Plenty of big projections imply big pieces of Lipschitz graphs

I prove that a closed $n$-regular set $E \subset \mathbb{R}^{d}$ with plenty of big projections has big pieces of Lipschitz graphs. This answers a question of David and Semmes.

A set E Ă R d is called s-regular if E is closed, and the restriction of s-dimensional Hausdorff measure H s on E is an s-regular Radon measure. An n-regular set E Ă R d has big pieces of Lipschitz graphs (BPLG) if the following holds for some constants θ, L ą 0: for every x P E and 0 ă r ă diampEq, there exists an n-dimensional L-Lipschitz graph Γ Ă R d , which may depend on x and r, such that H n pBpx, rq X E X Γq ě θr n . (1.1) H n pπ V pBpx, rq X Eqq ě δr n . (1. 3) The set E has plenty of big projections (PBP) if (1.3) holds for all V P BpV x,r , δq.
In [13,Definition 1.12], the PBP condition was called big projections in plenty of directions. As noted above Definition 1.2, sets with BPLG have PBP. Conversely, one of the main results in [13] states that even the weaker "single big projection" condition BP is sufficient to imply BPLG if it is paired with the following a priori geometric hypothesis: Definition 1.4 (WGL). An n-regular set E Ă R d satisfies the weak geometric lemma (WGL) if for all ǫ ą 0 there exists a constant Cpǫq ą 0 such that the following (Carleson packing condition) holds: H n ptx P E X Bpx 0 , Rq : βpBpx, rqq ě ǫuq dr r ď CpǫqR, x 0 P E, 0 ă R ă diampEq.
In the definition above, the quantity βpBpx, rqq could mean a number of different things without changing the class of n-regular sets satisfying Definition 1.4. In the current paper, the most convenient choice is βpBpx, rqq :" β 1 pBpx, rqq :" inf V PApd,nq 1 r n ż Bpx,rq distpy, V q r dµpyq with µ :" H n | E , and where Apd, nq is the "affine Grassmannian" of all n-dimensional planes in R d . The β-number above is an "L 1 -variant" of the original "L 8 -based β-number" introduced by Jones [20], namely If E Ă R d is n-regular, then the following relation holds between the two β-numbers: For a proof, see [11, p. 28]. This inequality shows that the WGL, a condition concerning all ǫ ą 0 simultaneously, holds for the numbers βpBpx, rqq if and only if it holds for the numbers β 8 pBpx, rqq. After these preliminaries, the result of David and Semmes [13,Theorem 1.14] can be stated as follows: Theorem 1.5 (David-Semmes). An n-regular set E Ă R d has BPLG if and only if E has BP and satisfies the WGL.
The four corners Cantor set has BP (find a direction where the projections of the four boxes tile an interval) but fails to have BPLG, being purely 1-unrectifiable. This means that the WGL hypothesis cannot be omitted from the previous statement. However, the four corners Cantor set fails to have PBP, by the Besicovitch projection theorem [5], which states that almost every projection of a purely 1-rectifiable set of σ-finite length has measure zero. The main result of this paper shows that PBP alone implies BPLG: Theorem 1.6. Let E Ă R d be an n-regular set with PBP. Then E has BPLG.
To prove Theorem 1.6, all one needs to show is that PBP ùñ WGL.
The rest then follows from the work of David and Semmes, Theorem 1.5.
1.1. Connection to uniform rectifiability. The BPLG property is a close relative of uniform n-rectifiability, introduced by David and Semmes [11] in the early 90s. An n-regular set E Ă R d is uniformly n-rectifiable, n-UR in brief, if (1.1) holds for some n-dimensional L-Lipschitz images Γ " f pBp0, rqq, with Bp0, rq Ă R n , instead of n-dimensional L-Lipschitz graphs. As shown by David and Semmes in [11,12], the n-UR property has many equivalent, often surprising characterisations: for example, singular integrals with odd ndimensional kernels are L 2 -bounded on an n-regular set E Ă R d if and only if E is n-UR. Since its conception, the study of uniform (and, more generally, quantitative) rectifiability has become an increasingly popular topic, for a good reason: techniques in the area have proven fruitful in solving long-standing problems on harmonic measure and elliptic PDEs [2,3,18,29], theoretical computer science [26], and metric embedding theory [27]. This list of references is hopelessly incomplete! Since n-dimensional Lipschitz graphs can be written as n-dimensional Lipschitz images, n-regular sets with BPLG are n-UR. In particular, Theorem 1.6 implies that nregular sets with PBP are n-UR. The converse is false: Hrycak (unpublished) observed in the 90s that a simple iterative construction can be used to produce 1-regular compact sets K ǫ Ă R 2 , ǫ ą 0, with the properties (a) H 1 pK ǫ q " 1 and H 1 pπ L pK ǫ qq ă ǫ for all L P Gp2, 1q, (b) K ǫ is 1-UR with constants independent of ǫ ą 0. This means that UR sets do not necessarily have PBP, or at least bounds for n-UR constants do not imply bounds for PBP constants. The details of Hrycak's construction are contained in the appendix of Azzam's paper [1], but they can also be outlined in a few words: pick n :" tǫ´1u. Sub-divide I 0 :" r0, 1sˆt0u Ă R 2 into n segments I 1 , . . . , I n of equal length, and rotate them individually counter-clockwise by 2π{n. Then, sub-divide each I j into n segments of equal length, and rotate by 2π{n again. Repeat this procedure n times to obtain a compact set K n " K ǫ consisting of n n segments of length n´n. It is not hard to check that (a) and (b) hold for K ǫ . In particular, to check (b), one can easily cover K ǫ by a single 1-regular continuum Γ Ă Bp0, 2q of length H 1 pΓq ď 10.

Previous and related work.
It follows from the Besicovitch-Federer projection theorem [5,17] that an n-regular set with PBP is n-rectifiable. The challenge in proving Theorem 1.6 is to upgrade this "qualitative" property to BPLG. For general compact sets in R 2 of finite 1-dimensional measure, a quantitative version of the Besicovitch projection theorem is due to Tao [34]. It appears, however, that Theorem 1.6 does not follow from his work, not even in R 2 . Another, more recent, result for general n-regular sets is due to Martikainen and myself [22]: the main result of [22] shows that BPLG is equivalent to a property (superficially) stronger than PBP. This property roughly states that the π Vprojections of the measure H n | E lie in L 2 pV q on average over V P B Gpd,nq pV 0 , δq. One of the main propositions from [22] also plays a part in the present paper, see Proposition 6.4. Interestingly, while the main result of the current paper is formally stronger than the result in [22], the new proof does not supersede the previous one: in [22], the L 2 -type assumption in a fixed ball was used to produce a big piece of a Lipschitz graph in the very same ball. Here, on the contrary, PBP needs to be employed in many balls, potentially much smaller than the "fixed ball" one is interested in. Whether this is necessary or not is posed as Question 1 below. Besides Tao's paper mentioned above, there is plenty of recent activity around the problem of quantifying Besicovitch's projection theorem, that is, showing that "quantitatively unrectifiable sets" have quantifiably small projections. As far as I know, Tao's paper is the only one dealing with general sets, while other authors, including Bateman, Bond, Łaba, Nazarov, Peres, Solomyak, and Volberg have concentrated on self-similar sets of various generality [4,6,7,8,21,28,30]. In these works, strong upper (and some surprising lower) bounds are obtained for the Favard length of the k th iterate of self-similar sets. In the most recent development [10], Cladek, Davey, and Taylor considered the Favard curve length of the four corners Cantor set.
Quantifying the Besicovitch projection theorem is related to an old problem of Vitushkin. The remaining open question is to determine whether arbitrary compact sets E Ă R 2 of positive Favard length have positive analytic capacity. It seems unlikely that the method of the present paper would have any bearing on Vitushkin's problem, but the questions are not entirely unrelated either: I refer to the excellent introduction in the paper [9] of Chang and Tolsa for more details.
Finally, Theorem 1.6 can be simply viewed as a characterisation of the BPLG property, of which there are not many available -in contrast to uniform rectifiability, which is charaterised by seven conditions in [11] alone! I already mentioned that BPLG is equivalent to BP+WGL by [13], and that with Martikainen [22], we characterised BPLG via the L 2 -norms of the projections π V 7 H n | E . Another, very recent, characterisation of BPLG, in terms of conical energies, is due to Dąbrowski [14].

1.
3. An open problem. An answer to the question below does not seem to follow from the method of this paper.

Question 1.
For all δ ą 0 and C 0 ě 1, do there exist L ě 1 and θ ą 0 such that the following holds? Whenever E Ă R d is an n-regular set with regularity constant at most C 0 , and then there exists an n-dimensional L-Lipschitz graph Γ Ă R d such that H n pE X Γq ě θ.
In addition to the "single scale" assumption (1.7), the proof of Theorem 1.6 requires information about balls much smaller than Bp0, 1q to produce the Lipschitz graph Γ.

1.4.
Notation. An open ball in R d with centre x P R d and radius r ą 0 will be denoted Bpx, rq. When x " 0, I sometimes abbreviate Bpx, rq ": Bprq. The notations radpBq and diampBq mean the radius and diameter of a ball B Ă R d , respectively, and λB :" Bpx, λrq for B " Bpx, rq and λ ą 0.
For A, B ą 0, the notation A p 1 ,...,p k B means that there exists a constant C ě 1, depending only on the parameters p 1 , . . . , p k , such that A ď CB. Very often, one of these parameters is either the ambient dimension "d", or then the PBP or n-regularity constant "δ" or "C 0 " of a fixed n-regular set E Ă R d having PBP, that is, satisfying the hypotheses of Theorem 1.6. In these cases, the dependence is typically omitted from the notation: in other words, A d,δ,C 0 B is abbreviated to A B. The two-sided inequality A p B p A is abbreviated to A " p B, and A p B means the same as B p A.
1.5. Acknowledgements. I would like to thank Michele Villa for useful conversations, and Alan Chang for pointing out a mistake in the proof of Lemma 5.17 in an earlier version of the paper. I'm also grateful to Damian Dąbrowski for reading the paper carefully and giving many useful comments. Finally, I am grateful to the anonymous reviewers for their careful reading, and for spotting a large number of small inaccuracies.

PRELIMINARIES ON THE GRASSMANNIAN
Before getting started, we gather here a few facts of the Grassmannian Gpd, nq of ndimensional subspaces of R d . Here 0 ď n ď d, and the extreme cases are Gpd, 0q " t0u and Gpd, dq " tR d u. We equip Gpd, nq with the metric where }¨} refers to operator norm. That "d" means two different things here is regrettable, but the correct interpretation should always be clear from context, and the metric "d" will only be used very occasionally. The metric space pGpd, nq, dq is compact, and open balls in Gpd, nq will be denoted B Gpd,nq pV, rq. An equivalent metric on Gpd, nq is given bȳ dpV 1 , V 2 q :" maxtdistpv 1 , V 2 q : v 1 P V 1 and |v 1 | " 1u. For a proof, see [25,Lemma 4.1]. With the equivalence of d andd in hand, we easily infer the following auxiliary result: Lemma 2.1. Let 0 ă n ă d, and let W 1 , W 2 P Gpd, n`1q, and let V 1 P Gpd, nq with V 1 Ă W 1 . Then, there exists V 2 P Gpd, nq such that V 2 Ă W 2 and dpV 1 , V 2 q dpW 1 , W 2 q.
Proof. By the equivalence of d andd, we have r :"dpW 1 , W 2 q dpW 1 , W 2 q. We may assume that r is small, depending on the ambient dimension, otherwise any n-dimensional subspace V 2 Ă W 2 satisfies dpV 1 , V 2 q ď diam Gpd, nq r. Now, let te 1 , . . . , e n u be an orthonormal basis for V 1 , and for all e j P V 1 Ă W 1 , pick someē j P W 2 with |e j´ēj | ď r. If r ą 0 is small enough, the vectorsē 1 , . . . ,ē n are linearly independent, hence span an n-dimensional subspace V 2 Ă W 2 . Since |e j´ēj | ď r for all 1 ď j ď n, an arbitrary unit vector v 1 " ř β j e j P V 1 lies at distance r from v 2 :" ř β jēj P V 2 , and consequently This completes the proof.
We will often use the standard "Haar" probability measure γ d,n on Gpd, nq. Namely, let θ d be the Haar measure on the orthogonal group Opdq, and define where V 0 P Gpd, nq is any fixed subspace. The measure γ d,n is the unique Opdq-invariant Radon probability measure on Gpd, nq, see [23, §3.9]. At a fairly late stage of the proof of Theorem 1.6, we will need the following "Fubini" theorem for the measure Gpd, nq: Then GpW, nq can be identified with Gpn`1, nq, and we equip GpW, nq with the Haar measure γ W,n`1,n :" γ n`1,n , constructed as above. Then, the following holds for all Borel sets B Ă Gpd, nq: γ d,n pBq " ż Gpd,n`1q γ W,n`1,n pBq dγ d,n`1 pW q.
Proof. This is the same argument as in [23,Lemma 3.13]: one simply checks that both sides of (2.3) define Opdq-invariant probability measures on γ d,n , and then appeals to the uniqueness of such measures.
Proof. The proof of [ [13, §2]) that an n-regular set E Ă R d supports a system D of "dyadic cubes", that is, a collection of subset of E with the following properties. First, D can be written as a disjoint union where the elements Q P D j are referred to as cubes of side-length 2´j. For j P Z fixed, the sets of D j are disjoint and cover E. For Q P D j , one writes ℓpQq :" 2´j. The sidelength ℓpQq is related to the geometry of Q P D j in the following way: there are constants 0 ă c ă C ă 8, and points c Q P Q Ă E (known as the "centres" of Q P D) with the properties Bpc Q , cℓpQqq X E Ă Q Ă Bpc Q , CℓpQqq. In particular, it follows from the n-regularity of E that µpQq " ℓpQq n for all Q P D. The balls Bpc Q , CℓpQqq containing Q are so useful that they will have an abbreviation: If we choose the constant C ě 1 is large enough, as we do, the balls B Q have the property The "dyadic" structure of the cubes in D is encapsulated by the following properties: ‚ For all Q, Q 1 P D, either Q Ă Q 1 , or Q 1 Ă Q, or Q X Q 1 " H. ‚ Every Q P D j has as parentQ P D j´1 with Q ĂQ. If Q P D j , the cubes in D j`1 whose parent is Q are known as the children of Q, denoted chpQq. The ancestry of Q consists of all the cubes in D containing Q.
A small technicality arises if diampEq ă 8: then the collections D j are declared empty for all j ă j 0 , and D j 0 contains a unique element, known as the top cube of D. All of the statements above hold in this scenario, except that the top cube has no parents.

Dyadic reformulations of PBP and WGL.
Let us next reformulate some of the conditions familiar from the introduction in terms of a fixed dyadic system D on E. Definition 3.1 (PBP). An n-regular set E Ă R d has PBP if there exists δ ą 0 such that the following holds. For all Q P D, there exists a ball S Q Ă Gpd, nq of radius radpS Q q ě δ such that H n pπ V pE X B Q qq ě δµpQq, V P S Q .
It is easy to see that the dyadic PBP is equivalent to the continuous PBP: in particular, the dyadic PBP follows by applying the continuous PBP to the ball B Q " Bpc Q , CℓpQqq centred at c Q P E. Only the dyadic PBP will be used below. Definition 3.2 (WGL). An n-regular set E Ă R d satisfies the WGL if for all ǫ ą 0, there exists a constant Cpǫq ą 0 such that the following holds: Here µ :" H n | E , βpQq :" βpB Q q, and DpQ 0 q :" tQ P D : Q Ă Q 0 u.
It is well-known, but takes a little more work to show, that the dyadic WGL is equivalent to the continuous WGL; this fact is stated without proof in numerous references, for example [13, (2.17)]. I also leave the checking to the reader.
One often wishes to decompose D, or subsets thereof, into trees: . Let E Ă R d be an n-regular set with associated dyadic system D.
A collection T Ă D is called a tree if the following conditions are met: ‚ T has a top cube QpT q P T with the property that Q Ă QpT q for all Q P T . ‚ T is consistent: if Q 1 , Q 3 P T , Q 2 P D, and Q 1 Ă Q 2 Ă Q 3 , then Q 2 P T . ‚ If Q P T , then either chpQq Ă T or chpQq X T " H.
The final axiom allows to define the leaves of T consistently: these are the cubes Q P T such that chpQq X T " H. The leaves of T are denoted LeavespT q. The collection LeavespT q always consists of disjoint cubes, and it may happen that LeavespT q " H. Some trees will be used to prove the following reformulation of the WGL: Lemma 3.4. Let E Ă R d be an n-regular set supporting a collection D of dyadic cubes. Let µ :" H n | E . Assume that for all ǫ ą 0, there exists N " N pǫq P N such that the following holds: µptx P Q : cardtQ 1 P D : x P Q 1 Ă Q and βpQ 1 q ě ǫu ě N uq ď 1 2 µpQq, Q P D. (3.5) Then E satisfies the WGL.
Proof of Lemma 3.4. Fix Q 0 P D and ǫ ą 0. We will show that Abbreviate D :" tQ P D : Q Ă Q 0 u, and decompose D into trees by the following simple stopping rule. The first tree T 0 has top QpT 0 q " Q 0 , and its leaves are the maximal cubes Q P D (if any should exist) such that cardtQ 1 P D : Q Ă Q 1 Ă Q 0 and βpQ 1 q ě ǫu " N. Here N " N pǫq ě 1, as in (3.5). All the children of previous generation leaves are declared to be new top cubes, under which new trees are constructed by the same stopping condition. Let T 0 , T 1 , . . . be the trees obtained by this process, with top cubes Q 0 , Q 1 , . . . Note that D " Ť jě0 T j , and cardtQ P T j : x P Q and βpQq ě ǫu ď N, x P Q j . Further, (3.5) implies that µpYLeavespT j qq ď 1 2 µpQ j q, j ě 0. On the other hand, the sets E j :" Q j z Y LeavespT j q are disjoint. Now, we may estimate as follows: This completes the proof of (3.7). By Theorem 1.5, the PBP condition together with the WGL implies BPLG, and the condition in Lemma 3.4 is a reformulation of the WGL. Therefore, our main result, Theorem 1.6, will be a consequence of the next proposition: Proposition 3.8. Assume that E Ă R d is an n-regular set with PBP. Then, for every ǫ ą 0, there exists N ě 1, depending on d, ǫ, and the n-regularity and PBP constants of E, such that the following holds. The sets E Q :" E Q pN, ǫq :" x P Q : cardtQ 1 P D : x P Q 1 Ă Q and βpQ 1 q ě ǫu ě N ( satisfy µpE Q q ď 1 2 µpQq for all Q P D.
Proving this proposition will occupy the rest of the paper.

CONSTRUCTION OF HEAVY TREES
The proof of Proposition 3.8 proceeds by counter assumption: there exists a cube Q 0 P D, a small number ǫ ą 0, and a large number N ě 1 of the form N " KM , where also K, M ě 1 are large numbers, with the property This will lead to a contradiction if both K and M are large enough, depending on d, ǫ, and the n-regularity and PBP constants of E. Precisely, M ě 1 gets chosen first within the proof of Proposition 4.2. The parameter K ě 1 is chosen second, and depends also on M . For the details, see the proof of Proposition 3.8, which can be found around (4.3).
From now on, we will restrict attention to sub-cubes of Q 0 , and we abbreviate D :" DpQ 0 q. We begin by using (4.1), and the definition of E Q 0 , to construct a number of heavy trees T 0 , T 1 , . . . Ă D with the following properties: (T1) µpE Q 0 X QpT j qq ě 1 4 µpQpT j qq for all j ě 0. (T2) E Q 0 X QpT j q Ă YLeavespT j q for all j ě 0. (T3) For every j ě 0 and Q P LeavespT j q it holds cardtQ 1 P T j : Q Ă Q 1 Ă QpT j q and βpQ 1 q ě ǫu " M.
(T4) The top cubes satisfy ř j µpQpT j qq ě K 4 µpQ 0 q. Before constructing the trees with properties (T1)-(T4), let us use them, combined with some auxiliary results, to complete the proof of Proposition 3.8. The first ingredient is the following proposition: Proposition 4.2. If the parameter M ě 1 is large enough, depending only on d, ǫ, and the nregularity and PBP constants of E, then widthpT j q ě τ µpQpT j qq, where τ ą 0 depends only on d, and the n-regularity and PBP constants of E.
Here widthpT j q " ř QPT j widthpQqµpQq is a quantity to be properly introduced in Section 5. For now, we only need to know that the coefficients widthpQq satisfy a Carleson packing condition, depending only on the n-regularity constant of E: We may then prove Proposition 3.8: Proof of Proposition 3.8. Let N " KM , where M ě 1 is chosen so large that the hypothesis of Proposition 4.2 is met: every heavy tree T j satisfies widthpT j q ě τ µpQpT j qq. According to (T4) in the construction of the heavy trees, this implies Now, the lower bound in (4.3) violates the Carleson packing condition for widthpDq if the constant K ě 1 is chosen large enough, depending on the admissible parameters. The proof of Proposition 3.8 is complete.
The rest of this section is spent constructing the heavy trees. We first construct a somewhat larger collection, and then prune it. In fact, the construction of the larger collection is already familiar from the proof of Lemma 3.4, with notational changes: the first tree T 0 has top QpT 0 q " Q 0 , and its leaves consist of the maximal cubes Q P D with the property that cardtQ 1 P D : Q Ă Q 1 Ă QpT 0 q and βpQ 1 q ě ǫu " M.
(4.4) The tree T 0 itself consists of the cubes in D which are not strict sub-cubes of some Q P LeavespT 0 q. It is easy to check that T 0 is a tree.
Assume then that some trees T 0 , . . . , T k have already been constructed. Let 0 ď j ď k be an index such that for some Q P LeavespT j q, at least one cube Q k`1 P chpQq has not yet been assigned to any tree. The cube Q k`1 then becomes the top cube of a new tree T k`1 , thus Q k`1 " QpT k`1 q. The tree T k`1 is constructed with the same stopping condition (4.4), just replacing QpT 0 q by Q k`1 " QpT k`1 q.
Note that if LeavespT j q " H for some j P N, then no further trees will be constructed with top cubes contained in QpT j q. As a corollary of the stopping condition, we record the uniform upper bound We next prune the collection of trees. Let Top be the collection of all the top cubes QpT j q constructed above, and let Top K Ă Top be the maximal cubes with the property We discard all the trees whose tops are strictly contained in one of the cubes in Top K , and we re-index the remaining trees as T 0 , T 1 , T 2 , . . . Thus, the remaining trees are the ones whose top cube contains some element of Top K . We record that We write T :" YT j for brevity. We claim that cardtQ P T : x P Q and βpQq ě ǫu " N, Indeed, fix x P E Q 0 , and recall that cardtQ P D : x P Q and βpQq ě ǫu ě N (4.8) by definition. We first claim that x is contained in ě K`1 cubes in Top. If x was contained in ď K cubes in Top, then x would be contained in ď K´1 distinct leaves, and the stopping condition (4.4) would imply that cardtQ P D : x P Q and βpQq ě ǫu ă pK´1qM`M " N, suffices to note that whenever x P Q j , 1 ď j ď K, then x is contained in some element of LeavespT j´1 q, which implies by the stopping condition that cardtQ P T j´1 : x P Q and βpQq ě ǫu " M. Since T j´1 Ă T for 1 ď j ď K, the claim (4.7) follows by summing up (4.10) over 1 ď j ď K and recalling that KM " N . We next verify that E Q 0 X QpT j q Ă YLeavespT j q for all j ě 0, as claimed in property (T2). Indeed, if x P E Q 0 X QpT j q for some j ě 0, then (4.8) holds, and QpT j q is contained in ď K elements of Top. This means that if x P QpT j q z Y LeavespT j q, then x is contained in ď K´1 distinct leaves, and hence satisfies (4.9). But this would imply x R E Q 0 . Hence x P LeavespT j q, as claimed.
The properties (T2)-(T3) on the list of requirements have now been verified (indeed (T3) holds by the virtue of the stopping condition). For (T1) and (T4), some further pruning will be needed. First, from (4.7), (4.5), and the assumption µpE Q 0 q ě 1 2 µpQ 0 q, we infer that Recalling that N " KM , this yields Now, we discard all light trees with the property µpE Q 0 X QpT j qq ă 1 4 µpQpT j qq. Then, by the uniform upper bound (4.6), we have Hence, the heavy trees with By definition of the heavy trees, the requirements (T1) and (T4) on our list are satisfied (and (T2)-(T3) were not violated by the final pruning, since they are statements about individual trees). After another re-indexing, this completes the construction of the heavy trees T 0 , T 1 , . . . We have now proven Proposition 3.8 modulo Proposition 4.2, which concerns an individual heavy tree T j . Proving Proposition 4.2 will occupy the rest of the paper.

A CRITERION FOR POSITIVE WIDTH
Let E Ă R d be a closed n-regular set, write µ :" H n | E , and let D be a system of dyadic cubes on E. I next discuss the notion of width, which appeared in the statement of Proposition 4.2. Width was first introduced in [16] in the context of Heisenberg groups, and [16, §8] contains the relevant definitions adapted to R n , but only in the case n " d´1. I start here with the higher co-dimensional generalisation.
The definition above is standard, see [23, §3.16]. We are interested in the case m " d´n, since we plan to slice sets by the fibres of projections to planes in Gpd, nq.
Definition 5.2 (Width). For Q P D and a plane W P Apd, d´nq, we define where we recall that B Q " Bpc Q , CℓpQqq is a ball centred at some point c Q P Q Ă E containing Q. Then, we also define Finally, if F Ă D is an arbitrary collection of dyadic cubes, we set widthpFq :" The µpQq-normalisation in (5.3) is the right one, because for V P Gpd, nq fixed, it is only possible that width Q pE, π´1 V twuq ‰ 0 if w P π V pB Q q Ă V , and H n pπ V pB Q qq " µpQq. As shown in [16,Theorem 8.8], width satisfies a Carleson packing condition. However, the proof in [16] was restricted to the case d " n´1, and a little graph-theoretic construction is needed in the higher co-dimensional situation. Details follow.
Proposition 5.5. There exists a constant C ě 1, depending only on the 1-regularity constant of E, such that widthpDpQ 0 qq ď CµpQ 0 q, Q 0 P D, where DpQ 0 q :" tQ P D : The main tool in the proof is Eilenberg's inequality where A Ă R d is Borel, see [23,Theorem 7.7]. In particular, we infer from (5.8) that for all V P Gpd, nq and for H n a.e. w P V . We continue our estimate of (5.7) for a fixed plane V P Gpd, nq, and for any w P V such that q :" q V,w ă 8. If q P t0, 1u, then Q P DpQ 0 q, so these pairs pV, wq contribute nothing to the integral in (5.7). So, assume that q ě 2, and enumerate the points in We will next need to construct a "spanning graph" whose vertices are the points x 1 , . . . , x q , and whose edges "E" are a (relatively small) subset of the " q 2 segments connecting the vertices. More precisely, we need the following properties from E: Property (E2) sounds like quasiconvexity, but is weaker: there are no restrictions on the length of the connecting E-path, as long as it is contained in Bpx i , 2|x i´xj |q. Let us then find the edges with the properties (E1)-(E2). Let ξ 1 , . . . , ξ p Ă S d´1 be a maximal 1 4 -separated set on S d´1 , with p " d 1, and let C j :" tre : e P Bpξ j , 1 2 q X S d´1 and r ą 0u, 1 ď j ď p, be a directed open cone around the half-line trξ j : r ą 0u. By the net property of ξ 1 , . . . , ξ p , We claim that the following holds: if y P x`C j , then Bpx, |x´y|q X px`C j q Ă Bpy, |x´y|q. (5.10) First, use translations and dilations to reduce to the case x " 0 and |x´y| " 1: To check this case, one first verifies by explicit computation that if y P S d´1 , then the set C y :" tre : e P Bpy, 1q X S d´1 and 0 ă r ď 1u is contained in Bpy, 1q. Consequently, We are then prepared to define the edge set E. Fix one of the points x i , 1 ď i ď q. For every of 1 ď j ď p, draw an edge (that is, a segment) between x i and one of the points closest to x i in the finite set if the intersection on the left hand side is non-empty; this is the case for at least one j P t1, . . . , pu by (5.9). Thus, for every x i , one draws " d 1 edges. Let E be the collection of all edges so obtained. Then card E " d q, so requirement (E1) is met.
To prove (E2), fix s 0 :" x i and t :" x j with 1 ď i ă j ď q. The plan is to find, recursively, a collection of segments I j :" rs j´1 , s j s P E, 1 ď j ď k, whose union is connected, contains ts 0 , tu (indeed s k " t) and is contained in Bpt, |s 0´t |q ĂBps, 2|s 0´t |q.
By (5.9), there is a half-cone C j 1 with t P s 0`Cj 1 . Let I 1 " rs 0 , s 1 s P E be the edge connecting s 0 to one of the nearest points s 1 P tx 1 , . . . , x q u X ps 0`Cj 1 q. Evidently |s 0ś 1 | ď |s 0´t |, since t P tx 1 , . . . , x q uX ps 0`Cj 1 q itself is one of the candidates among which s 1 is chosen. Hence, applying (5.10) with x " s 0 and y " t, we find that s 1 PBps 0 , |s 0´t |q X ps 0`Cj 1 q Ă Bpt, |s 0´t |q. (5.11) In particular, Also, we see from (5.11) that BI 1 " ts 0 , s 1 u ĂBpt, |s 0´t |q, and hence I 1 ĂBpt, |s 0´t |q by convexity. We then replace "s 0 " by "s 1 " and repeat the procedure above: by (5.9), there is a half-cone C j 2 with the property t P s 1`Cj 2 (unless s 1 " t and we are done already), and we let I 2 " rs 1 , s 2 s P E be the edge connecting s 1 to the nearest point s 2 P tx 1 , . . . , x q u X ps 1`Cj 2 q. Then |s 1´s2 | ď |s 1´t | (otherwise we chose t over s 2 ), so Ă Bpt, |s 1´t |q (5.12) ĂBpt, |s 0´t |q.
We proceed inductively, finding further segments rs i , s i`1 s P E, which are contained in Bpt, |s 0´t |q, and with the property that |s j`1´t | ă |s j´t | ă . . . ă |s 0´t |. Since the points s j are drawn from the finite set tx 1 , . . . , x q u, these strict inequalities eventually force s k " t for some k ě 1, and at that point the proof of property (E2) is complete. Let us then use the edges E constructed above to estimate the integrand in (5.7). I claim that To see this, fix Q P DpQ 0 q, and let x i , x j P B Q X E X π´1 V twu Ă tx 1 , . . . , x q u be points such that According to property (E2) of the edge family E, there exists a connected union of segments in E which is contained in and which contains tx i , x j u. Since the union is connected, the total length of the segments involved exceeds |x i´xj |: Swapping the order of summation proves (5.13). To complete the proof of the proposition, fix I P E, and consider the inner sum in (5.13). Note that the inclusion I Ă 4B Q is only possible if ℓpQq |I|. On the other hand, for a fixed side-length 2´j |I|, there are 1 cubes Q P DpQ 0 q with ℓpQq " 2´j and I Ă 4B Q . Putting these observations together, From this, (5.13), and the cardinality estimate card E d q from (E1) it follows that Plugging this estimate into (5.7) and using Eilenberg's inequality (5.8), one finds that This completes the proof of the proposition.
Recall that our objective, in Proposition 4.2, is to prove that each heavy tree T j satisfies widthpT j q µpQpT j qq if the parameter M ě 1 was chosen large enough. To accomplish this, we start by recording a technical criterion which guarantees that a general tree T Ă D satisfies widthpT q µpQpT qq. Afterwards, the criterion will need to be verified for heavy trees.
Proposition 5.14. For every c, δ ą 0 and C 0 ě 1 there exists N ě 1 such that the following holds. Assume that the n-regularity constant of E is at most C 0 . Let T Ă D be a tree with top cube Q 0 :" QpT q. Assume that there is a subset G Ă LeavespT q with the following properties.
‚ All the cubes in G have PBP with common plane V 0 P Gpd, nq and constant δ: Assume that there is a subset S G Ă BpV 0 , δq such that the "high multiplicity" sets H V :" tx P V : Here Mf V is the (centred) Hardy-Littlewood maximal function of f V . Then where the implicit constant only depends on "d" and the n-regularity constant of E.
The proof of Proposition 5.14 would be fairly simple if all the leaves in G had approximately the same generation in D. In our application, this cannot be assumed, unfortunately, and we will need another auxiliary result to deal with the issue: Then, there exists a collection R heavy of disjoint cubes such that the "sub-functions" satisfy the following properties: The lemma is easy in the case where the balls in B have common radius, say r. Then one can take R heavy to be a suitable collection of disjoint cubes of side-length " r. In the application to Proposition 5.14, this case corresponds to the situation where ℓpQq " ℓpQ 1 q for all Q, Q 1 P G. In the general case, the elementary but lengthy proof of Lemma 5.17 is contained in Appendix A.
We then prove Proposition 5.14, taking Lemma 5.17 for granted: Proof of Proposition 5.14. The plan is to show that The proposition then follows by recalling the definitions of widthpQq and widthpT q from (5.3)-(5.4) and integrating (5.19) over V P S G .
To prove (5.19), we assume, to avoid a rescaling argument, that ℓpQ 0 q " 1. Then, we begin by re-interpreting (5.16) in such a way that we may apply Lemma 5.17. Namely, we identify V P S G with R n , and consider the collection of balls B :" tπ V pB Q q : Q P Gu.
More precisely, let B be an index set for the balls π V pB Q q such that if some ball B " π V pB Q q arises from multiple distinct cubes Q P G, then B has equally many indices in B.
Note that the balls in B are all contained in We then define f :" ř BPB 1 B and H N :" tx P V : Mf pxq ě N u. It follows from (5.16), and the assumption ℓpQ 0 q " 1, that In other words, the hypotheses of Lemma 5.17 are met with γ " 1. We fix M :" Cδ´1, where C ě 1 is a large constant to be specified soon, depending only on the n-regularity constant of E. We then assume that N ą AM 3 {c, in accordance with (5.18). Lemma 5.17 now provides us with a collection R " R heavy of disjoint cubes in R n -V such that ÿ RPR }f R } 1 cN´1 and }f R } 1 ě M |R| for R P R. (5.20) In this proof we abbreviate |¨| :" H n | V . We recall that where T pRq :" π´1 V pRq. Therefore, the conditions in (5.20) are equivalent to where the implicit constants depend on the n-regularity constant of µ. We now make a slight refinement to the set G: for R P R fixed, we apply the 5r-covering theorem to the balls t2B Q : Q P G and B Q Ă T pRqu. As a result, we obtain a sub-collection G R Ă G with the properties and ď In particular, by (5.21), by (5.21). We also write B R :" tπ V pB Q q : Q P G R u, R P R, so B R Ă B is a collection of balls contained in R satisfying Just like B, the set B R should also, to be precise, be defined as a set of indices, accounting for the possibility that B " π V pB Q q arises from multiple cubes Q P G R . Next, recall a key assumption of the proposition, namely that all the cubes in G have PBP with common ball BpV 0 , δq Ă Gpd, nq. In particular, for our fixed plane V P S G Ă BpV 0 , δq, we have Since the balls B Q , Q P G, are all contained in B 0 :" B Q 0 , the ball associated with the top cube of the tree, the conclusion of (5.26) persists if we replace Then, for R P R fixed, we infer from (5.25) that We now choose the constant C ě 1 so large that Then, if we consider the "set of multiplicity ď 1", we may infer from (5.27) that Consequently, if P R :" R z L R is the "positive multiplicity set", we have (If the sum happens to equal 8, pick m ě 2 arbitrary; eventually one will have to let m Ñ 8 in this case). Unraveling the definitions, the pd´nq-plane W :" W w :" π´1 V twu contains m points of E X B 0 inside m distinct balls B Q , with Q P G R . Let P Ă E X W be the set of these m points, and define the following set E of edges connecting (some) pairs of points in P : for every point p P P , pick exactly one of the points q P P z tpu at minimal distance from p, and add the edge pp, qq to E. Note that card E " m, since E contains precisely one edge of the form pp, qq for every p P P . We have now used the assumption m ě 2: otherwise we could not have drawn any edges in the preceding manner! Note that the edges in the graph pP, Eq are directed: pp, qq P E does not imply pq, pq P E. Now that the edge set E has been constructed, define the following relation between edges I P E and the cubes Q P T : write I ă Q if I Ă B Q , and |I| ě ρℓpQq. Slightly abusing notation, here I also refers to the segment rp, qs, for an edge pp, qq P E. The choice of the constant ρ ą 0 will become apparent soon, and it will only depend on the n-regularity constant of E. We now claim that We already know that card E " m, so it remains to prove the first inequality. Fix I " pp, qq P E, with p, q P P . Then, by the definition of P , the points p and q are contained in two balls B p :" B Qp and B q :" B Qq , respectively, with Q p , Q q P G R and Q p ‰ Q q . In particular, we recall from (5.22) that 2B p X 2B q " H. Hence p R 2B q , and |I| ℓpQ q q. On the other hand, p, q P B 0 , so |I| ℓpQ 0 q. Let Q 1 Ą Q q be the smallest cube in the ancestry of Q q such that p, q P B Q 1 . Then Q q Q 1 Ă Q 0 , hence Q 1 P T , and Since p, q P B Q 1 , by convexity also I Ă B Q 1 . If the constant "ρ" in the definition of "ă" was chosen appropriately, we infer from I Ă B Q 1 and (5.30) that I ă Q 1 . This proves the lower bound in (5.29). Next, we claim that Indeed, fix Q P T and assume that there is at least one edge I P E such that I ă Q.
Then I Ă B Q X W , and both endpoints of I lie in E, so diampE X B Q X W q ě |I|. Thus, (5.31) boils down to showing that cardtI P E : I ă Qu d 1. Let P Q :" tp P P : pp, qq P E and pp, qq ă Q for some q P P z tpuu. Then since E contains precisely one edge of the form pp, qq for all p P P , i.e. the map I " pp, qq Þ Ñ p is injective tI P E : I ă Qu Ñ P Q . So, it remains to argue that card P Q d 1.
Otherwise, if card P Q " d 1, there exist two distinct points p 1 , p 2 P P Q with |p 1´p2 | ă ρℓpQq. However, if q P P is such that I :" pp 1 , qq ă Q, then |I| ě ρℓpQq, and since pp 1 , qq P E, the point q must be one of the nearest neighbours of p in P z tpu. This is not true, however, since |p 1´p2 | ă |p 1´q |. We have proven (5.31). A combination of (5.29) and (5.31) leads to Here P R is the subset of R introduced above (5.28). Integrating over w P R next gives Finally, summing the result over the (disjoint) cubes R P R, and using (5.23), we find that This completes the proof of (5.19), and the proof of the proposition.

FROM BIG β NUMBERS TO HEAVY CONES
Proposition 5.14 contains criteria for showing that widthpT q µpQpT j qq. To prove Proposition 4.2, these criteria need to be verified for the heavy trees T j . The selling points (T1)-(T4) of a heavy tree T j were that all of its leaves are contained in M cubes in T j with non-negligible β-number (see (T3)), and the total µ measure of the leaves is at least 1 4 µpQpT j qq (see (T1)-(T2)). We will use this information to show that if a reasonably wide cone is centred at a typical point x contained in one of the leaves of T j , then the cone intersects many other leaves at many different (dyadic) distances from x.
We first need to set up our notation for cones: Definition 6.1 (Cones). Let V 0 P Gpd, nq, α ą 0, and x P R d . We write Xpx, V, αq " ty P R d : |π V px´yq| ď α|x´y|u.
Note the non-standard notation: Xpx, V, αq is a cone with axis V K P Gpd, d´nq! The next proposition extracts "conical" information from many big β-numbers: Proposition 6.2. Let α, d, ǫ, θ ą 0 and C 0 , H ě 1. Then, there exists M ě 1, depending only on the previous parameters, such that the following holds. Let E 0 Ă R d be a n-regular set with regularity constant at most C 0 , and let E Ă E 0 X Bp0, 1q be a subset of measure H n pEq ě θ ą 0 with the following property: for every x P E, there exist M distinct dyadic scales 0 ă r ă 1 such that Then, there exists a subset G Ă E of measure H 1 pGq ě θ{2 such that for all x P G, The key point of Proposition 6.2 is that information about the β-numbers relative to the "ambient" set E 0 is sufficient to imply something useful about cones intersecting the subset E. The proof is heavily based on [22, Proposition 1.12], which we quote here: Proposition 6.4. Let α, d, θ ą 0 and C 0 , H ě 1. Then, there exist constants τ ą 0 and L ě 1, depending only on the previous parameters, such that the following holds. Let E 0 Ă R d be an n-regular set with regularity constant at most C 0 , and let B Ă E 0 X Bp0, 1q be a subset with H n pBq ě θ satisfying the following: there exists V P Gpd, nq such that for every x P B, cardtj ě 0 : Xpx, V, α, 2´j´1, 2´j q X B ‰ Hu ď H.
Then, there exists a subset B 1 Ă B with H n pB 1 q ě τ which is contained on an L-Lipschitz graph over V . In fact, one can take L " 2 H {α.
We may then prove Proposition 6.2.
Proof of Proposition 6.2. It suffices to show that the subset B Ă E such that (6.3) fails has measure H n pBq ă θ{2 if M ě 1 was chosen large enough. Assume to the contrary that H n pBq ě θ{2. By definition, for every x P B, there exists an plane V x P Gpd, nq such that We observe that the dependence of V x on x P B can be removed, at the cost of making B and α slightly smaller. Indeed, choose an α 2 -net V 1 , . . . , V k Ă Gpd, nq with k " α,d,n 1, and note that for every x P B, there exists 1 ď j ď k such that cardti ě 0 : Xpx, V j , α 2 , 2´i´1, 2´iq X E ‰ Hu ă H. By the pigeonhole principle, there is a subset B 1 Ă B of measure H n pB 1 q α,d,n H n pBq ě θ{2 such that the choice of V :" V j is common for x P B 1 . It follows that (6.5) holds for this V , for all x P B 1 , with α 2 in place of α. We replace B by B 1 without altering notation, that is, we assume that (6.5) holds for all x P B, and for some fixed V P Gpd, nq. Now Proposition 6.4 can be applied to the set B, and the plane V . The conclusion is that there is a further subset B 1 Ă B of measure H n pB 1 q " α,d,C 0 ,θ,H 1 (6.6) which is contained in Γ X Bp0, 1q, where Γ Ă R d is an L-Lipschitz graph over V for some L " 2 H {α " α,H 1. We will derive a contradiction, using that B 1 Ă E and, consequently, for all x P B 1 , and for M distinct dyadic scales 0 ă r ă 1 (which may depend on x P B 1 ). For technical convenience, we prefer to work with a lattice D of dyadic cubes on E 0 . As usual, we define β E 0 pQq :" β E 0 pB Q q, Q P D. Then, reducing "M " by a constant factor if necessary, it follows from (6.7) that every x P B 1 is contained in ě M distinct cubes Q P D of side-length 0 ă ℓpQq ď 1 satisfying β E 0 pQq ě ǫ. Moreover, since B 1 Ă E Ă E 0 X Bp0, 1q, we may assume that B Q Ă Bp0, Cq for all the cubes Q P D, for some C " C 0 1.
The main tool is that since Γ is an n-dimensional L-Lipschitz graph in R d , it satisfies the WGL with constants depending only on L and d. This follows from a more quantitative result -a strong geometric lemma for Lipschitz graphs -of Dorronsoro [15, Theorem 2] (or see [11,Lemma 10.11]). As a corollary of the WGL, the subset Γ bad of points x P ΓXBp0, 1q for which for ě M {2 distinct dyadic scales 0 ă r ă 1 has measure H n pΓ bad q ! 1, and in particular H n pΓ bad q ď H n pB 1 q{2, assuming that M ě 1 is large enough, depending only on L, c, C 0 , d, H, ǫ, and θ. In (6.8), c ą 0 is a constant so small that B Q Ă Bpx, c´1ℓpQq{100q for all x P Q. (6.9) In particular, c only depends on the n-regularity constant of E. Further, in (6.8), the quantity β Γ,8 pBpx, rqq is the L 8 -type β-number As pointed out after Definition 1.4, the WGL holds for the L 8 -type β-numbers if and only if it does for the L 1 -type β-numbers β Γ pBpx, rqq (Dorronsoro's strong geometric lemma holds for the latter, hence implies the WGL for the former). We then focus attention on B 2 :" B 1 z Γ bad Ă Γ X Bp0, 1q, which still satisfies H n pB 2 q ě 1 2 H n pB 1 q " α,d,C 0 ,θ,H 1, (6.10) recalling (6.6). Comparing (6.7) and (6.8), we find that every point x P B 2 has the following property: there exist M {2 cubes Q P D such that x P Q, β E 0 pQq ě ǫ and β Γ,8 pBpx, c´1ℓpQq{100qq ă cǫ. (6.11) Consider now a cube Q P D containing at least one point x P B 2 such that (6.11) holds. In particular, recalling the choice of c ą 0 from (6.9), the intersection is contained in a slab T Ă R d (a neighbourhood of an n-plane) of width ď cc´1ǫℓpQq{100 " ǫℓpQq{100. Since β E 0 pQq ě ǫ, however, we have H n pty P E 0 X B Q : y R 2T uq ǫH n pQq.
In other words, for every Q P D containing some x P B 2 such that (6.11) holds, there exists a subset E Q Ă E 0 X B Q Ă Bp0, Cq ‚ of measure H n pE Q q ǫH n pQq which is contained ‚ in the " ℓpQq-neighbourhood of Γ, yet ‚ outside the " ǫℓpQq-neighbourhood of Γ.
The collection of such cubes in D will be denoted G. As observed above (6.11), we have On the other hand, the sets E Q have bounded overlap in the sense since y P R d can only lie in the sets E Q associated to cubes Q P D with ℓpQq " ǫ distpy, Γq. Combining (6.12)-(6.13), we find that We have shown that H n pB 2 q ǫ M´1. This inequality contradicts (6.10) if M ě 1 is large enough, depending on α, d, ǫ, C 0 , θ, and H. The proof of Proposition 6.2 is complete.

HEAVY TREES HAVE POSITIVE WIDTH
We are equipped to prove Proposition 4.2. Fix a heavy tree T :" T j , and recall from the heavy tree property (T3) that if Q P LeavespT q, then cardtQ 1 P T : Q Ă Q 1 Ă QpT q and βpQ 1 q ě ǫu " M, Moreover, by (T1)-(T2), the total measure of LeavespT q is µpYLeavespT qq ě 1 4 µpQpT qq. Based on this information, we seek to verify the hypotheses of Proposition 5.14, which will eventually guarantee that widthpT q 1 and finish the proof of Proposition 4.2. We split the argument into three parts. 7.1. Part I: Finding heavy cones. Abbreviate Q 0 :" QpT q and L :" LeavespT q. To avoid a rescaling argument later on, we assume with no loss of generality that µpQ 0 q " ℓpQ 0 q " 1. For every Q P L, the PBP condition implies the existence of a plane V Q P Gpd, nq such that H n pπ V pB Q X Eqq ě δµpQq, V P BpV Q , δq. (7.2) We would prefer that all the planes V Q are the same, and this can be arranged with little cost. Namely, pick a δ 2 -net tV 1 , . . . , V m u Ă Gpd, nq with m " δ,d,n 1, and note that for all Q P L, there is some V j such that S j :" BpV j , δ 2 q Ă BpV Q , δq ": S Q . Therefore, by the pigeonhole principle, there is a fixed index 1 ď j ď m with the property Let L G be the good leaves satisfying S j Ă S Q for this j, and write S :" S j and V 0 :" V j .
We have just argued that µpYL G q " δ,d,n 1, and (7.2) holds for all Q P L G , for all V P S " B Gpd,nq pV 0 , δ 2 q. From this point on, I cease recording the dependence of the " " notation on the nregularity and PBP constants C 0 and δ.
For technical purposes, let us prune the set of good leaves a little further. Namely, apply the 5r-covering theorem to the balls 10B Q , Q P L G . As a result, we obtain a subcollection of the good leaves, still denoted L G , with the separation property and such that the lower bound µpYL G q " 1 remains valid.
Next we arrive at some geometric arguments. We may and will assume, with no loss of generality, and without further mention, that the radius of the ball S " B Gpd,nq pV 0 , δ 2 q is "small enough", in a manner depending only on d.
For every Q P L G , pick an n-dimensional disc D Q Ă B Q which is parallel to the plane V 0 and which satisfies H n pD Q q " µpQq and H n pD Q X Eq " 0. Such discs are pairwise disjoint by the separation property (7.3). We will also use frequently that the restrictions π V | D Q : D Q Ñ V are bilipschitz for all Q P L G and V P S " B Gpd,nq pV 0 , δ 2 q if δ ą 0 is small enough, as we assume. Therefore, the projections π V pD Q q Ă V are n-regular ellipsoids which contain, and are contained in, n-dimensional balls of radius " radpD Q q.
We then consider the slightly augmented set E`, where we have added the discs corresponding to all good leaves: The point behind the set E D can already be explained. Compare the two statements (a) The Hardy-Littlewood maximal function of π V 7 pH n | E q is large at x P V P S, (b) The Hardy-Littlewood maximal function of π V 7 pH n | E D q is large at x P V P S. Statement (b) contains much more information! Statement (a) could e.g. be true because a single cube Q P L G satisfies π V pQq " txu. But since π V | D Q is bilipschitz for all Q P L G and V P S, statement (b) forces π´1 V txu to intersect many distinct balls B Q Ą D Q . Recalling Proposition 5.14, this is helpful for finding a lower bound for widthpT q. Let us verify that E`is n-regular, with n-regularity constant 1. We leave checking the lower bound to the reader. To check the upper bound, fix x P E`and a radius r ą 0. Since E itself is n-regular, it suffices to show that ÿ QPL G H n pD Q X Bpx, rqq r n .
(7.4) Write L ď G :" tQ P L G : D Q X Bpx, rq ‰ H and radpD Q q ď ru and L ą G :" tQ P L G : D Q X Bpx, rq ‰ H and radpD Q q ą ru.
Here we used that the leaves L consist of disjoint cubes. To finish the proof of (7.4), we claim that card L ą G ď 1. Assume to the contrary that D Q , D Q 1 P L ą G with Q ‰ Q 1 . Then certainly 2B Q X Bpx, rq ‰ H ‰ 2B Q 1 X Bpx, rq, and both B Q , B Q 1 have diameters ě r. This forces 10B Q X 10B Q 1 ‰ H, violating the separation condition (7.3). This completes the proof of (7.4).
We next claim that for every x P E D there exist M distinct dyadic radii 0 ă r 1 such that β`pBpx, rqq ǫ. This follows easily by recalling that if x P D Q with Q P L Q Ă L, then cardtQ 1 P T : Q Ă Q 1 Ă Q 0 and βpQ 1 q ě ǫu " M by the definition of good leaves, but let us be careful: let x P D Q , and let Q 1 P T be one of the ancestors of Q with Then, if V P Apd, nq is arbitrary, we simply have which proves that β`pBpx, rqq ǫ. A fixed radius "r" can only be associated to 1 cubes Q 1 in the ancestry of Q, so M of them arise in the manner above. The claim follows. We note that µ`pE D q µpYL G q " 1.
We aim to apply Proposition 6.2 to the set E D , but we will perform a final pruning before doing so. Let c ą 0 be a small constant to be determined soon, and let L G,light Ă L G consist of the good leaves with the following property: there exists a point x Q P D Q and a radius 0 ă r Q ď 1 such that We also observe that since x Q P D Q Ă B Q Ă B Q 0 , and r Q ď 1 " ℓpQ 0 q, we have Bpx Q , r Q q Ă 2B Q 0 for all Q P L G,light . Now, use the 5r covering theorem to find a subset L 1 Ă L G,light such that the associated balls Bpx Q , r Q {5q are disjoint, and It follows from (7.6), and the n-regularity of µ`, that Comparing this upper bound with (7.5), we find that if c ą 0 was chosen small enough, depending only on the PBP and n-regularity constants of E, then where L G,heavy " L G z L G,light . Let E D,dense be the union of the discs D Q with Q P L G,heavy . We summarise the properties of E D,dense Ă E D Ă E`: (1) µ`pE D,dense q " 1, (2) If x P E D,dense , there are M dyadic scales 0 ă r 1 such that β`pBpx, rqq ǫ, (3) If x P E D,dense , then µ`pE D X Bpx, rqq r for all 0 ă r ď 1. We then apply Proposition 6.2 to the set E D,dense with a "multiplicity" parameter H ě 1 to be chosen later. As usual, the choice of the parameter H will eventually only depend on the n-regularity and PBP constants of E. The parameters α and θ in the statement of the proposition are set to be such that α " d,δ 1 (specifics to follow later), and θ " 1 is so small that H n pE D,dense q ě θ, which is possible by (1) above. As a good first approximation of how to choose α, recall from Lemma 2.4 that if x P R d and |π V 0 pxq| ď α|x|, where α " αpd, δq ą 0 is small enough, then there exists a plane V P B Gpd,nq pV 0 , δ 2 q " S such that π V pxq " 0. In symbols, the previous statement is equivalent to In fact, in the case n " d´1, this would be a suitable definition for α, and the reader may think that α is at least so small that (7.7) holds. In the case n ă d´1, additional technicalities force us to pick α slightly smaller. Proposition 6.2 then states that if M ě 1 is chosen large enough, in a manner depending only on α, H, d, δ, ǫ, θ, and the n-regularity constant of E, the following holds: there exists a subset G Ă E D,dense of measure 1 H n pGq θ " 1 (7.8) with the property cardtj ě 0 : Xpx, V 0 , α 2 , 2´j´1, 2´j q X E D,dense ‰ Hu ě H, x P G. (7.9) (The upper bound in (7.8) follows from G Ă E D and diampE D q ℓpQ 0 q " 1). We next upgrade (7.9) to a measure estimate, using the definition of E D,dense . Namely, recall from (3) above that if y P E D,dense , then µ`pE D X Bpy, rqq r n for all 0 ă r ď 1. By definitions and a few applications of the triangle inequality, y P Xpx, V 0 , α 2 , 2´j´1, 2´jq ùñ Bpy, α2´j´1 0 q Ă Xpx, V 0 , α, 2´j´2, 2´j`1q, and hence H n pE D X Xpx, V 0 , α, 2´j´2, 2´j`1qq µ`pE D X Bpy, α2´j´1 0 qq 2´j n (7.10) for all those scales 2´j such that Xpx, V 0 , α 2 , 2´j´1, 2´j q contains some y P E D,dense . (Here we used that α " d,δ 1.) For x P G, the number of such scales "2´j" is no smaller than H, by (7.9), for every such "2´j", it follows from (7.10) that at least one of the three scales 2´i P t2´j´1, 2´j, 2´j`1u satisfies H n pE D X Xpx, V 0 , α, 2´i´1, 2´iqq ě c2´i n . Here c " 1 is a constant which records for the implicit constants in (7.10). Therefore, replacing "H" by "H{3" without altering notation, we have just proven the following: cardtj ě 0 : H n pE D X Xpx, V 0 , α, 2´j´1, 2´j qq ě c2´j n u ě H, x P G.
(7.11) 7.2. Part II: Besicovitch-Federer argument. By following the classical argument of Besicovitch and Federer, we aim to use (7.11) to show that the projections of E D to planes close to V 0 have plenty of of overlap. This part of the argument will be quite familiar to readers acquainted with the proof of the Besicovitch-Federer projection theorem. For V P S " B Gpd,nq pV 0 , δ 2 q, write interpreted as a function on R n , and let Mf V stand for the centred Hardy-Littlewood maximal function of f V . We will prove the following claim: Claim 7.12. For every x P G, there exists a subset S x Ă S of measure γ d,n pS x q 1{ ? H with the following property: As usual, the implicit constants here are allowed to depend on d, and the n-regularity and PBP constants of E. During the proof of the claim, we use the abbreviation E j,x :" E D X Xpx, V 0 , α, 2´j´1, 2´jq, j ě 0. (7.14) By (7.11), there exist H distinct indices j ě 0 such that H n pE j,x q ě c2´j n . The proof of the claim splits into two cases: either there is at least one of these indices "j" such that E j,x meets only a few planes π´1 V tπ V pxqu, V P S, or then E j,x meets fairly many of the planes π´1 V tπ V pxqu, V P S, for every one of the H indices "j". Case 1. Fix x P G, assume with no loss of generality that x " 0. This has the notational benefit that π´1 V tπ V pxqu " V K for V P Gpd, nq. Assume that there exists at least one index j ě 0 such that H n pE j,x q ě c2´j n , and γ d,n ptV P S : V K X E j,0 ‰ Huq ď 1 ? H . (7.15) Fix such an index j ě 0, and abbreviate E j,0 :" E 0 . Then (7.15) will imply that most of the (non-negligible) H n mass of E 0 Ă Xp0, V 0 , αq is contained in narrow slabs around pd´nq-planes with "high density". As in the classical proof of the Besicovitch-Federer projection theorem, the case n ă d´1 requires integralgeometric considerations, whose necessity will only become clear at the very end of Case 1. Fortunately, they also make technical sense in the case n " d´1 (they just become trivial), so the case n " d´1 does not require separate treatment. As in Section 2, we define GpW, nq :" tV P Gpd, nq : V Ă W u -Gpn`1, nq, W P Gpd, n`1q, and we write γ W,n`1,n for the Opdq-invariant probability measure on GpW, nq. The metric on GpW, nq is inherited from Gpd, nq. Recall the Fubini formula established in Lemma 2.2: γ W,n`1,n pBq dγ d,n`1 pW q (7.16) for B Ă Gpd, nq Borel. We will need to find a Borel set W Ă Gpd, n`1q, in fact a ball, which may depend on j and x, with the following properties: (W1) γ d,n`1 pWq " d,δ 1, (W2) For every W P W, the set S X GpW, nq contains a ball S W " B GpW,nq pV W , δ 4 q, (W3) There exists a subset E W,0 Ă E 0 of measure H n pE W,0 q ě c2´j n with the property The "c" appearing in property (W3) may be a constant multiple (depending on δ, d) of the constant in H n pE 0 q ě c2´j n . Finding W with the properties (W1)-(W3) is easy if n " d´1, so let us discuss this case first to get some intuition. Simply take W :" Gpd, dq " tR d u.
We next verify (W2). Let W P W, that is, dpW, W 0 q ď ρ. Then, since V 0 Ă W 0 , Lemma 2.1 implies that there exists a plane V W P GpW, nq with dpV W , V 0 q ρ. In particular, V W P B Gpd,nq pV 0 , δ 4 q if ρ is chosen small enough, and consequently S W :" B GpW,nq pV W , δ 4 q Ă S. This completes the proof of (W2).
To prove (W3), we need to check that if W P W and z P E W,0 , then there exists a plane V P S W with π V pzq " 0. This will be accomplished by an application of Lemma 2.4 inside W -R n`1 . First, since z P E W,0 Ă E 0 , V W Ă W , and dpV W , V 0 q ρ ď α, we have Second, |π W pzq| ě |π W 0 pz 0 q|´dpW, W 0 q¨|z 0 |´|z´z 0 | |z|, (7.19) using that z 0 P W 0 , and z P Bpz 0 , ρ2´jq Ă Bpz 0 , |z 0 |{2q, and dpW, W 0 q ď ρ. Combining (7.18)- (7.19), and setting z W :" π W pzq P W , we find that Finally, the estimate (7.20) allows us to apply Lemma 2.4 to the point z W P W in the space GpW, nq -Gpn`1, nq. The conclusion is that if α is small enough, depending only on δ, n, then there exists a plane V P B GpW,nq pV W , δ 4 q " S W such that π V pz W q " 0. But now V Ă W , and π W pz´z W q " 0, so also π V pzq " π V pz W q`π V pz´z W q " 0. This is what we claimed, so the proof of (W3) is complete.
After the preparations (W1)-(W3), we can get to the business of verifying Claim 7.12 in Case 1. Recall from the main assumption (7.15) that γ d,n ptV P S : V K X E 0 ‰ Huq ď 1{ ? H. Combined with the Fubini formula (7.16), this implies that the set of planes W P Gpd, n`1q such that γ W,n`1,n ptV P S W : has γ d,n`1 -measure at most C´1, for C ě 1. Choose C " δ 1 here so large that the planes W P Gpd, n`1q in question have total measure ď 1 2 γ d,n`1 pWq. After discarding these "bad" planes from W, we may assume that the opposite of (7.21) holds for all W P W: γ W,n`1,n ptV P S W : V K X E 0 ‰ Huq ď C ? H . (7.22) Fix W P W, so (7.22) holds, and abbreviate γ W,n`1,n ": γ n`1,n . Then, let S be a system of dyadic cubes on the (n-regular) ball S W Ă GpW, nq, with top cube S W . Then, cover the setS W :" tV P S W : V K X E 0 ‰ Hu by a disjoint collection Q Ă S of these cubes such that ÿ QPQ γ n`1,n pQq ď 2C ? H .
For Q P Q, write CpQq :" YtV K : V P Qu, generalising the notation CpSq introduced in (7.7). SinceS W is covered by the cubes Q P Q, the set E W,0 Ă E 0 X Ť V PS W V K is covered by the cones CpQq, Q P Q. Now, let Q light be the cubes Q P Q satisfying H¨2´j n¨γ n`1,n pQq. Recalling from (W3) that H n pE W,0 q ě c2´j n , and that E W,0 is covered by the union of the cones CpQq, Q P Q, we infer that there is a subsetĒ W,0 Ă E W of measure H n pĒ W,0 q ě c 2¨2´j n which is covered by the union of the cones CpQq, Q P Q z Q light . Every cube Q P Q z Q light satisfies the inequality reverse to (7.23), and is consequently contained in some maximal cube in S with this property. Let Q heavy be the collection of such maximal (hence disjoint) cubes. Then, since Q Ă Q 1 implies CpQq Ă CpQ 1 q, we see thatĒ W,0 is also covered by the union of the cones CpQq, Q P Q heavy , and consequently ÿ QPQ heavy H n pCpQq XĒ W,0 q ě c 2¨2´j n . (7.24) We moreover claim that the union of the heavy cubes, denoted H W , satisfies This completes the proof of (7.25).
We are now ready to prove Claim 7.12 in Case 1, that is, define the set S x " S 0 Ă S such that (7.13) holds for all V P S 0 . Define Then, by the Fubini formula (7.16), and the uniform lower bound (7.25), we have γ d,n pS 0 q ě ż W γ W,n`1,n pH W q dγ d,n`1 pW q (7.25) H. Fix V P S 0 , let first W P W be such that V P H W , and then let Q P Q W,heavy " Q heavy be the unique cube with V P Q (we do not claim, however, that the choice of W would be unique). By definitions, especially H¨2´j n¨γ n`1,n pQq, (7.27) where of course CpQ, r, Rq :" CpQq XBpRq z Bprq, and we recall that CpQq " tV K : V P Qu. Note that CpQ, 2´j´1, 2´jq Ă T " T V , where T Ă R d is a slab of the form T :" π´1 V rBp0, C2´j ℓpQqqs of width " d 2´jℓpQq around the plane V K P Gpd, d´nq. Indeed, if x P CpQ, 2´j´1, 2´jq, then π V 1 pxq " 0 for some V 1 P Q. Then dpV, V 1 q d ℓpQq, and |π V pxq| ď dpV, V 1 q¨|x| 2´jℓpQq, which means that x P T if the constant C ě 1 is chosen appropriately.
Write B V :" Bp0, C2´jℓpQqq Ă V . With this notation, recalling that D Q Ă B Q , and using that the projections π V | D Q : D Q Ñ V are bilipschitz for Q P L G and V P S 0 Ă S, we infer that H¨2´j n¨γ n`1,n pQq radpB V q n " ? H.
In final estimate, we used that γ n`1,n pQq " ℓpQq n . This is the whole point of the integralgeometric argument: without splitting Gpd, nq into a "product" of Gpd, n`1q and GpW, nq, we could have, more easily, reached the penultimate estimate with "γ d,n pQq" in place of "γ n`1,n pQq". But γ d,n pQq " ℓpQq npd´nq ! ℓpQq n if n ă d´1, and the final estimate would have failed. We have now proved Claim 7.12 in Case 1.

Case 2.
Again, fix x P G, assume with no loss of generality that x " 0, and let j 1 , . . . , j H ě 0 be distinct scale indices such that H n pE j i ,0 q ě c2´j i n for all 1 ď i ď H, recall the notation from (7.14). This time, we assume that Then, it follows by splitting the integration in (7.29) to S z S 0 and S 0 , that ? H ď ? H¨γ d,n pS z S 0 q`H¨γ d,n pS 0 q.
Recalling that γ d,n pSq ď 1 2 (that is, S " B Gpd,nq pV 0 , δ 2 q is a fairly small ball), we find that γ d,n pS 0 q 1{ ? H, as required by Claim 7.12. It remains to check that ? H whenever V P S 0 . Fixing V P S 0 , it follows by definition that there are ě ? H indices i P t1, . . . , Hu with the property that V PS 0,i , which meant by definition that For each of these indices i, the plane V K intersects at least one of the discs D Q with Q P L G , whose union is E D . Moreover, since the sets E j,0 ĂBp2´j q zBp2´j´1q are disjoint for distinct indices j ě 0, we conclude that V K meets ě ? H distinct discs D Q . Consequently, recalling also that D Q Ă B Q for all Q P L G , Moreover, by their definition below (7.2), all the cubes Q P L G satisfy the PBP condition with common plane V 0 : H n pπ V pE X B Q qq ě δµpQq, Q P L Q , V P S " B Gpd,nq pV 0 , δ 2 q. Consequently, Proposition 5.14 states that if the parameter H 1 is chosen large enough, depending only on C 0 and δ, then widthpT q cδpH 1 q´1¨γ d,n pS G q " 1{H. (7.33) As explained above (7.8), choosing H 1 " ? H this big means forces us to choose the parameter M ě 1 large enough in a manner depending on α " d,δ 1, C 0 , H " C 0 ,δ 1, d, δ, ǫ, θ " C 0 ,d,δ 1. So, in fact M " C 0 d,δ,ǫ 1, as claimed in Proposition 4.2. Since the lower bound for widthpT q in (7.33) only depends on the n-regularity and PBP constant of E, the proof of Proposition 4.2 is complete.
Since Proposition 3.8 follows from Proposition 4.2, and the construction of heavy trees in Section 4, we have now proved Proposition 3.8. As we recorded in Lemma 3.4, this implies that n-regular sets E Ă R d having PBP satisfy the WGL, and then the BPLG property follows from Theorem 1.5. This completes the proof of Theorem 1.6.

APPENDIX A. A VARIANT OF THE LEBESGUE DIFFERENTIATION THEOREM
Here we prove Lemma 5.17, which we restate below for the reader's convenience: Then, there exists a collection R heavy of disjoint cubes such that the "sub-functions" valid for f P L 1 pR d q, every λ ą 0, and a certain constant C " C d ě 1. The first inequality in (A.7) is stated in [32, (6)], but we provide the short details. Let C " C d ě 1 be a constant to be specified in a moment. Write Ω h :" tMf ą hu for h ą 0. For every x P Ω Cλ , choose a radius r x ą 0 such that, denoting B x :" Bpx, r x q, we have This is possible, since f P L 1 pR d q. For example, one can take r x ą 0 to be the supremum of the (non-empty and bounded set of) radii such that the left hand inequality in (A.8) holds. The radii "r x " are uniformly bounded, again by f P L 1 pR d q. We then apply the 5rcovering lemma to the balls 1 5 B x to obtain a countable sub-sequence tB i u iPN Ă tB x u xPΩ Cλ with the properties that (i) the balls 1 5 B i are disjoint, and (ii) the balls B i cover Ť t 1 5 B x : x P Ω Cλ u Ą Ω Cλ . We observe that if C " C d ě 1 is large enough, it follows from (A.8) that 1 5 as desired. For the second inequality in (A.7), see [33, (5), p. 7].
Proof of Lemma A.1. We begin with an initial reduction. If f R L 1 pr0, 1q d q, there is nothing to prove: then R heavy :" tr0, 1q d u satisfies the conclusions (A.4). So, assume that f P L 1 pr0, 1q d q, and hence f P L 1 pR d q, since spt f Ă r0, 1q d . Let With this in mind, we replace N by N {p2Cq, and we re-define H N to be the set H N :" tx : f pxq ě N u. As we just argued, the hypothesis (A.3) remains valid with the new notation, possibly with slightly worse constants.
Fix N ě 1 and abbreviate θ :" cN´γ ą 0. It would be helpful if the elements in B were dyadic cubes instead of arbitrary balls, so we first perform some trickery to reduce (essentially) to this situation. There exist d`1 dyadic systems D 1 , D 2 , . . . , D d`1 with the following property: every cube Q Ă r0, 1q d , and consequently every ball B Ă r0, 1q d , is contained in a dyadic cube R P D 1 Y. . .YD d`1 with |R| ď C d |Q| (resp. |R| ď C d |B|). The constant "d`1" is not crucial -any dimensional constant would do. The fact that d`1 systems in R d suffice was shown by Mei [24], but such "adjacent" dyadic systems can even be produced in metric spaces, see [19].
In particular, for every B P B, we may assign an index i " i B P t1, . . . , d`1u, possibly in a non-unique way, such that B Ă Q 1 for some Q 1 P D i with |Q 1 | ď C d |B|. We let B i be the set of balls in B with fixed index i P t1, . . . , d`1u, and we write pxqf i pxq ě f pxq{pd`1q for this particular i, and This implies (A.9). We now fix i P t1, . . . , d`1u satisfying (A.9). Then f i satisfies the hypothesis (A.3) with the slightly worse constants "θ{pd`1q 2 " and "N {pd`1q". Also, it evidently suffices to prove the claimed lower bounds in (A.4) for "f i " and its "subfunctions" f i R :" in place of f and the "sub-functions" f R . Let us summarise the findings: by passing from B to B i and from f to f i if necessary, we may assume that every ball in the original collection "B" is contained in an element "R" of some dyadic system "D" with |R| ď C d |B|.
We make this a priori assumption in the sequel. For every dyadic cube R P D, we define the weight Here the relation B " R means that B Ă R, and |R| ď C d |B|. By the previous arrangements, for every B P B there exist " d 1 dyadic cubes R P D such that B " R. It is worth pointing out that because if x P B P B, then B " R for some R P D. It follows that x P R, and w B is one of the terms in the sum defining w R . We now begin the proof in earnest. If }f } 1 ą M there is nothing to prove: then we simply declare R heavy :" tr0, 1q d u, and (A.4) is satisfied. So, we may assume that }f } 1 ď M. (A.10) We will next perform k P N successive stopping time constructions, for some 1 ď k ď γ`1, which will generate a families R 1 , R 2 , . . . , R k Ă D of disjoint dyadic cubes. The cubes in R k`1 will be contained in the union of the cubes in R k . A subset of one of these families will turn out to be the family "R heavy " whose existence is claimed.
Let R 1 Ă D be the maximal (hence disjoint) dyadic cubes with the property Note that the definition is well posed, since the sum on the left hand side of (A.11) is constant on R. We first record the easy observation It then follows from the definition of the coefficients w R (and the fact that every B P B is contained in some R P D) that there exist dyadic cubes R P D containing x such that (A.11) holds, and in particular x P R for some R P R 1 . Next, we calculate that since the cubes in R 1 are disjoint. Moreover, by (A.10), for some constant A " A d ě 1. The precise relation between this "A" and the dimensional constant appearing in the main assumption (A.2) is that, in the end, we will need N ą p2Aq γ`1 3 pγ`1q 2 M γ`2 {c. Next, we claim that if x P R P R 1 , then The second inequality follows directly from the definition of the maximal cubes R P R 1 .
Regarding the first inequality, note that if B P B is a ball satisfying x P B X R and B Ć R, then B Ă R 1 for some strict ancestor R 1 P D of R. Then the coefficient w B appears in the sum defining w R 1 for this ancestor R 1 R. As a corollary of (A. 15), and recalling that f pxq ě N for all x P H N , we record that The proof now splits into two cases: in the first one, we are actually done, and in the second one, a new stopping family R 2 will be generated. The case distinction is based on examining the following "heavy" cubes in R 1 : In this case, we set R heavy :" R 1,heavy , and the proof terminates, because (A.4) is satisfied.
Case 2. Assume next that (A.17) fails, and recall from (A.12) that H N is contained in the union of the cubes in R 1 . Therefore, where R 1,light " R 1 z R 1,heavy . We now proceed to define the next generation stopping cubes R 2 . Fix R 0 P R 1,light , and consider the maximal dyadic sub-cubes R Ă R 0 with the property ÿ RĂR 1 ĂR 0 w R 1 1 R 1 pxq ě N 2 :" tN {4u, x P R, (A. 19) Again, the left hand side of (A.19) is constant on R, so the stopping condition is wellposed. The cubes so obtained are denoted R 2 pR 0 q, and we set We claim that the (fairly large) part of H N covered by cubes in R 1,light is remains covered by the cubes in R 2 . Indeed, fix x P R 0 X H N , where R 0 P R 1,light Ă R 1 . Then by definitions of R 1 and N 1 , so and hence x is contained in some (maximal) dyadic cube R Ă R 0 satisfying (A. 19). Arguing as in (A.15), we infer the following: if x P R P R 2 , then Indeed, the first inequality follows exactly as in (A.15). To see the second inequality, split the cubes R 1 R into the ranges R R 1 Ă R 0 and R 0 R Ă r0, 1q d , where R 0 P R 1 .
cubes R k´1,light , and they are defined as the maximal sub-cubes "R" of R 0 P R k´1,light satisfying One of these cases must occur because of (A.26), and the covering property stated above (A.26). In the first case, (A.27) shows that ÿ RPR k,heavy }f R } 1 ě 2´k ÿ RPR k,heavy ż RXH N f pxq dx ě 2´2 k θ, and the proof of (A.4) concludes if k ď γ`1. So, the only remaining task is to show that the first case must occur for some k ď γ`1. Indeed, if the second case of (A.29) occurs for any k ě 1, we have Recalling that N k " tN {2 k u ě N {3 k , hence N 1¨¨¨Nk ě N k 3´k 2 , this yields N k´γ ď 3 k 2 p2Aq k M k`1 c .
Assuming that N ą 3 pγ`1q 2 p2Aq γ`1 M γ`2 {c (in agreement with (A.2)), the inequality above cannot hold for k " γ`1. Thus, the "heavy" case of (A.29) occurs latest at step k " γ`1. The proof of the lemma is complete.