p − 1, set v t+1 = y, which also satisfies |v t+1 − w t+1 | = |y − w y | < C 2 −(k−1) by (V III ). We will now show that v t and v t+1 are connected in Γ k , and thus, v 0 and v t+1 are connected in Γ k . The proof splits into two cases, depending on whether the vertices w t and w t+1 in V k−1 are connected by a bridge or an edge.
First, suppose that w t , w t+1 ∈ B[j, u , u ] for some k 0 ≤ j ≤ k − 1 and some u , u ∈ V j with w t ∈ E[j, u ] and w t+1 ∈ E[j, u ]. Let z denote the unique point in V k ∩ E[j, u ] and let z denote the unique point in V k ∩ E[j, u ]. Then z , z ∈ B[j, u , u ]. Hence z and z are connected in Γ k , because B[j, u , u ] ⊆ Γ k . Next, by definition of the extensions, |z − w t | < C 2 −(k−1) and |z − w t+1 | < C 2 −(k−1) . Thus, and similarly, |v t+1 − z | < 30C 2 −k . It follows that v t is connected to z in Γ k and v t+1 is connected to z in Γ k . Therefore, concatenating paths, v t is connected to v t+1 in Γ k .
Secondly, suppose that [w t , w t+1 ] is an edge in Γ k−1 . Then |w t − w t+1 | < 30C 2 −(k−1) . Hence Because |v t − v t+1 | < 65C 2 −k , it follows that v t is connected to v t+1 in V k if α k,vt ≥ ε by Case I in the definition of Γ k,vt . On the other hand, suppose that α k,vt < ε. Then V k ∩ B(v t , 64C 2 −k ) may be arranged linearly according to their relative order under orthogonal projection onto k,vt . Label the vertices in V k ∩ B(v t , 64C 2 −k ) lying between v t and v t+1 inclusively, according to this order, say Figure 8.1). Thus, v t and v t+1 are connected in V k if α k,z i ≥ ε for some 1 ≤ i ≤ q by Case I in the definition of Γ k,z i . Finally, suppose that α k,z i < ε for all 1 ≤ i ≤ q. Because Γ k−1 contains the edge [w t , w t+1 ], the set Γ k,z i contains B[k, z i , z i+1 ] or [z i , z i+1 ] for each 0 ≤ i ≤ q − 1, depending on whether z i is terminal or z i is not terminal in the direction from z i to z i+1 . (In particular, in each instance alternative T1 does not occur.) Hence z i and z i+1 are connected in Γ k for all 0 ≤ i ≤ q − 1. Therefore, concatenating paths, we see that v t = z 0 and v t+1 = z q are connected in Γ k in this case, as well.
By induction, v 0 and v t are connected in V k for all 1 ≤ t ≤ p. In particular, x = v 0 and y = v p are connected in V k . Since x and y were arbitrary vertices in V k , it follows that V k is connected in Γ k . Therefore, by induction, Γ k is connected for all k ≥ k 0 .

Drawing rectifiable curves II: length estimates
We continue to adopt the notation and assumptions of § §8.1-8.3. Our goal in this section is to verify that Γ k 0 +1 , Γ k 0 +2 , . . . satisfy the estimate (8.6). Roughly speaking, we would like to bound the length of Γ k 0 by C2 −k 0 and to bound H 1 (Γ k ) by H 1 (Γ k−1 ) + C v∈V k α 2 k,v 2 −k for all k ≥ k 0 + 1, for some C independent of k. In other words, we want to "pay" for the length of the "new curve" Γ k with the length of "old curve" Γ k−1 and the sum C v∈V k α 2 k,v 2 −k . This plan works more or less, except that more work is required to pay for an edge [v , v ] in Γ k when the vertex v or v is close to a terminal vertex in Case II of the construction, because the old curve may not "span" the new edge [v , v ]. To handle this extra complication, we introduce a mechanism to "prepay" length called phantom length. The idea for phantom length comes from Jones' original traveling salesman construction (see [Jon90]). 9.1. Phantom length. Below it will be convenient to have notation to refer to the vertices appearing in a bridge. For each extension For all generations k ≥ k 0 and for all vertices v ∈ V k , we define the phantom length p k,v := 3C 2 −k . If B[k, v , v ] is a bridge between vertices v , v ∈ V k , then the totality p k,v ,v of phantom length associated to pairs in I[k, v , v ] is given by p k,v ,v := 3C 2 −k + 2 −(k+1) + · · · + 3C 2 −k + 2 −(k+1) + · · · = 12C 2 −k .
During the proof we will keep tally of phantom length at certain pairs (k, v) with v ∈ V k as an accounting tool.
We initialize Phantom(k 0 ), the index set of phantom length tracked at stage k 0 , to be the set of all pairs (j, u) such that the vertex u ∈ V j appears in the definition of Γ k 0 , including all vertices in V k 0 and all vertices in extensions in bridges in Γ k 0 . That is, Suppose that Phantom(k 0 ), . . . , Phantom(k − 1) have been defined for some k ≥ k 0 + 1, where the index set Phantom(k − 1) satisfies the following two properties.
Arrange V k−1 ∩ B(w, 30C 2 −(k−1) ) linearly with respect to the orthogonal projection π onto . If there is no vertex w ∈ V k−1 ∩ B(w, 30C 2 −(k−1) ) to the "left" of w or to the "right" of w, then (k − 1, w) ∈ Phantom(k − 1). That is, identifying with R, if (Note that Phantom(k 0 ) satisfies the terminal vertex property trivially, since Phantom(k 0 ) includes (k 0 , v) for every v ∈ V k 0 .) We form Phantom(k) starting from Phantom(k − 1), as follows. Initialize the set Phantom(k) to be equal to Phantom(k − 1). Next, delete all pairs (k − 1, w) and (k,ṽ) appearing in Phantom(k − 1) from Phantom(k). Lastly, for each vertex v ∈ V k , include additional pairs in Phantom(k) according to the following rules. , v] as a subset of Phantom(k). In particular, note that (k, v) is included in Phantom(k).
The phantom length associated to deleted pairs will be available to pay for the length of edges in Γ k near terminal vertices in V k and to pay for the phantom length of pairs in Phantom(k) \ Phantom(k − 1).
It is clear that Phantom(k) satisfies the bridge property. To check that Phantom(k) satisfies the terminal vertex property, let v ∈ V k and let be a line such that Identify with R and arrange V k ∩ B(v, 30C 2 −k ) linearly with respect to the orthogonal projection π onto . Assume that there is no vertex v ∈ V k ∩ B(v, 30C 2 −k ) to the "left" of v or to the "right" of v with respect the ordering under π . If α k,v ≥ ε, then (k,ṽ) was included in Phantom(k) for everyṽ ∈ V k ∩ B(v, 65C 2 −k ). In particular, (k, v) is in Phantom(k). Otherwise, if α k,v < ε, then V k ∩ B(v, 30C 2 −k ) is also linearly ordered with respect to the orthogonal projection onto k,v . By Lemma 8.3, the two orderings agree modulo the choice of orientation for and k,v . In this case, the assumption that there is no vertex v ∈ V k ∩ B(v, 30C 2 −k ) to the "left" of v or to the "right" of v translates to the statement that Γ L k,v or Γ R k,v is defined by Case II-T1 or Case II-T2, whence (k, v) was included in Phantom(k). Therefore, Phantom(k) satisfies the terminal vertex property.

Core of a bridge. For every bridge
That is, C[k, v , v ] is the interval of length 9/10 of the length of included in the construction, the corresponding core has significant length: For all k ≥ k 0 + 1, let Cores II (k) denote the set of cores C[k, v , v ] of bridges B[k, v , v ] between vertices v , v ∈ V k that were included in Γ k in Case II-T2 for some Γ R k,v or Γ L k,v . We claim that cores in ∞ j=k 0 +1 Cores II (j) are pairwise disjoint. To see this, suppose that C[k, v , v ] ∈ Cores II (k) and C[j, w , w ] ∈ Cores II (j) for some j ≥ k ≥ k 0 + 1. Because the bridges B[k, v , v ] and B[j, w , w ] arise in Case II-T2 of the construction, we have where the upper bounds follow from the bound Figure 9.1). Suppose to get a contradiction that C[k, v , v ] and C[j, w , w ] are distinct cores that intersect. Note that the intersection of the cores implies that the two end points w and w of B[j, w , w ] lie in opposite shaded regions in Figure 9.1. There are now several cases to consider, but we can reach a contradiction in each one. First, if j ≥ k + 2, then the intersection of C[k, v , v ] and C[j, w , w ] implies (by length considerations, see (9.2)) that w or w lies in the empty space of the figure, where no vertex exists. Next, if j = k + 1, then the intersection of the cores would imply that [v , v ] is included as an edge in Γ k , violating the bound |v − v | ≥ 30C 2 −k . Lastly, if j = k, then the intersection of the cores contradicts that fact that v is terminal in the direction from v to v and w (or w ) is terminal in the direction from w (w ) to w (w ). We leave the details to the reader.
where C depends on at most n and C . Since the cores in k 9.5. Proof of (9.4). Edges and bridges forming the curve Γ k and "new" phantom length associated to pairs in Phantom(k) \ Phantom(k − 1) may enter the local picture Γ k,v of Γ k near v for several vertices v ∈ V k , but only need to be accounted for once each to estimate the left hand side of (9.4). We prioritize as follows: bridge B[j, u , u ] included in Γ j for some k 0 ≤ j ≤ k − 1 and some u , u ∈ V j . Letṽ be the unique point in V k ∩ E[j, u ]. By the bridge property, (k,ṽ) ∈ Phantom(k − 1). Then Third Estimate (Case II-T2): Suppose that α k,v < ε and v is terminal to the right with alternative T2. (The case when v is terminal to the left is handled analogously.) Write v 1 ∈ V k and w v,r , w v,r+1 ∈ V k−1 for the vertices appearing in the definition of Γ R k,v . In this case, we will pay for p k,v,v 1 , the length of the bridge B[k, v, v 1 ], and the length of edges in Γ k ∩ B(v, 2C 2 −k ) and Γ k ∩ B(v 1 , 2C 2 −k ). Assume that α k,v 1 < ε (otherwise, everything is paid for by the first set of estimates). In §9.4, we noted that Because |v − w v,r | < 2C 2 −k and |v 1 − w v,r+1 | < 2C 2 −k , it follows that The totality p k,v,v 1 of phantom length associated to vertices in B[k, v, v 1 ] is 12C 2 −k . Finally, since α k,v < ε and α k,v 1 < ε, the total length of (parts of) edges in does not exceed 5C 2 −k by Lemma 8.3 since (1 + 3ε 2 )2 < 2.5. Altogether, where [w v,r , w v,r+1 ] ∈ Edges(k − 1) and C[k, v, v 1 ] ∈ Cores II (k). Fourth Estimate (Case II-NT): Let [v , v ] be an edge between vertices v , v ∈ V k , which is not yet wholly paid for. Then α k,v < ε and α k,v < ε. Let [u , u ] be the largest closed subinterval of [v , v ] such that u and u lie at distance at least 2C 2 −k away from Case II-T1 and Case II-T2 terminal vertices (see Figure 9.2). By Lemma 8.3, Without loss of generality, suppose that u lies to the left of u relative to the order of their projection onto k,v . Let z denote the first vertex in V k ∩ B(v , 65C 2 −k ) to the left of u (relative to the order of their projection onto k,v ) such that π k,v (z ) < π k,v (u ) − C 2 −k .
Let z denote the first vertex in V k ∩ B(v , 65C 2 −k ) to the right of u (relative to the order of their projection onto k,v ) such that π k,v (u ) + C 2 −k < π k,v (z ). The vertices z and z exist, because u and u stay a distance of at least 2C 2 −k away from Case II-T1 and Case II-T2 terminal vertices and α k,v < ε. By (V III ), we can find w , w ∈ V k−1 such that |w − z | < C 2 −k and |w − z | < C 2 −k . It follows that π k,v (w ) < π k,v (u ) < π k,v (u ) < π k,v (w ).
All that remains is to estimate the length of overlaps of sets of the form S k,v [u , u ] := E k−1 (v ) ∩ π −1 k,v ([π k,v (u ), π k,v (u )]). Since S k,v [u , u ] ⊆ S k,v [v , v ], it clearly suffices to estimate the length of the overlaps of sets S k,v [v , v ]. Suppose that v, v , v are consecutive vertices in V k ∩ B(v , 65C 2 −k ) such that v, v , and v lie at distance at least 65C 2 −k from Case I vertices and distance at least 2C 2 −k from Case II-T1 and Case II-T2 terminal vertices. We will show that (9.5) To start, let 1 and 2 be lines chosen so that 1 is parallel to k,v , 2 is parallel to k,v , and 1 and 2 pass through v . Note that, by the triangle inequality, (9.6) dist(x, i ) ≤ 2α2 −k for all x ∈ (V k−1 ∪ V k ) ∩ B(v , 65C 2 −k ) and i ∈ {1, 2}.
Let π i denote the orthogonal projection onto i and let N i denote the closed tubular neighborhood of i of radius 2α2 −k . Also let E k−1 (v, v ) := E k−1 (v) ∩ E k−1 (v ). By (9.6), (9.7) Figure 9.3. Inside the 2-plane containing the lines 1 and 2 , the diamondshaped region S is the union of two congruent triangles. The length of the red shadow π 1 (S) is no greater than 20α 2 2 −k .
Appendix A. Proof of Lemmas 8.2 and 8.3 The proof of Lemma 8.2 uses elementary properties of excess and Hausdorff distance; for a comprehensive reference, we recommend the monograph [Bee93] by Beer. Recall that for nonempty sets S, T ⊆ R n , the excess ex(S, T ) of S over T is defined by The excess satisfies the triangle inequality in the sense that ex(S, T ) ≤ ex(S, U )+ex(U, T ) for all nonempty S, T, U ⊆ R n . The set of nonempty compact subsets of R n equipped with the Hausdorff distance is a metric space. Thus, when restricted to the nonempty compact sets, we may refer to the Hausdorff distance as the Hausdorff metric.
Thus, by the triangle inequality for excess, ex(V k , V ) ≤ ex(V k , V k j ) + ex(V k j , V ) < C 2 −k j r 0 + HD(V k j , V ) and Therefore, HD(V k , V ) < C 2 −k j r 0 + max HD(V k j , V ), HD(V, V k j+1 ) whenever k j < k < k j+1 . We conclude that the whole sequence V k converges to V in the Hausdorff metric as k → ∞.
The proof of Lemma 8.3 that we give uses the area formula for Lipschitz graphs; for a nice presentation, see §3.3 of the book [EG15] by Evans and Gariepy.
Proof of Lemma 8.3. Let V ⊆ R n be a 1-separated set with at least two points. Assume that there exist straight lines 1 and 2 in R n and a number 0 ≤ α ≤ 1/16 such that dist(v, i ) ≤ α for all v ∈ V and i = 1, 2.
Let π i denote the orthogonal projection onto i . Let π ⊥ i denote the orthogonal projection onto an orthogonal complement of i . For any distinct pair of points v 1 , v 2 ∈ V , because V is 1-separated and the distance of points in V to i is bounded by α. Hence It follows that V belongs to the graph of a piecewise linear function g i : i → ⊥ i such that By the area formula for Lipschitz maps, for any line segment [u 1 , u 2 ] in the graph of g i , 1 + |∇g i | 2 dH 1 .