Cones, rectifiability, and singular integral operators

Let $\mu$ be a Radon measure on $\mathbb{R}^d$. We define and study conical energies $\mathcal{E}_{\mu,p}(x,V,\alpha)$, which quantify the portion of $\mu$ lying in the cone with vertex $x\in\mathbb{R}^d$, direction $V\in G(d,d-n)$, and aperture $\alpha\in (0,1)$. We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that $\mu$ has polynomial growth, we give a sufficient condition for $L^2(\mu)$-boundedness of singular integral operators with smooth odd kernels of convolution type.


Introduction
Let m < d be positive integers. Given an m-plane V ∈ G(d, m), a point x ∈ R d , and α ∈ (0, 1), we define That is, K(x, V, α) is an open cone centered at x, with direction V , and aperture α.
Let 0 < n < d. It is well-known that if a set E ⊂ R d satisfies for some V ∈ G(d, d−n), α ∈ (0, 1), the condition x ∈ E ⇒ E ∩ K(x, V, α) = ∅, (1.1) then E is contained in some n-dimensional Lipschitz graph Γ, and Lip(Γ) ≤ 1 α , see e.g. To what extent can we weaken the condition (1.1) and still get meaningful information about the geometry of E? It depends on what we mean by "meaningful information", naturally. One could ask for the rectifiability of E, or if E contains big pieces of Lipschitz graphs, or whether nice singular integral operators are bounded on L 2 (E). The aim of this paper is to answer these three questions.
Given a Radon measure µ on R d and x ∈ supp µ, the lower and upper densities of µ at x are defined as Θ n * (µ, x) = lim inf r→0 µ(B(x, r)) r n and Θ n, * (µ, x) = lim sup r→0 µ(B(x, r)) r n .
Theorem 1.2 ([Fed47, Theorem 9.1]). Let µ be finite Radon measure on R d satisfying 0 < Θ n, * (µ, x) < ∞ for µ-a.e. x ∈ R d . Then, the following are equivalent: a) µ is n-rectifiable, b) for µ-a.e. x ∈ R d there is a unique approximate tangent plane to µ at x, c) for µ-a.e. x ∈ R d there is W x ∈ G(d, n) and α x ∈ (0, 1) such that where ε(n) is a small dimensional constant.
The results we prove in this paper are of similar nature. More precisely, we introduce and study conical energies. Definition 1.3. Suppose µ is a Radon measure on R d , and x ∈ supp µ. Let V ∈ G(d, d − n), α ∈ (0, 1), 1 ≤ p < ∞ and R > 0. We define the (V, α, p)-conical energy of µ at x up to scale R as For E ⊂ R d we set also E E,p (x, V, α, R) = E H n | E ,p (x, V, α, R).
Note that the definition above depends on the dimension parameter n, so it would be more precise to say that E µ,p (x, V, α, R) is the n-dimensional (V, α, p)-conical energy. For the sake of brevity, throughout the paper we will consider n to be fixed, and we will usually not point out this dependence. The same applies to other definitions.
We are ready to state our first result.

1.4)
Then, µ is n-rectifiable. Conversely, if µ is n-rectifiable, then for µ-a.e. x ∈ R d there exists V x ∈ G(d, d − n) such that for all α ∈ (0, 1) we have E µ,p (x, V x , α, 1) < ∞. (1.5) Remark 1.5. The "necessary" part of Theorem 1.4 improves on Theorem 1.2 in the following way. Existence of approximate tangents means that the conical density simply converges to 0, while (1.5) means that the conical density satisfies a Dini-type condition, and converges to 0 rather fast.
Remark 1.6. Concerning the "sufficient" part of Theorem 1.4: clearly, condition (1.3) is weaker than (1.4). However, Theorem 1.4 has the following advantage over Theorem 1.2: we only require Θ n, * (µ, x) > 0 and Θ n * (µ, x) < ∞ for our criterion to hold. In particular, we do not assume µ ≪ H n . It is not clear to the author how to show a criterion involving (1.3) or (1.2) without assuming a priori µ ≪ H n .
Question 1.7. Suppose µ is a Radon measure on R d satisfying Θ n, * (µ, x) > 0 and Θ n * (µ, x) < ∞ for µ-a.e. x ∈ R d . Assume that for µ-a.e. x ∈ R d there is an approximate tangent plane to µ at x. Does this imply that µ is n-rectifiable?
Let us mention that in recent years many similar characterizations of rectifiable measures have been obtained. By "similar" we mean: the pointwise finiteness of a square function involving some flatness quantifying coefficients. The most famous coefficients of this type are β numbers, first introduced in [Jon90] and further developed by David and Semmes [DS91,DS93a]. A necessary condition for rectifiability that uses β p numbers was shown in [Tol15], see Theorem 9.3 for the precise statement. Its sufficiency (under various assumptions on densities of the measure) was proved in [Paj97,AT15,ENV16,BS16]. Measures carried by rectifiable curves are studied using β numbers in [Ler03, BS15, BS16, AM16, BS17, MO18a,Nap20], see also the survey [Bad19].
Finiteness of a square function involving α coefficients (defined in [Tol09]) is shown to be necessary for rectifiability in [Tol15]. The opposite implication is studied in [ADT16,Orp18,ATT20]. In [Dąb20a,Dąb21] rectifiable measures were characterized using α 2 numbers, first defined in [Tol12]. Square functions involving centers of mass are studied in [MV09] and [Vil19]. Finally, [TT15,Tol17] are devoted to a square function involving ∆ numbers, where ∆ µ (x, r) = | µ(B(x,r)) For related characterizations of rectifiable measures in terms of tangent measures, see [Mat95,Chapter 16] and [Pre87,Section 5]. For a study of tangent points of Jordan curves in terms of β numbers see [BJ94], and for a generalization of this result for lower content regular sets of arbitrary dimension see [Vil20].
The behaviour of conical densities on purely unrectifiable sets is studied in [CKRS10] and [Käe10,§5]. In [Mat88,KS08,CKRS10,KS11] the relation between conical densities for higher dimensional sets and their porosity is investigated.
Higher order rectifiability in terms of approximate differentiability of sets is studied in [San19]. In [DNI19] the authors characterize C 1,α rectifiable sets using approximate tangents paraboloids, essentially obtaining a C 1,α counterpart of Theorem 1.2. See also [Ghi20] and [GG20] for related results.
We would also like to mention recent results of Badger and Naples that nicely complement Theorem 1.4. In [Nap20, Theorem D] Naples showed that a modified version of (1.2) can be used to characterize pointwise doubling measures carried by Lipschitz graphs, that is measures vanishing outside of a countable union of n-dimensional Lipschitz graphs. In an even more recent paper [BN21] the authors completely describe measures carried by ndimensional Lipschitz graphs on R d . They use a Dini condition imposed on the so-called conical defect, and their condition is closely related to (1.4). Note the absence of densities in the assumptions (and conclusion) of their results. If one adds an assumption Θ n * (µ, x) < ∞ for µ-a.e. x ∈ R d , then it actually follows from [BN21] that µ-a.e. finiteness of their conical Dini function implies that µ is n-rectifiable. We would like to stress however that neither Theorem 1.4 implies the results from [BN21], nor the other way around.
1.2. Big pieces of Lipschitz graphs. Before stating our next theorem, we need to recall some definitions.
Definition 1.8. We say that E ⊂ R d is n-Ahlfors-David regular (abbreviated as n-ADR) if there exist constants C 0 , C 1 > 0 such that for all x ∈ E and 0 < r < diam(E) Constants C 0 , C 1 will be referred to as ADR constants of E. Definition 1.9. We say that an n-ADR set E ⊂ R d has big pieces of Lipschitz graphs (BPLG) if there exist constants κ, L > 0, such that the following holds.
For all balls B centered at Sets with BPLG were studied e.g. in [Dav88,DS93a,DS93b] as one of the possible quantitative counterparts of rectifiability. Let us point out that the class of sets with BPLG is strictly smaller than the class of uniformly rectifiable sets, introduced in the seminal work of David and Semmes [DS91,DS93a]. An example of a uniformly rectifiable set that does not contain BPLG is due to Hrycak, although he never wrote it down, see [Azz21,Appendix].
While there are available many characterizations of uniformly rectifiable sets, the sets containing BPLG are not as well understood. David and Semmes showed in [DS93b] that a set contains BPLG if and only if it has big projections and satisfies the weak geometric lemma. We refer the reader to [DS93b] or [DS93a, §I.1.5] for details.
Very recently, Orponen characterized the BPLG property in terms of the big projections in plenty of directions property [Orp21], answering an old question of David and Semmes. A little before that, Martikainen and Orponen [MO18b] characterized sets with BPLG in terms of L 2 norms of their projections. Interestingly, the authors use the information about projections of an n-ADR set E to draw conclusions about intersections with cones of some subset E ′ ⊂ E with H n (E ′ ) ≈ H n (E). This in turn allows them to find a Lipschitz graph intersecting an ample portion of E ′ . We will use some of their techniques to prove a characterization of sets containing BPLG in terms of the following property.
Definition 1.10. Let 1 ≤ p < ∞. We say that a measure µ has big pieces of bounded energy for p, abbreviated as BPBE(p), if there exist constants α, κ, M 0 > 0 such that the following holds.
For all balls B centered at supp µ, Remark 1.12. In particular, for n-ADR sets, the condition BPBE(p) is equivalent to BPBE(q) for all 1 ≤ p, q < ∞.
Remark 1.13. In fact, one can show that an a priori slightly weaker condition than BPBE is already sufficient for BPLG. To be more precise, in (1. We show that this "weak" BPBE is sufficient for BPLG in Proposition 10.1. It is obvious that (1.7) is also necessary for BPLG: if E contains BPLG, then choosing G B = Γ B as in Definition 1.9, one can pick the corresponding V B and α so that It is tempting to consider also the following definition.
Definition 1.14. Let 1 ≤ p < ∞. We say that a measure µ has bounded mean energy (BME) for p if there exist constants α, M 0 > 0, and for every x ∈ supp µ there exists a direction V x ∈ G(d, d − n), such that the following holds. For all balls B centered at supp µ, 0 < r(B) < diam(supp µ), we have In other words we require µ(K(x, V x , α, r)) p r −np dr r dµ(x) to be a Carleson measure. This condition looks quite natural due to many similar characterizations of uniform rectifiability, e.g. the geometric lemma of [DS91,DS93a] or the results from [Tol09,Tol12].
It is easy to see, using the compactness of G(d, d − n) and Chebyshev's inequality, that BME for p implies BPBE(p). However, the reverse implication does not hold. In [Dąb20b] we give an example of a set containing BPLG that does not satisfy BME. The problem is the following. In the definition above, the plane V x is fixed for every x ∈ supp µ once and for all, and we do not allow it to change between different scales. This is too rigid.
Question 1.15. Can one modify the definition of BME, allowing the planes V x to depend on the scale r, so that the modified BME could be used to characterize BPLG, or uniform rectifiability?
It seems likely that every uniformly rectifiable measure would satisfy such relaxed BME (the idea would be similar to what is done in Section 9: use the β-numbers characterization of UR to get an upper bound for β-numbers, and then estimate the measure of cones from above by the β-numbers). It is less clear whether this relaxed BME would imply uniform rectifiability. Perhaps additional control for the oscillation of V x,r would be needed.
1.3. Boundedness of SIOs. We will be concerned with singular integral operators of convolution type, with odd C 2 kernels k : R d \ {0} → R satisfying for some constant C k > 0 |∇ j k(x)| ≤ C k |x| n+j for x = 0 and j ∈ {0, 1, 2}. (1.8) We will denote the class of all such kernels by K n (R d ). Note that these kernels are particularly nice examples of Calderón-Zygmund kernels (see [Tol14,p. 48] for definition), which will let us use many tools from the Calderón-Zygmund theory. Since the measures we work with may be non-doubling, our main reference will be [Tol14, Chapter 2]. For the more classical theory, we refer the reader to [Gra14a, Chapter 5], [Gra14b, Chapter 4].
Definition 1.16. Given a kernel k ∈ K n (R d ), a constant ε > 0, and a (possibly complex) Radon measure ν, we set For a fixed positive Radon measure µ and all functions f ∈ L 1 loc (µ) we define We say that T µ is bounded in L 2 (µ) if all T µ,ε are bounded in L 2 (µ), uniformly in ε > 0. Let M (R d ) denote the space of all finite real Borel measures on R d . When endowed with the total variation norm · T V , this is a Banach space. We say that T is bounded from M (R d ) to L 1,∞ (µ) if there exists a constant C such that for all ν ∈ M (R d ) and all λ > 0 The main motivation for developing the theory of quantitative rectifiability was finding necessary and/or sufficient conditions for boundedness of singular integral operators. David and Semmes showed in [DS91] that, for an n-ADR set, the L 2 boundedness of all singular integral operators with smooth and odd kernels is equivalent to uniform rectifiability. The famous David-Semmes problem asks whether the L 2 boundedness of a single SIO, the Riesz transform, is already sufficient for uniform rectifiability. It was shown that the answer is affirmative for n = 1 in [MMV96], for n = d − 1 in [NTV14a], and the problem is open for other n.
In the non-ADR setting less is known. A necessary condition for the boundedness of SIOs in L 2 (µ), where µ is Radon and non-atomic, is the polynomial growth condition:  [ENV14] that if µ is a measure on R 2 , H 1 (supp µ) < ∞, and µ has vanishing lower 1-density, then the Riesz transform is unbounded. Their result was generalized to SIOs associated to gradients of single layer potentials in [CAMT19]. Nazarov, Tolsa and Volberg proved in [NTV14b] that if E ⊂ R n+1 satisfies H n (E) < ∞ and the n-dimensional Riesz transform is bounded in L 2 (H n | E ), then E is n-rectifiable. That the same is true for gradients of single layer potentials was shown by Prat, Puliatti and Tolsa in [PPT21]. Concerning sufficient conditions for boundedness of SIOs, in [AT15] Azzam and Tolsa estimated the Cauchy transform of a measure using its β numbers. Their method was further developed by Girela-Sarrión [GS19]. He gives a sufficient condition for boundedness of singular integral operators with kernels in K n (R d ) in terms of β numbers. We use the main lemma from [GS19] to prove the following criterion involving 2-conical energy.
Theorem 1.17. Let µ be a Radon measure on R d satisfying the polynomial growth condition (1.9). Suppose that µ has BPBE(2). Then, all singular integral operators T µ with kernels k ∈ K n (R d ) are bounded in L 2 (µ), with norm depending only on BPBE constants, the polynomial growth constant C 1 , and the constant C k from (1.8).
Remark 1.19. Recall that for n-ADR sets the condition BPBE(p) was equivalent to BPLG, regardless of p. By the remark above, it is clear that if we replace the n-ADR condition with polynomial growth (i.e. if we drop the lower regularity assumption), then the condition BPBE(p) is no longer independent of p. In general we only have one implication: for 1 ≤ p < q < ∞

BP BE(p) ⇒ BP BE(q).
Remark 1.20. Theorem 1.17 is sharp in the following sense. If one tried to weaken the assumption BPBE(2) to BPBE(p) for some p > 2, then the theorem would no longer hold. The reason is that for any p > 2 one may construct a Cantor-like probability measure µ, say on a unit square in R 2 , that has linear growth and such that for all x ∈ supp µ 1 0 µ(B(x, r)) r p dr r 1, (that is, a much stronger version of BPBE(p) holds), but nevertheless, the Cauchy transform is not bounded on L 2 (µ). See [Tol14,Chapter 4.7].
Sadly, the implication of Theorem 1.17 cannot be reversed. Let E ⊂ R 2 be the previously mentioned example of a 1-ADR uniformly rectifiable set that does not contain BPLG. In particular, by Theorem 1.11 E does not satisfy BPBE(p) for any p. Nevertheless, by the results of David and Semmes [DS91], all nice singular integral operators are bounded on L 2 (E).

Cones and projections.
Let us note that [CT20, Theorem 10.2] was merely a tool to prove the main result of [CT20]: a lower bound on analytic capacity involving L 2 norms of projections. Chang and Tolsa proved also an interesting inequality showing the connection between 1-conical energy and L 2 norms of projections. We introduce additional notation before stating their result.
We define also We will often suppress the arguments V, α, and write simply E µ,p (B), E µ,p (R d ). (1.10) and so (1.11) In their paper Chang and Tolsa were working with the expression from the right hand side above.
where · op is the operator norm. We write π V µ to denote the image measure of µ by the projection π V . If π V µ ≪ H n | V , then we identify π V µ with its density with respect to H n | V , and π V µ L 2 (V ) denotes the L 2 norm of this density. Otherwise, we set π V µ L 2 (V ) = ∞. Proposition 1.23 ([CT20, Corollary 3.11]). Let V 0 ∈ G(d, n) and α > 0. Then, there exist constants λ, C > 1 such that for any finite Borel measure µ in R d , Let us note that a variant of this estimate was also proved in [MO18b], for a measure of the form µ = H n | E , with E a suitable set.
The inequality converse to that of Proposition 1.23 in general is not true, but it is not far off. Additional assumptions on µ are necessary, and one has to add another term to the left hand side. See [CT20, Remark 3.12, Appendix A].
In the light of results mentioned above, as well as the characterization of sets with BPLG from [MO18b], the connection between L 2 norms of projections and cones is quite striking. Note that the proof of the Besicovitch-Federer projection theorem also involves careful analysis of measure in cones, see [Mat95,Chapter 18]. Exploring further the relationship between cones and projections would be very interesting.
Question 1.24. Is it possible to obtain an inequality similar to that of Proposition 1.23, but with E µ,2 on the left hand side, and some quantity involving π V µ on the right hand side?
1.5. Organization of the article. In Section 2 we introduce additional notation, and recall the properties of the David-Mattila lattice D µ . In Section 3 we state our main lemma, a corona decomposition-like result. Roughly speaking, it says that if a measure µ has polynomial growth, and for some then we can decompose D µ into a family of trees such that: • for every tree, µ is "well-behaved" at the scales and locations of the tree, • we have a good control on the number of trees (see (3.2)). We prove the main lemma in Sections 4-6. Let us point out that in the case p = 1 an analogous corona decomposition was already shown in [CT20, Lemma 5.1]. Our proof follows the same general strategy, but some key estimates had to be done differently (most notably the estimates in Section 5).
In Section 7 we show how to use the main lemma and results from [GS19] to get Theorem 1.17. Sections 8 and 9 are dedicated to the proof of Theorem 1.4. The "sufficient part" follows from our main lemma, while the "necessary part" is deduced from the corresponding β 2 result of Tolsa [Tol15]. Finally, we prove Theorem 1.11 in Sections 10 and 11. To show the "sufficient part" we use the results from [MO18b], whereas the "necessary part" follows from a simple geometric argument.
Acknowledgements. I would like to thank Xavier Tolsa for all his help and patience. I am also grateful to the anonymous referee for carefully reading the article, and for many helpful suggestions. I received support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445), and also partial support from the Catalan Agency for Management of University and Research Grants (2017-SGR-0395), and from the Spanish Ministry of Science, Innovation and Universities (MTM-2016-77635-P).

Additional notation. We will write
If the constant C depends on some parameter t, we will write A t B. We usually omit the dependence on n and d.
B(x, r) stands for the open ball {y ∈ R d : |y − x| < r}. On the other hand, if B is a ball, then r(B) denotes its radius.
A characteristic function of a set E ⊂ R d will be denoted by 1 E . Given a Radon measure µ and a ball B = B(x, r), we set If T is a singular integral operator as in Definition 1.16, then the associated maximal operator T * is defined as Given an n-plane L, π L will denote the orthogonal projection onto L, and π ⊥ L will denote the orthogonal projection onto L ⊥ .
Given two bounded sets E, F ⊂ R d , dist H (E, F ) will stand for the Hausdorff distance between E and F .

David-Mattila lattice.
In the proof of Theorem 1.17 we will use the lattice of "dyadic cubes" constructed by David and Mattila [DM00]. Their construction depends on parameters C 0 > 1 and A 0 > 5000C 0 . The parameters can be chosen in such a way that the following lemmas hold.
The general position of the cubes Q can be described as follows. For each k ≥ 0 and For any Q ∈ D µ we denote by D µ (Q) the family of P ∈ D µ such that P ⊂ Q. Given and S∈Dµ:Q⊂S⊂R Let us remark that the constant 9d in the exponent of (2.2) could be replaced by any other positive constant, if C 0 and A 0 are chosen suitably, see [DM00,(5 From now on we will treat C 0 and A 0 as absolute constants, and we will not track the dependence on them in our estimates.

Main lemma
In order to formulate our main lemma we need to introduce some vocabulary. Let µ be a compactly supported Radon measure with polynomial growth (1.9). Suppose D µ is the associated David-Mattila lattice, and assume that Given a family of cubes Top ⊂ D db µ satisfying R 0 ∈ Top we define the following families associated to each R ∈ Top: • Next(R) is the family of maximal cubes Q ∈ Top strictly contained in R, • Tr(R) is the family of cubes Q ∈ D µ contained in R, but not contained in any P ∈ Next(R).
Then, there exists a family of cubes Top ⊂ D db µ , and a corresponding family of Lipschitz graphs {Γ R } R∈Top , satisfying: (i) the Lipschitz constants of Γ R are uniformly bounded by a constant depending on α, Moreover, the following packing condition holds: (3. 2) The implicit constant does not depend on r 0 .
We prove the lemma above in Sections 4-6. From this point on, until the end of Section 6, we assume that µ is a compactly supported Radon measure satisfying the growth condition (3.1), and that there exist For simplicity, in our notation we will suppress the parameters V and α. That is, we Parameters. In the proof of Lemma 3.1 we will use a number of parameters. To make it easier to keep track of what depends on what, and at which point the parameters get fixed, we list them below. Recall that "C 1 = C 1 (C 2 )" means that "the value of C 1 depends on the value of C 2 ." • A = A(p) > 1 is the "HD" constant, it is fixed in Lemma 6.1.
It is fixed just below (5.7), but depends also on Lemma 4.5 and Lemma 4.7.
In this section we will construct a corresponding tree of cubes Tree(R), and a Lipschitz graph Γ R that "approximates µ at scales and locations from Tree(R)"; see Lemma 4.8.
4.1. Stopping cubes. Consider constants A ≫ 1, 0 < ε ≪ τ ≪ 1, and 0 < η ≪ 1, which will be fixed later on. Given Q ∈ D µ we set For any R ∈ D db µ we define the following families of cubes: • BCE 0 (R), the family of big conical energy cubes, consisting of Q ∈ D µ (R) such that • HD 0 (R), the high density family, consisting of Q ∈ D db • LD 0 (R), the low density family, consisting of Q ∈ D µ (R) \ BCE 0 (R) such that Note that the cubes in HD(R) are doubling (by the definition), while the cubes from LD(R) and BCE(R) may be non-doubling.
We define Tree(R) as the family of cubes from D µ (R) which are not strictly contained in any cube from Stop(R) (in particular, Stop(R) ⊂ Tree(R)). Note that it may happen that R ∈ BCE(R), in which case Tree(R) = {R}.
Basic properties of cubes in Tree(R) are collected in the lemma below.
Lemma 4.1. Suppose Q ∈ Tree(R). Then, Proof. First, note that if R ∈ Stop(R), then Tree(R) = {R} and the lemma above is trivial. Assume that R ∈ Stop(R). Inequalities (4.3) and (4.4) are obvious by the definition LD(R) and BCE(R).
by the high density stopping condition. In general, given Q ∈ Tree(R), let P (Q) be the smallest doubling cube containing Q, other than Q. Since R ∈ D db µ and R ∈ Stop(Q), On the other hand, a minor modification of the computation above shows that It follows that k A,τ 1.
The following estimate of the measure of cubes in BCE(R) will be used later on in the proof of the packing estimate (3.2).

Lemma 4.2. We have
(4.5) Proof. We use the fact that for Q ∈ BCE(R) we have 4.2. Key estimate. We introduce some additional notation. Given x ∈ R d and λ > 0 set For Q ∈ D µ , we denote If λ = 1, we will write K Q instead of K 1 Q .
Lemma 4.3. There exists a constant M = M (α) > 1 such that, if Q ∈ Tree(R) and P ∈ D µ (R) satisfy and then P ∈ Tree(R).
Proof. Taking M = M (α) > 1 big enough, we can choose cubes P ′ , Q ′ ∈ D µ (R) such that Thus, if η is taken small enough (say, η ≪ M −1 ), we have Since Q ∈ Tree(R) and Q Q ′ , we have Q ′ ∈ Tree(R) \ Stop(R), and so It follows that, for ε small enough, P ′ ∈ LD 0 (R). Since P P ′ , we get that P / ∈ Tree(R).
We set

Lemma 4.4.
For all x, y ∈ G R we have y ∈ K 1/2 (x). Thus, G R is contained in an n-dimensional Lipschitz graph with Lipschitz constant depending only on α.
Proof. Proof by contradiction. Suppose that x, y ∈ G R and x − y ∈ K 1/2 . Let Q, P ∈ Tree(R) be such that x ∈ 2M B Q , y ∈ 2M B P , with sidelength so small that P ∩ (K 1/2 Q \ M B Q ) = ∅ and dist(Q, P ) ≥ M r(P ) (note that this can be done because K 1/2 is an open cone, and so x ′ − y ′ ∈ K 1/2 also for x ′ ∈ B(x, ε ′ ) and y ′ ∈ B(y, ε ′ ), assuming ε ′ > 0 small enough). It follows by Lemma 4.3 that P / ∈ Tree(R), and so we reach a contradiction.

4.3.
Construction of Γ R . The Lipschitz graph from Lemma 4.4 can be thought of as a first approximation of Γ R . It contains the "good set" G R , but we would also like for Γ R to lie close to cubes from Tree(R). In this subsection we show how to do it. Given t > 1, we say that cubes Q, P ∈ D µ are t-neighbours if they satisfy and dist(Q, P ) ≤ t(r(Q) + r(P )). (4.10) If at least one of the conditions above does not hold, we say that Q and P are t-separated. We will also say that a family of cubes is t-separated if the cubes from that family are pairwise t-separated. Consider a big constant t = t(M, α) > M which will be fixed later on. We denote by Sep(R) a maximal t-separated subfamily of Stop(R) (it exists by Zorn's lemma). Clearly, for every Q ∈ Stop(R) there exists some P ∈ Sep(R) which is a t-neighbour of Q.
Furthermore, we define Sep * (R) as the family of all cubes Q ∈ Sep(R) satisfying the following two conditions: and for all P ∈ Sep(R), P = Q, we have Proof. Suppose Q ∈ Sep * (R), and Q ⊂ 1.5M B P . We will show that P / ∈ Sep * (R). Firstly, if r(Q) > t −1 r(P ), then Q ⊂ 1.5M B P implies that Q and P are t-neighbours (for t big enough), and so P / ∈ Sep * (R). On the other hand, if r(Q) ≤ t −1 r(P ), then (if t is big enough) Q ⊂ 1.5M B P implies 2M B Q ⊂ 2M B P , contradicting (4.12).
Lemma 4.6. For every Q ∈ Sep(R) at least one of the following is true: Proof. If Q ∈ Sep * (R), then of course (b) holds (with P = Q). Suppose that Q / ∈ Sep * (R), and that (a) does not hold (i.e. 2M B Q ∩ G R = ∅). We will find P ∈ Sep * (R) such that 2M B P ⊂ 2M B Q .
Since Q / ∈ Sep * (R) and (4.11) holds, condition (4.12) must be false. Thus, we get a cube Otherwise, we continue as follows.
Reasoning as before, Iterating this process, we get a (perhaps infinite) sequence of cubes If the algorithm never stops, then ∞ j=0 2M B Q j = ∅. But, by the definition of G R (4.8) we have ∞ j=0 2M B Q j ⊂ G R , and so we get a contradiction with 2M B Q ∩ G R = ∅. Thus, the algorithm stops at some cube Q m , which means that Q m ∈ Sep * (R). Setting P = Q m finishes the proof. (a) for all Q, P ∈ Sep * (R), Q = P, we have for all x ∈ G R and for all Q ∈ Sep * (R) we have (4.14) Proof of (a). Proof by contradiction. Suppose Q ∩ K 1/2 P = ∅ (which by symmetry of cones implies P ∩ K 1/2 Q = ∅). Without loss of generality, assume r(Q) ≤ r(P ). Since Q and P are t-separated, at least one of the conditions (4.9), (4.10) fails, i.e. r(Q) ≤ t −1 r(P ) or dist(Q, P ) > t(r(Q) + r(P )).
We know by Lemma 4.5 that Q ⊂ 1.5M B P . It is easy to see that in either of the cases considered above, this implies Q ∩ 1.2M B P = ∅. It follows that Q ∩ (K 1/2 P \ M B P ) = ∅ and r(Q) ≤ r(P ) ≤ M −1 dist(Q, P ). Hence, we can use Lemma 4.3 to conclude that Q / ∈ Tree(R). This contradicts Q ∈ Sep * (R).
Proof of (b). Proof by contradiction. Suppose x ∈ K 1/2 by (4.11). Since x ∈ G R , we can find an arbitrarily small cube P ∈ Tree(R) such that x ∈ 2M B P . Taking r(P ) small enough we will have r(P ) ≤ M −1 dist(Q, P ) and P ∩ K is an open set). Lemma 4.3 yields P / ∈ Tree, a contradiction.

Moreover, there exists a big constant Λ = Λ(M, t) > 1 such that for every
Proof. Recall that for each cube Q ∈ D µ we have a "center" denoted by x Q ∈ Q. Set F = {x Q : Q ∈ Sep * (R)} ∪ G R . It follows by Lemma 4.4 and Lemma 4.7 that for any x, y ∈ F we have x − y / ∈ K 1/2 . Thus, there exists a Lipschitz graph Γ R , with slope depending only on α, such that F ⊂ Γ R .
Concerning the second statement, it is clearly true for Q ∈ Sep * (R) (even with Λ = 1). For Q ∈ Sep(R), we have by Lemma 4.6 that either 2M B Q ∩ G R = ∅ or there exists P ∈ Sep * (R) with 2M B P ⊂ 2M B Q . Thus, (4.15) holds if Λ ≥ 2M .
If Q ∈ Stop(R), there exists some P ∈ Sep(R) which is a t-neighbour of Q, so that for some Λ = Λ(t, M ) > 1 we have ΛB Q ⊃ 2M B P , and 2M B P intersects Γ R . Finally, for a general Q ∈ Tree(R), either Q contains some cube from Stop(R), or Q ⊂ G R . In any case, Remark 4.9. Note that while for a general cube Q ∈ Tree(R) we only have ΛB Q ∩ Γ R = ∅, we have a better estimate for the root R: (4.16) Indeed, (4.16) is clear if the set G R is non-empty. If G R = ∅, then Sep * (R) = ∅, so that for some P ∈ Sep * (R) we have x P ∈ Γ R ∩ B R .

Small measure of cubes from LD(R)
In the proof of the packing estimate (3.2) it will be crucial to have a bound on the measure of low density cubes.

Lemma 5.2. There exists a t-separated family LD
Proof. We construct the family LD Sep (R) in the following way. Define LD 1 (R) as a maximal t-separated subfamily of LD(R). Next, define LD 2 (R) as a maximal t-separated subfamily of LD(R) \ LD 1 (R). In general, having defined LD j (R), we define LD j+1 (R) to be a maximal t-separated subfamily of LD(R) \ (LD 1 (R) ∪ · · · ∪ LD j (R)). We claim that there is only a bounded number of non-empty families LD j (R), with the bound depending on t. Indeed, if Q ∈ LD j (R), then Q has at least one t-neighbour in each family LD k (R), k ≤ j. It follows easily from the definition of t-neighbours that the number of t-neighbours of any given cube is bounded by a constant C(t). Hence, j ≤ C(t).
Set LD Sep (R) to be the family LD j (R) maximizing Q∈LD j (R) µ(Q). Then, We define also a family LD * Sep (R) ⊂ LD Sep (R) in the following way: we remove from LD Sep (R) all the cubes P for which there exists some Q ∈ LD Sep (R) such that 1.1B Q ∩ 1.1B P = ∅ and r(Q) < r(P ).
(5.2) Lemma 5.3. For each Q ∈ LD Sep (R) at least one of the following is true: There exists some P ∈ LD * Sep (R) such that 1.2B P ⊂ 1.2B Q . Proof. Suppose Q ∈ LD Sep (R), and that (a) does not hold. We will find P such that (b) is satisfied.

(5.3)
Since Q and Q 1 are t-separated, and (4.10) holds, it follows that t r(Q 1 ) < r(Q). Thus, Q 1 is tiny compared to Q and we have 1.2B Q 1 ⊂ 1.2B Q . If Q 1 ∈ LD * Sep (R), we set P = Q 1 and we are done. Otherwise, we iterate as in Lemma 4.6 (with 2M replaced by 1.2) to find a finite sequence Q 1 , Q 2 , . . . , Q m satisfying 1.2B Q j+1 ⊂ 1.2B Q j , and such that Q m ∈ LD * Sep (R).
In particular, if ε is small enough, then for each Q ∈ LD * Sep (R) we can choose a point Proof. Suppose Q ∈ LD * Sep (R) and that we have Q ∩ K 1/2 P \ M B P = ∅ for some P ∈ LD * Sep (R). Note that if we had M r(Q) ≤ dist(Q, P ), then the assumptions of Lemma 4.3 would be satisfied, and we would arrive at Q ∈ Tree(R), a contradiction. Thus,

dist(Q, P ) ≤ M r(Q) < t r(Q).
(5.6) It follows that (4.10) -one of the t-neigbourhood conditions -is satisfied. Since Q and P are t-separated, we necessarily have t r(Q) ≤ r(P ) or t r(P ) ≤ r(Q).
If we had t r(Q) ≤ r(P ), then (5.6) implies dist(Q, P ) ≤ r(P ). Hence, 1.1B Q ∩1.1B P = ∅. But this cannot be true, by the definition of LD * Sep (R). It follows that t r(P ) ≤ r(Q). (5.7) Let S ⊃ P be the biggest ancestor of P satisfying r(S) ≤ δr(Q) for some small constant δ = δ(α) which will be fixed in a few lines. If t is big enough, then S = P . Thus, r(S) ≈ δ r(Q), and S ∈ Tree(R) \ Stop(R). Recall that by the definition of LD * Sep (R) we have 1.1B Q ∩ 1.1B P = ∅. It follows that if δ < 0.001, then 4B S ∩ 1.05B Q = ∅. Now, using this separation, it is not difficult to check that for δ = δ(α) small enough, for any x ∈ K 1/2 . Observe also that, due to (5.6) and the fact that r(S) ≤ δr(Q), we have provided that η is small enough (say, η −1 ≫ t). Putting together the two estimates above, we get that for any x ∈ K 1/2 P ∩ Q ⊃ Q ∩ K 1/2 P \ M B P and all r ∈ (η −1 r(Q)/2, η −1 r(Q)). Integrating the above over all x ∈ A, where A ⊂ Q ∩ K 1/2 P \ M B P is an arbitrary measurable subset, yields Now, let P i be some ordering of cubes P ∈ LD * Sep (R) satisfying Q ∩ K 1/2 Observe that A i are pairwise disjoint and their union is Note that since Q / ∈ BCE(R), we have E µ,p (Q)µ(Q) ≤ εΘ µ (2B R ) p µ(Q). So the estimate (5.4) holds.
Lemma 5.5. There exists an n-dimensional Lipschitz graph Γ LD passing through all the points w P , P ∈ LD * Sep (R). The Lipschitz constant of Γ LD depends only on α. Proof. It suffices to show that for any Q, P ∈ LD * Sep (R), Q = P, we have Without loss of generality assume r(P ) ≤ r(Q). By (5.5) we have That is, (4.10) holds. But Q and P are t-separated, and so (4.9) must fail. Hence, Q and P belong to LD * Sep (R), so by (5.2) we have 1.
We can finally finish the proof of Lemma 5.1.
Proof of Lemma 5.1. By Lemma 5.2 it suffices to estimate the measure of cubes from LD Sep (R). Let G denote an arbitrary finite subfamily of LD Sep (R). We use the covering lemma [Tol14,Theorem 9.31] to choose a subfamily F ⊂ G such that The above and the LD stopping condition give Now, it follows from Lemma 5.3 and Lemma 5.5 that for each Q ∈ G ⊂ LD Sep (R) there exists either Now, using the bounded superposition property of F we get Together with (5.11), this gives Q∈G µ(Q) α τ µ(R).
Since G was an arbitrary finite subfamily of LD Sep (R), we finally arrive at 6. Top cubes and packing estimate 6.1. Definition of Top. In order to define the Top family, we need to introduce some additional notation. Given Q ∈ D µ , let MD(Q) denote the family of maximal cubes from D db µ (Q) \ {Q}. It follows from Lemma 2.1 (c) that the cubes from MD(Q) cover µ-almost all of Q.
Given R ∈ D db µ set Next(R) =

Since we always have MD(Q) = {Q}, it is clear that Next(R) = {R}.
Observe that if P ∈ Next(R), then by Lemma 4.1 and Lemma 2.2 we have for all intermediate cubes S ∈ D µ , P ⊂ S ⊂ R, (6.1) We are finally ready to define Top. It is defined inductively as Top = k≥0 Top k . First, set where R 0 was defined as supp µ. Having defined Top k , we set Note that for each k ≥ 0 the cubes from Top k are pairwise disjoint.
6.2. Definition of ID. We distinguish a special type of Top cubes. We say that R ∈ Top is increasing density, R ∈ ID, if Lemma 6.1. If A is big enough, then for all R ∈ ID Proof. The definition of ID and the HD stopping condition imply that for any R ∈ ID Note that all Q ∈ HD(R) are doubling, and so by Lemma 2.3 If A is taken big enough, then the estimates above yield (6.2).
6.3. Packing condition. We will now establish the packing condition (3.2). For S ∈ Top set Top(S) = Top ∩ D µ (S) and Top j (S) = Top j ∩ D µ (S). For k ≥ 0 we also define Top j (S), Recall that µ satisfies the following polynomial growth condition: there exist C 1 > 0 and r 0 > 0 such that for all x ∈ supp µ, 0 < r ≤ r 0 , we have µ (B(x, r)) ≤ C 1 r n . (6.3)

Lemma 6.2. For all S ∈ Top we have
The implicit constant does not depend on r 0 .
Proof. First, we deal with ID cubes. Note that where the last inequality follows from the fact that Note that for small cubes Q ∈ Top k 0 (S) (i.e. satisfying r(2B Q ) ≤ r 0 ) we have (6.5), while for big cubes the trivial estimate Θ µ (2B Q ) ≤ µ(2B S )r −n 0 holds. It follows that and so we may deduce from (6.6) that Letting k → ∞ we arrive at Now, we need to estimate the sum from the right hand side. By the definition of ID we have for all R ∈ Top(S) \ ID and so by Lemma 2.1 (c) we get The measure of low density cubes is small due to (5.1), and so for τ small enough we have Thus, Concerning the first sum, notice that if µ R\ Q∈Next(R) Q > 0, then we have arbitrarily small cubes P belonging to Tree(R). In particular, by (4.3) and (6.3), we have Θ µ (2B R ) ≤ τ −1 Θ µ (2B P ) ≤ τ −1 C 1 , taking P ∈ Tree(R) \ Stop(R) small enough. Recall also that for R ∈ Top(S), the sets R \ Q∈Next(R) Q are pairwise disjoint. Hence, (6.9) To estimate the second sum from (6.8), we apply (4.5) to get By the definition of E µ,p (P ), and the bounded intersection property of the balls 2B P for cubes P of the same generation, we have Consequently, Together with (6.7), (6.8), and (6.9), this gives (6.4).
Let us put together all the ingredients of the proof of the main lemma.
Proof of Lemma 3.1. Let Top ⊂ D db µ be as above, and {Γ R } R∈Top be as in Lemma 4.8. Then, properties (i) and (ii) are ensured by Lemma 4.8. Property (iii) follows from (6.1). We get the packing estimate (3.2) from (6.4) by taking S = R 0 .

Application to singular integral operators
To prove Theorem 1.17, we will use geometric characterizations of boundedness of operators from K n (R d ) shown in [GS19,Sections 4,5,9]. For n = 1, d = 2, a variant of this characterization valid for the Cauchy transform was already proved in [Tol05].
For Q, S ∈ D µ , Q ⊂ S, we set The notation Good(R), Tr(R), Next(R) used below was introduced in Section 3. (i) Lipschitz constants of Γ R are uniformly bounded by some absolute constant, (iv) for all Q ∈ Next(R) there exists S ∈ D µ , Q ⊂ S, such that δ µ (Q, S) Θ µ (2B R ), and 2B S ∩ Γ R = ∅. Then, for every singular integral operator T with kernel k ∈ K n (R d ) we have with the implicit constant depending on C 1 and the constant C k from (1.8).
The result above is not explicitly stated in [GS19], but it is essentially [GS19, Section 5, Lemma 1]. The "corona decomposition" assumptions of Lemma 7.1 come from [GS19, Lemma D], which is treated there as a black-box. The proof of [GS19, Lemma 1] is concluded in [GS19, Section 9], and it is evident from its last line that we may replace the β-number right hand side of [GS19, Lemma 1] by the sum-over-Top-cubes right hand side of Lemma 7.1.
We are going to use Lemma 3.1 together with Lemma 4.8 and Lemma 7.1 to get the following.
Lemma 7.2. Let µ be a compactly supported Radon measure on R d satisfying the growth condition (1.9). Assume further that for some V ∈ G(d, d − n), α ∈ (0, 1), we have Then, for every singular integral operator T with kernel k ∈ K n (R d ) we have with the implicit constant depending on C 1 , α and the constant C k from (1.8).
Proof. Using Lemma 3.1 (with p = 2), it is clear that the assumptions (i)-(iii) of Lemma 7.1 are satisfied. We still have to check if (iv) holds. Once we do that, the packing estimate (3.2) together with Lemma 7.1 will ensure that (7.1) holds. Suppose R ∈ Top, Q ∈ Next(R). We are looking for S ∈ D µ such that δ µ (Q, S) Θ µ (2B R ), and 2B S ∩ Γ R = ∅. Let P ∈ Stop(R) be such that Q ⊂ P . By Lemma 4.8 we have some constant Λ such that ΛB P ∩ Γ R = ∅. Together with (4.16), this implies that there exists S ∈ Tree(R) such that P ⊂ S, r(S) ≈ Λ r(P ), and 2B S ∩ Γ R = ∅. We split Concerning the first integral, for y ∈ 2B S \ 2B P we have |y − x Q | ≈ r(S) ≈ Λ r(P ), and so To deal with the second integral, observe that there are no doubling cubes between Q and P . Then, it follows from Lemma 2.2 that If P = R, then P is doubling and we have Θ µ (100B(P )) Θ µ (2B R ). Otherwise, the parent of P , denoted by P ′ , belongs to Tree(R) \ Stop(R). Since 100B(P ) ⊂ 2B P ′ , we get Either way, we get that δ µ (Q, S) A Θ µ (2B R ), and so the assumption (iv) of Lemma 7.1 is satisfied.
Lemma 7.2 allows us to use the non-homogeneous T 1 theorem of Nazarov, Treil and Volberg [NTV97] to prove a version of Theorem 1.17 in the case of a fixed direction V , i.e. if for all x ∈ supp µ we have V x ≡ V . Lemma 7.3. Let µ be a Radon measure on R d satisfying the polynomial growth condition (1.9). Suppose that there exist M 0 > 1, α ∈ (0, 1), V ∈ G(d, d − n), such that for every ball Then, all singular integral operators T µ with kernels in K n (R d ) are bounded in L 2 (µ). The bound on the operator norm of T µ depends only on C 1 , α, M 0 , and the constant C k from (1.8).
Proof. We apply Lemma 7.2 to µ| B , where B is an arbitrary ball, and get that It is easy to see that, using the assumptions (1.9) and (7.2), we have 3) The L 2 boundedness of T µ follows by the non-homogeneous T 1 theorem from [NTV97]. The condition (7.3) is slightly weaker than the original assumption in [NTV97], but this is not a problem, see the discussion in [Tol14, §3.7.2].
We are ready to finish the proof of Theorem 1.17.
Proof of Theorem 1.17. Let B be an arbitrary ball intersecting supp µ. Recall that, by the definition of BPBE(2), there exist By the polynomial growth condition (1.9) we also have The estimate above implies that for all balls B ′ ⊂ R d we have Clearly, ν = µ| G B has polynomial growth, and so we may apply Lemma 7.3 to conclude that all singular integral operators T ν with kernels in K n (R d ) are bounded in L 2 (ν). Thus, the corresponding maximal operators T * are bounded from M (R d ) to L 1,∞ (ν), see [Tol14,Theorem 2.21].
Recall that for all balls B we have µ(G B ) ≈ κ µ(B). For any fixed T , the operator norm of T µ| G B ,ε : L 2 (µ| G B ) → L 2 (µ| G B ) is bounded uniformly in B and ε, and so the same is true for the operator norm of T * : M (R d ) → L 1,∞ (µ| G B ). Hence, we may use the good lambda method [Tol14,Theorem 2.22] to conclude that T µ is bounded in L 2 (µ).

Sufficient condition for rectifiability
The aim of this section is to prove the following sufficient condition for rectifiability.
We reduce the proposition above to the following lemma.
Proof of Proposition 8.1 using Lemma 8.2. To show that µ is rectifiable, it suffices to prove that for any bounded E ⊂ supp µ of positive measure there exists F ⊂ E, µ(F ) > 0, such that µ| F is rectifiable. Given any such E we may rescale it and translate it, so without loss of generality E ⊂ B(0, 1). Since 0 < Θ n, * (µ, x) and Θ n * (µ, x) < ∞ for µ-a.e. x ∈ E, choosing C * > 1 big enough, we get that the set has positive µ-measure.
Let {V k } k∈N be a countable and dense subset of G(d, d − n). It is clear that for any α ∈ (0, 1), It is a simpl exercise to check that for each k ∈ N the set E k is Borel. Moreover, it follows Using the Lebesgue differentiation theorem and (8.3), it is easy to see that for µ-a.e. x ∈ F we have Θ n, * (µ| F , x) = Θ n, * (µ, x) > 0 and Θ n * (µ| F , x) = Θ n * (µ, x) ≤ C * . Hence, µ| F satisfies the assumptions of Lemma 8.2, and so it is n-rectifiable.
Using similar tricks as in the proof of Proposition 8.1, it is easy to see that we may actually replace Θ n, * (µ, x) < ∞ by a stronger condition: without loss of generality, we can assume that there exist C 1 > 0 and r 0 > 0 such that for all x ∈ supp µ and all 0 < r ≤ r 0 we have µ(B(x, r)) ≤ C 1 r n .
Then, the assumptions of Lemma 3.1 are satisfied, and we get a family of cubes Top ⊂ D db µ and an associated family of Lipschitz graphs Γ R , R ∈ Top. The cubes from Top satisfy the packing condition 0, 1)).
It follows that for µ-a.e. x ∈ R d we have Fix some x for which the above holds. Denote by R 0 ⊃ R 1 ⊃ . . . the sequence of cubes from Top containing x. We claim that for µ-a.e. x this sequence is finite. Indeed, if the sequence is infinite, we have Θ µ (2B R i ) → 0. On the other hand, let i ≥ 0 and r(R i+1 ) ≤ r ≤ r(R i ). Since R i+1 ∈ Next(R i ), we get from (6.1) which may happen only on a set of µ-measure 0 because Θ n, * (µ, x) > 0 for µ-a.e. x ∈ R d . Hence, for µ-a.e. x ∈ R d the sequence {R i } is finite. This means that if R k denotes the smallest Top cube containing x, then x ∈ Good(R k ). It follows that and so µ is n-rectifiable.  µ(B(x, r k )) ≤ 20 C * C n+d r n k . (8.5) Let λ < 1 2 be a small constant depending on α, to be chosen later. By the lemma above (used with C = λ −1 ) and Vitali's covering theorem (see [Mat95, Theorem 2.8]), there exists a family of pairwise disjoint closed balls B i , i ∈ I, centered at x i ∈ supp µ ⊂ B(0, 1), which cover µ-almost all of B(0, 1), and which satisfy for some arbitrary fixed ρ > 0. We may assume that (8.2) holds for all the centers x i . Choose I 0 ⊂ I a finite subfamily such that where ε > 0 is some small constant. Clearly, I 0 = I 0 (ρ, ε). For each i ∈ I 0 we consider an n-dimensional disk D i , centered at x i , parallel to V ⊥ ∈ G(d, n), with radius λr(B i ). We define an approximating measure Moreover, since I 0 is a finite family, the definition of ν and (8.6) imply that ν satisfies the polynomial growth condition (3.1) with r 0 = min i∈I 0 r(B i )/2 and C 1 = C(λ)C * , i.e. for 0 < r < r 0 and x ∈ supp ν ν(B(x, r)) ≤ C(λ)C * r n . (8.7) Lemma 8.4. For λ = λ(α) < 1 2 small enough, we have µ(B(0, 1)) p )µ (B(0, 1)). The implicit constant does not depend on ρ, ε.
where K is independent of ρ and ε.
Set ε k = 2 −k . Observe that, for a fixed ρ > 0, we have µ( Since the sequence of sets G j is increasing, we easily get that for µ-a.e. x ∈ B(0, 1) and the convergence is monotone. Hence, by monotone convergence theorem, The estimate is uniform in ρ, and so once again monotone convergence gives Taking into account Lemma 8.6 and Section 8.1, the proof of Lemma 8.2 is finished.

Necessary condition for rectifiability
In this section we will prove the following. First, we recall the definition of β 2 numbers, as defined by David and Semmes [DS91].
Definition 9.2. Given a Radon measure µ, x ∈ supp µ, r > 0, and an n-plane L, define where the infimum is taken over all n-planes intersecting B(x, r).
Tolsa showed the following necessary condition for rectifiability in terms of β 2 numbers. For a fixed n-rectifiable measure µ, let L x,r denote a plane minimizing β µ,2 (x, r) (it may be non-unique, in which case we simply choose one of the minimizers).
Recall that in Definition 1.1 we defined the approximate tangent to µ to be an n-plane W ′ x ∈ G(d, n). Let W x := x + W ′ x , whenever the approximate tangent exists and is unique (that is for µ-a.e. x, by Theorem 1.2). In order to apply Tolsa's result in our setting, we need the following intuitively clear result.
Lemma 9.5. Let µ be a rectifiable measure. Then for µ-a.e. x ∈ supp µ we have A relatively simple (although lengthy) proof can be found in Appendix A. Before proving Proposition 9.1 we need one more lemma. Recall that if α > 0, W is an n-plane, and 0 < r < R, then K(x, W ⊥ , α, r, R) = K(x, W ⊥ , α, R) \ K(x, W ⊥ , α, r).
We need to show that dist(y, L) > ηr.
Letπ W andπ L denote the orthogonal projections onto the n-planes parallel to W and L passing through the origin. It follows from (9.3) that π W −π L op ≤ ε. Thus, Hence, using the fact that |π W (x − y)| = |x − π W (y)| < α|x − y|, we get from the two estimates above Proof of Proposition 9.1. Let µ be n-rectifiable. For r > 0 and x ∈ supp µ let L x,r be the n-plane minimizing β µ,2 (x, r). We know that for µ-a.e. x ∈ supp µ we have (9.1) and (9.2) (in particular, the approximate tangent plane W x exists). Fix such x. Set V x = W ⊥ x , let α ∈ (0, 1) be arbitrary, and for 0 < r < R set K(r) = K(x, V x , α, r), K(r, R) = K(x, V x , α, r, R). We will show that Let ε > 0 be a constant so small that η := 1 − α − 3ε > 0. Use Lemma 9.5 to find r 0 > 0 such that for 0 < r ≤ r 0 we have Then, it follows from Lemma 9.6 that for all 0 < r ≤ r 0 K(r, 2r) ⊂ B(x, 2r) \ B ηr (L x,r ).

Note that by Chebyshev's inequality
Hence, for 0 < r ≤ r 0 we have µ (K(r, 2r)) r n η β µ,2 (x, 2r) 2 , and so Now, observe that for any integer N > 0 To prove the above we will use techniques developed in [MO18b]. Fix V ∈ G(d, d − n). Let θ > 0 and M ∈ {0, 1, 2 . . . }. In the language of Martikainen and Orponen, a set E ⊂ R d has the n-dimensional (θ, M )-property if for all x ∈ E #{j ∈ Z : It is easy to see that if E has the n-dimensional (θ, 0)-property, then E is contained in a Lipschitz graph with Lipschitz constant bounded by 1/θ, see [MO18b,Remark 1.11].
The main proposition of [MO18b] reads as follows.
Remark 10.3. It follows immediately from the proposition above that if we construct F 1 ⊂ E ∩ B(0, 1) with H n (F 1 ) ≈ κ satisfying the n-dimensional (α/2, M )-property, then we will get a Lipschitz graph Γ such that (10.2) holds. Hence, we will be done with the proof of Proposition 10.1.
To construct F 1 we will use another lemma from [MO18b]. Note that the set F \ F ε does not have to be AD-regular. Nevertheless, we gain some extra regularity that will prove useful. Now, let E and F ⊂ E ∩ B(0, 1) be as in the assumptions of Proposition 10.1. We apply Lemma 10.4 to conclude that for some ε, depending on κ and the AD-regularity constant of E, we have H n (F \ F ε ) ≥ κ 2 .
We will show that if η is chosen small enough (depending on ε, the constant from (A.6)), then the estimate above leads to a contradiction. Roughly speaking, (A.8) means that a lot of measure is concentrated in the intersection of B ηr k (V ) and B ηr k (W ), but since V and W are somewhat well-separated by (A.6), this intersection behaves approximately like an (n − 1)-dimensional set.
Let us start by exploiting (A.6). By the definition of Hausdorff distance and the fact that V and W are n-planes, it follows from easy linear algebra that there exists some w ∈ W ⊥ with |w| = 1 and |π V (w)| ≥ ε. Let v 1 = π V (w)/|π V (w)|, and let V 0 ⊂ V be the orthogonal complement of span(v 1 ) in V .
We define T = T (k, η) to be a tube-like set defined as We claim that S(k, η) ⊂ T (k, η). Indeed, let z ∈ S. The estimate |π V 0 (z)| ≤ r k is trivial since S ⊂ B(0, r k ). The estimate |π ⊥ V (z)| ≤ ηr k follows from the fact that z ∈ B ηr k (V ). Concerning |z · v 1 |, note that since z ∈ B ηr k (W ) and w ∈ W ⊥ , we have |z · w| ≤ ηr k . We can use our choice of w and v 1 = π V (w)/|π V (w)| to get where in the last inequality we used again z ∈ B ηr k (V ). Thus, we have |z · v 1 | ≤ 2ηε −1 r k , and the proof of S(k, η) ⊂ T (k, η) is finished.
Choose η = γε for some tiny γ = γ(M ) > 0, and let k be large enough for (A.8) to hold. It follows from the definition of T that we can cover T with a family of balls {B i } i∈I such that r(B i ) = ηr k and #I ε −1 η −(n−1) . It is well-known that (A.2) implies that for all y ∈ R d and r > 0 we have ν(B(y, r)) ≤ M r n . In particular, for each i ∈ That is, This is a contradiction for γ = γ(M ) small enough. Hence, (A.4) is false, and so (A.1) holds for µ-a.e. x ∈ E M . Taking M → ∞ finishes the proof.