Abstract
I prove that closed n-regular sets \(E \subset {\mathbb {R}}^{d}\) with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993.
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1 Introduction
I start by introducing the key concepts of the paper. A Radon measure \(\mu \) on \({\mathbb {R}}^{d}\) is called s-regular, \(s \geqslant 0\), if there exists a constant \(C_{0} \geqslant 1\) such that
A set \(E \subset {\mathbb {R}}^{d}\) is called s-regular if E is closed, and the restriction of s-dimensional Hausdorff measure \({\mathcal {H}}^{s}\) on E is an s-regular Radon measure. An n-regular set \(E \subset {\mathbb {R}}^{d}\) has big pieces of Lipschitz graphs (BPLG) if the following holds for some constants \(\theta ,L > 0\): for every \(x \in E\) and \(0< r < {\text {diam}}(E)\), there exists an n-dimensional L-Lipschitz graph \(\Gamma \subset {\mathbb {R}}^{d}\), which may depend on x and r, such that
By an n-dimensional L-Lipschitz graph, I mean a set of the form \(\Gamma = \{v + f(v) : v \in V\}\), where \(V \subset {\mathbb {R}}^{d}\) is an n-dimensional subspace, and \(f :V \rightarrow V^{\perp }\) is L-Lipschitz. Sometimes it is convenient to call \(\Gamma = \{v + f(v): v \in V\}\) an L-Lipschitz graph over V. The BPLG property is stronger than uniform n-rectifiability, see Sect. 1.1 for more discussion.
Let G(d, n) be the Grassmannian of all n-dimensional subspaces of \({\mathbb {R}}^{d}\), equipped with a natural metric which is invariant under the action of the orthogonal group \({\mathcal {O}}(d)\). See Sect. 2 for details. For \(V \in G(d,n)\), let \(\pi _{V}\) be the orthogonal projection to V. It is straightforward to check, see [22, Proposition 1.4], that if \(E \subset {\mathbb {R}}^{d}\) is an n-regular set with BPLG, then E has many projections of positive \({\mathcal {H}}^{n}\) measure: more accurately, if \(\Gamma \) in (1.1) is an L-Lipschitz graph over \(V_{0} \in G(d,n)\), then there is a constant \(\delta > 0\), depending only on \(d,L,\theta \), such that
David and Semmes asked in their 1993 paper [13] whether a converse holds: are sets with BPLG precisely the ones with plenty of big projections? The problem is also mentioned in the monograph [12, p. 29] and, less precisely, in the 1994 ICM lecture of Semmes [31].
Definition 1.2
(BP and PBP) An n-regular set \(E \subset {\mathbb {R}}^{d}\) has big projections (BP) if there exists a constant \(\delta > 0\) such that the following holds. For every \(x \in E\) and \(0< r < {\text {diam}}(E)\), there exists at least one plane \(V = V_{x,r} \in G(d,n)\) such that
The set E has plenty of big projections (PBP) if (1.3) holds for all \(V \in B(V_{x,r},\delta )\).
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 \(E \subset {\mathbb {R}}^{d}\) satisfies the weak geometric lemma (WGL) if for all \(\epsilon > 0\) there exists a constant \(C(\epsilon ) > 0\) such that the following (Carleson packing condition) holds:
In the definition above, the quantity \(\beta (B(x,r))\) 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
with \(\mu := {\mathcal {H}}^{n}|_{E}\), and where \({\mathcal {A}}(d,n)\) is the “affine Grassmannian” of all n-dimensional planes in \({\mathbb {R}}^{d}\). The \(\beta \)-number above is an “\(L^{1}\)-variant” of the original “\(L^{\infty }\)-based \(\beta \)-number” introduced by Jones [20], namely
If \(E \subset {\mathbb {R}}^{d}\) is n-regular, then the following relation holds between the two \(\beta \)-numbers:
For a proof, see [11, p. 28]. This inequality shows that the WGL, a condition concerning all \(\epsilon > 0\) simultaneously, holds for the numbers \(\beta (B(x,r))\) if and only if it holds for the numbers \(\beta _{\infty }(B(x,r))\).
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 \(E \subset {\mathbb {R}}^{d}\) 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 \(\sigma \)-finite length has measure zero. The main result of this paper shows that PBP alone implies BPLG:
Theorem 1.6
Let \(E \subset {\mathbb {R}}^{d}\) be an n-regular set with PBP. Then E has BPLG.
To prove Theorem 1.6, all one needs to show is that
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 \(E \subset {\mathbb {R}}^{d}\) is uniformly n-rectifiable, n-UR in brief, if (1.1) holds for some n-dimensional L-Lipschitz images \(\Gamma = f(B(0,r))\), with \(B(0,r) \subset {\mathbb {R}}^{n}\), 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 n-dimensional kernels are \(L^{2}\)-bounded on an n-regular set \(E \subset {\mathbb {R}}^{d}\) 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 n-regular 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 \(K_{\epsilon } \subset {\mathbb {R}}^{2}\), \(\epsilon > 0\), with the properties
-
(a)
\({\mathcal {H}}^{1}(K_{\epsilon }) = 1\) and \({\mathcal {H}}^{1}(\pi _{L}(K_{\epsilon })) < \epsilon \) for all \(L \in G(2,1)\),
-
(b)
\(K_{\epsilon }\) is 1-UR with constants independent of \(\epsilon > 0\).
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 \(n := \lfloor \epsilon ^{-1}\rfloor \). Sub-divide \(I_{0} := [0,1] \times \{0\} \subset {\mathbb {R}}^{2}\) into n segments \(I_{1},\ldots ,I_{n}\) of equal length, and rotate them individually counter-clockwise by \(2\pi /n\). Then, sub-divide each \(I_{j}\) into n segments of equal length, and rotate by \(2\pi /n\) again. Repeat this procedure n times to obtain a compact set \(K_{n} = K_{\epsilon }\) consisting of \(n^{n}\) segments of length \(n^{-n}\). It is not hard to check that (a) and (b) hold for \(K_{\epsilon }\). In particular, to check (b), one can easily cover \(K_{\epsilon }\) by a single 1-regular continuum \(\Gamma \subset B(0,2)\) of length \({\mathcal {H}}^{1}(\Gamma ) \leqslant 10\).
1.2 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 \({\mathbb {R}}^{2}\) 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 \({\mathbb {R}}^{2}\). 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 \(\pi _{V}\)-projections of the measure \({\mathcal {H}}^{n}|_{E}\) lie in \(L^{2}(V)\) on average over \(V \in B_{G(d,n)}(V_{0},\delta )\). 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 kth 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 \(E \subset {\mathbb {R}}^{2}\) 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 \(\pi _{V\sharp }{\mathcal {H}}^{n}|_{E}\). 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 \(\delta > 0\) and \(C_{0} \geqslant 1\), do there exist \(L \geqslant 1\) and \(\theta > 0\) such that the following holds? Whenever \(E \subset {\mathbb {R}}^{d}\) is an n-regular set with regularity constant at most \(C_{0}\), and
then there exists an n-dimensional L-Lipschitz graph \(\Gamma \subset {\mathbb {R}}^{d}\) such that \({\mathcal {H}}^{n}(E \cap \Gamma ) \geqslant \theta \).
In addition to the “single scale” assumption (1.7), the proof of Theorem 1.6 requires information about balls much smaller than B(0, 1) to produce the Lipschitz graph \(\Gamma \).
1.4 Notation
An open ball in \({\mathbb {R}}^{d}\) with centre \(x \in {\mathbb {R}}^{d}\) and radius \(r > 0\) will be denoted B(x, r). When \(x = 0\), I sometimes abbreviate \(B(x,r) =: B(r)\). The notations \(\mathrm {rad}(B)\) and \({\text {diam}}(B)\) mean the radius and diameter of a ball \(B \subset {\mathbb {R}}^{d}\), respectively, and \(\lambda B := B(x,\lambda r)\) for \(B = B(x,r)\) and \(\lambda > 0\).
For \(A,B > 0\), the notation \(A \lesssim _{p_{1},\ldots ,p_{k}} B\) means that there exists a constant \(C \geqslant 1\), depending only on the parameters \(p_{1},\ldots ,p_{k}\), such that \(A \leqslant CB\). Very often, one of these parameters is either the ambient dimension “d”, or then the PBP or n-regularity constant “\(\delta \)” or “\(C_{0}\)” of a fixed n-regular set \(E \subset {\mathbb {R}}^{d}\) 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, \(A \lesssim _{d,\delta ,C_{0}} B\) is abbreviated to \(A \lesssim B\). The two-sided inequality \(A \lesssim _{p} B \lesssim _{p} A\) is abbreviated to \(A \sim _{p} B\), and \(A > rsim _{p} B\) means the same as \(B \lesssim _{p} A\).
2 Preliminaries on the Grassmannian
Before getting started, we gather here a few facts of the Grassmannian G(d, n) of n-dimensional subspaces of \({\mathbb {R}}^{d}\). Here \(0 \leqslant n \leqslant d\), and the extreme cases are \(G(d,0) = \{0\}\) and \(G(d,d) = \{{\mathbb {R}}^{d}\}\). We equip G(d, n) with the metric
where \(\Vert \cdot \Vert \) 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 (G(d, n), d) is compact, and open balls in G(d, n) will be denoted \(B_{G(d,n)}(V,r)\). An equivalent metric on G(d, n) is given by
For a proof, see [25, Lemma 4.1]. With the equivalence of d and \({\bar{d}}\) in hand, we easily infer the following auxiliary result:
Lemma 2.1
Let \(0< n < d\), and let \(W_{1},W_{2} \in G(d,n + 1)\), and let \(V_{1} \in G(d,n)\) with \(V_{1} \subset W_{1}\). Then, there exists \(V_{2} \in G(d,n)\) such that \(V_{2} \subset W_{2}\) and \(d(V_{1},V_{2}) \lesssim d(W_{1},W_{2})\).
Proof
By the equivalence of d and \({\bar{d}}\), we have \(r := {\bar{d}}(W_{1},W_{2}) \lesssim d(W_{1},W_{2})\). We may assume that r is small, depending on the ambient dimension, otherwise any n-dimensional subspace \(V_{2} \subset W_{2}\) satisfies \(d(V_{1},V_{2}) \leqslant {\text {diam}}G(d,n) \lesssim r\). Now, let \(\{e_{1},\ldots ,e_{n}\}\) be an orthonormal basis for \(V_{1}\), and for all \(e_{j} \in V_{1} \subset W_{1}\), pick some \({\bar{e}}_{j} \in W_{2}\) with \(|e_{j} - {\bar{e}}_{j}| \leqslant r\). If \(r > 0\) is small enough, the vectors \({\bar{e}}_{1},\ldots ,{\bar{e}}_{n}\) are linearly independent, hence span an n-dimensional subspace \(V_{2} \subset W_{2}\). Since \(|e_{j} - {\bar{e}}_{j}| \leqslant r\) for all \(1 \leqslant j \leqslant n\), an arbitrary unit vector \(v_{1} = \sum \beta _{j}e_{j} \in V_{1}\) lies at distance \(\lesssim r\) from \(v_{2} := \sum \beta _{j}{\bar{e}}_{j} \in V_{2}\), and consequently
This completes the proof. \(\square \)
We will often use the standard “Haar” probability measure \(\gamma _{d,n}\) on G(d, n). Namely, let \(\theta _{d}\) be the Haar measure on the orthogonal group \({\mathcal {O}}(d)\), and define
where \(V_{0} \in G(d,n)\) is any fixed subspace. The measure \(\gamma _{d,n}\) is the unique \({\mathcal {O}}(d)\)-invariant Radon probability measure on G(d, n), 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 G(d, n):
Lemma 2.2
Let \(0< n < d\). For \(W \in G(d,n + 1)\), let \(G(W,n) := \{V \in G(d,n) : V \subset W\}\). Then G(W, n) can be identified with \(G(n + 1,n)\), and we equip G(W, n) with the Haar measure \(\gamma _{W,n + 1,n} := \gamma _{n + 1,n}\), constructed as above. Then, the following holds for all Borel sets \(B \subset G(d,n)\):
Proof
This is the same argument as in [23, Lemma 3.13]: one simply checks that both sides of (2.3) define \({\mathcal {O}}(d)\)-invariant probability measures on \(\gamma _{d,n}\), and then appeals to the uniqueness of such measures. \(\square \)
We record one final auxiliary result:
Lemma 2.4
For all \(0< n < d\), \(\delta > 0\), there exists an “angle” \(\alpha = \alpha (d,\delta ) > 0\) such that the following holds. If \(z \in {\mathbb {R}}^{d}\), and \(V \in G(d,n)\) satisfy \(|\pi _{V}(z)| \leqslant \alpha |z|\), then there exists a plane \(V' \in G(d,n)\) with \(d(V,V') < \delta \) such that \(\pi _{V'}(z) = 0\).
Proof
The proof of [22, Lemma A.1] begins by establishing exactly this claim, although the statement of [22, Lemma A.1] does not mention it explicitly. \(\square \)
3 Dyadic reformulations
3.1 Dyadic cubes
It is known (see for example [13, §2]) that an n-regular set \(E \subset {\mathbb {R}}^{d}\) supports a system \({\mathcal {D}}\) of “dyadic cubes”, that is, a collection of subset of E with the following properties. First, \({\mathcal {D}}\) can be written as a disjoint union
where the elements \(Q \in {\mathcal {D}}_{j}\) are referred to as cubes of side-length \(2^{-j}\). For \(j \in {\mathbb {Z}}\) fixed, the sets of \({\mathcal {D}}_{j}\) are disjoint and cover E. For \(Q \in {\mathcal {D}}_{j}\), one writes \(\ell (Q) := 2^{-j}\). The side-length \(\ell (Q)\) is related to the geometry of \(Q \in {\mathcal {D}}_{j}\) in the following way: there are constants \(0< c< C < \infty \), and points \(c_{Q} \in Q \subset E\) (known as the “centres” of \(Q \in {\mathcal {D}}\)) with the properties
In particular, it follows from the n-regularity of E that \(\mu (Q) \sim \ell (Q)^{n}\) for all \(Q \in {\mathcal {D}}\). The balls \(B(c_{Q},C\ell (Q))\) containing Q are so useful that they will have an abbreviation:
If we choose the constant \(C \geqslant 1\) is large enough, as we do, the balls \(B_{Q}\) have the property
The “dyadic” structure of the cubes in \({\mathcal {D}}\) is encapsulated by the following properties:
-
For all \(Q,Q' \in {\mathcal {D}}\), either \(Q \subset Q'\), or \(Q' \subset Q\), or \(Q \cap Q' = \emptyset \).
-
Every \(Q \in {\mathcal {D}}_{j}\) has as parent \({\hat{Q}} \in {\mathcal {D}}_{j - 1}\) with \(Q \subset {\hat{Q}}\).
If \(Q \in {\mathcal {D}}_{j}\), the cubes in \({\mathcal {D}}_{j + 1}\) whose parent is Q are known as the children of Q, denoted \({{\mathbf {c}}}{{\mathbf {h}}}(Q)\). The ancestry of Q consists of all the cubes in \({\mathcal {D}}\) containing Q.
A small technicality arises if \({\text {diam}}(E) < \infty \): then the collections \({\mathcal {D}}_{j}\) are declared empty for all \(j < j_{0}\), and \({\mathcal {D}}_{j_{0}}\) contains a unique element, known as the top cube of \({\mathcal {D}}\). All of the statements above hold in this scenario, except that the top cube has no parents.
3.2 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 \({\mathcal {D}}\) on E.
Definition 3.1
(PBP) An n-regular set \(E \subset {\mathbb {R}}^{d}\) has PBP if there exists \(\delta > 0\) such that the following holds. For all \(Q \in {\mathcal {D}}\), there exists a ball \(S_{Q} \subset G(d,n)\) of radius \(\mathrm {rad}(S_{Q}) \geqslant \delta \) such that
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 \(B_{Q} = B(c_{Q},C\ell (Q))\) centred at \(c_{Q} \in E\). Only the dyadic PBP will be used below.
Definition 3.2
(WGL) An n-regular set \(E \subset {\mathbb {R}}^{d}\) satisfies the WGL if for all \(\epsilon > 0\), there exists a constant \(C(\epsilon ) > 0\) such that the following holds:
Here \(\mu := {\mathcal {H}}^{n}|_{E}\), \(\beta (Q) := \beta (B_{Q})\), and \({\mathcal {D}}(Q_{0}) := \{Q \in {\mathcal {D}}: Q \subset Q_{0}\}\).
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 \({\mathcal {D}}\), or subsets thereof, into trees:
Definition 3.3
(Trees) Let \(E \subset {\mathbb {R}}^{d}\) be an n-regular set with associated dyadic system \({\mathcal {D}}\). A collection \({\mathcal {T}} \subset {\mathcal {D}}\) is called a tree if the following conditions are met:
-
\({\mathcal {T}}\) has a top cube \(Q({\mathcal {T}}) \in {\mathcal {T}}\) with the property that \(Q \subset Q({\mathcal {T}})\) for all \(Q \in {\mathcal {T}}\).
-
\({\mathcal {T}}\) is consistent: if \(Q_{1},Q_{3} \in {\mathcal {T}}\), \(Q_{2} \in {\mathcal {D}}\), and \(Q_{1} \subset Q_{2} \subset Q_{3}\), then \(Q_{2} \in {\mathcal {T}}\).
-
If \(Q \in {\mathcal {T}}\), then either \({{\mathbf {c}}}{{\mathbf {h}}}(Q) \subset {\mathcal {T}}\) or \({{\mathbf {c}}}{{\mathbf {h}}}(Q) \cap {\mathcal {T}} = \emptyset \).
The final axiom allows to define the leaves of \({\mathcal {T}}\) consistently: these are the cubes \(Q \in {\mathcal {T}}\) such that \({{\mathbf {c}}}{{\mathbf {h}}}(Q) \cap {\mathcal {T}} = \emptyset \). The leaves of \({\mathcal {T}}\) are denoted \(\mathbf {Leaves}({\mathcal {T}})\). The collection \(\mathbf {Leaves}({\mathcal {T}})\) always consists of disjoint cubes, and it may happen that \(\mathbf {Leaves}({\mathcal {T}}) = \emptyset \).
Some trees will be used to prove the following reformulation of the WGL:
Lemma 3.4
Let \(E \subset {\mathbb {R}}^{d}\) be an n-regular set supporting a collection \({\mathcal {D}}\) of dyadic cubes. Let \(\mu := {\mathcal {H}}^{n}|_{E}\). Assume that for all \(\epsilon > 0\), there exists \(N = N(\epsilon ) \in {\mathbb {N}}\) such that the following holds:
Then E satisfies the WGL.
Remark 3.6
Chebyshev’s inequality applied to the set \(\{x \in Q : \sum _{Q' \subset Q, \beta (Q') \geqslant \epsilon } {\mathbf {1}}_{Q'}(x) \geqslant N\}\) shows that the WGL implies (3.5). Therefore (3.5) is equivalent to the WGL.
Proof of Lemma 3.4
Fix \(Q_{0} \in {\mathcal {D}}\) and \(\epsilon > 0\). We will show that
Abbreviate \({\mathcal {D}}:= \{Q \in {\mathcal {D}}: Q \subset Q_{0}\}\), and decompose \({\mathcal {D}}\) into trees by the following simple stopping rule. The first tree \({\mathcal {T}}_{0}\) has top \(Q({\mathcal {T}}_{0}) = Q_{0}\), and its leaves are the maximal cubes \(Q \in {\mathcal {D}}\) (if any should exist) such that
Here \(N = N(\epsilon ) \geqslant 1\), 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 \({\mathcal {T}}_{0},{\mathcal {T}}_{1},\ldots \) be the trees obtained by this process, with top cubes \(Q_{0},Q_{1},\ldots \) Note that \({\mathcal {D}}= \bigcup _{j \geqslant 0} {\mathcal {T}}_{j}\), and
Further, (3.5) implies that
On the other hand, the sets \(E_{j} := Q_{j} \, \setminus \, \cup \mathbf {Leaves}({\mathcal {T}}_{j})\) are disjoint. Now, we may estimate as follows:
This completes the proof of (3.7). \(\square \)
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 \(E \subset {\mathbb {R}}^{d}\) is an n-regular set with PBP. Then, for every \(\epsilon > 0\), there exists \(N \geqslant 1\), depending on d, \(\epsilon \), and the n-regularity and PBP constants of E, such that the following holds. The sets
satisfy \(\mu (E_{Q}) \leqslant \tfrac{1}{2}\mu (Q)\) for all \(Q \in {\mathcal {D}}\).
Proving this proposition will occupy the rest of the paper.
4 Construction of heavy trees
The proof of Proposition 3.8 proceeds by counter assumption: there exists a cube \(Q_{0} \in {\mathcal {D}}\), a small number \(\epsilon > 0\), and a large number \(N \geqslant 1\) of the form \(N = KM\), where also \(K,M \geqslant 1\) are large numbers, with the property
This will lead to a contradiction if both K and M are large enough, depending on d, \(\epsilon \), and the n-regularity and PBP constants of E. Precisely, \(M \geqslant 1\) gets chosen first within the proof of Proposition 4.2. The parameter \(K \geqslant 1\) 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 \({\mathcal {D}} := {\mathcal {D}}(Q_{0})\). We begin by using (4.1), and the definition of \(E_{Q_{0}}\), to construct a number of heavy trees \({\mathcal {T}}_{0},{\mathcal {T}}_{1},\ldots \subset {\mathcal {D}}\) with the following properties:
-
(T1)
\(\mu (E_{Q_{0}} \cap Q({\mathcal {T}}_{j})) \geqslant \tfrac{1}{4}\mu (Q({\mathcal {T}}_{j}))\) for all \(j \geqslant 0\).
-
(T2)
\(E_{Q_{0}} \cap Q({\mathcal {T}}_{j}) \subset \cup \mathbf {Leaves}({\mathcal {T}}_{j})\) for all \(j \geqslant 0\).
-
(T3)
For every \(j \geqslant 0\) and \(Q \in \mathbf {Leaves}({\mathcal {T}}_{j})\) it holds
$$\begin{aligned} {\text {card}}\{Q' \in {\mathcal {T}}_{j} : Q \subset Q' \subset Q({\mathcal {T}}_{j}) \text { and } \beta (Q') \geqslant \epsilon \} = M. \end{aligned}$$ -
(T4)
The top cubes satisfy \(\sum _{j} \mu (Q({\mathcal {T}}_{j})) \geqslant \tfrac{K}{4} \mu (Q_{0})\).
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 \(M \geqslant 1\) is large enough, depending only on d, \(\epsilon \), and the n-regularity and PBP constants of E, then \(\mathrm {width}({\mathcal {T}}_{j}) \geqslant \tau \mu (Q({\mathcal {T}}_{j}))\), where \(\tau > 0\) depends only on d, and the n-regularity and PBP constants of E.
Here \(\mathrm {width}({\mathcal {T}}_{j}) = \sum _{Q \in {\mathcal {T}}_{j}} \mathrm {width}(Q)\mu (Q)\) is a quantity to be properly introduced in Sect. 5. For now, we only need to know that the coefficients \(\mathrm {width}(Q)\) 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 \(M \geqslant 1\) is chosen so large that the hypothesis of Proposition 4.2 is met: every heavy tree \({\mathcal {T}}_{j}\) satisfies \(\mathrm {width}({\mathcal {T}}_{j}) \geqslant \tau \mu (Q({\mathcal {T}}_{j}))\). 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 \(\mathrm {width}({\mathcal {D}})\) if the constant \(K \geqslant 1\) is chosen large enough, depending on the admissible parameters. The proof of Proposition 3.8 is complete. \(\square \)
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 \({\mathcal {T}}_{0}\) has top \(Q({\mathcal {T}}_{0}) = Q_{0}\), and its leaves consist of the maximal cubes \(Q \in {\mathcal {D}}\) with the property that
The tree \({\mathcal {T}}_{0}\) itself consists of the cubes in \({\mathcal {D}}\) which are not strict sub-cubes of some \(Q \in \mathbf {Leaves}({\mathcal {T}}_{0})\). It is easy to check that \({\mathcal {T}}_{0}\) is a tree.
Assume then that some trees \({\mathcal {T}}_{0},\ldots ,{\mathcal {T}}_{k}\) have already been constructed. Let \(0 \leqslant j \leqslant k\) be an index such that for some \(Q \in \mathbf {Leaves}({\mathcal {T}}_{j})\), at least one cube \(Q_{k + 1} \in {{\mathbf {c}}}{{\mathbf {h}}}(Q)\) has not yet been assigned to any tree. The cube \(Q_{k + 1}\) then becomes the top cube of a new tree \({\mathcal {T}}_{k + 1}\), thus \(Q_{k + 1} = Q({\mathcal {T}}_{k + 1})\). The tree \({\mathcal {T}}_{k + 1}\) is constructed with the same stopping condition (4.4), just replacing \(Q({\mathcal {T}}_{0})\) by \(Q_{k + 1} = Q({\mathcal {T}}_{k + 1})\).
Note that if \(\mathbf {Leaves}({\mathcal {T}}_{j}) = \emptyset \) for some \(j \in {\mathbb {N}}\), then no further trees will be constructed with top cubes contained in \(Q({\mathcal {T}}_{j})\). As a corollary of the stopping condition, we record the uniform upper bound
We next prune the collection of trees. Let \(\mathbf {Top}\) be the collection of all the top cubes \(Q({\mathcal {T}}_{j})\) constructed above, and let \(\mathbf {Top}_{K} \subset \mathbf {Top}\) be the maximal cubes with the property
We discard all the trees whose tops are strictly contained in one of the cubes in \(\mathbf {Top}_{K}\), and we re-index the remaining trees as \({\mathcal {T}}_{0},{\mathcal {T}}_{1},{\mathcal {T}}_{2},\ldots \) Thus, the remaining trees are the ones whose top cube contains some element of \(\mathbf {Top}_{K}\). We record that
We write \({\mathcal {T}}:= \cup {\mathcal {T}}_{j}\) for brevity. We claim that
Indeed, fix \(x \in E_{Q_{0}}\), and recall that
by definition. We first claim that x is contained in \(\geqslant K + 1\) cubes in \(\mathbf {Top}\). If x was contained in \(\leqslant K\) cubes in \(\mathbf {Top}\), then x would be contained in \(\leqslant K - 1\) distinct leaves, and the stopping condition (4.4) would imply that
contradicting \(x \in E_{Q_{0}}\). Therefore, x is indeed contained in \(K + 1\) cubes in \(\mathbf {Top}\). Let the largest such top cubes be \(Q_{0} \supset Q_{1} \supset \cdots \supset Q_{K - 1} \supset Q_{K}\), so \(Q_{K - 1} \in \mathbf {Top}_{K}\). Now, it suffices to note that whenever \(x \in Q_{j}\), \(1 \leqslant j \leqslant K\), then x is contained in some element of \(\mathbf {Leaves}({\mathcal {T}}_{j - 1})\), which implies by the stopping condition that
Since \({\mathcal {T}}_{j - 1} \subset {\mathcal {T}}\) for \(1 \leqslant j \leqslant K\), the claim (4.7) follows by summing up (4.10) over \(1 \leqslant j \leqslant K\) and recalling that \(KM = N\).
We next verify that \(E_{Q_{0}} \cap Q({\mathcal {T}}_{j}) \subset \cup \mathbf {Leaves}({\mathcal {T}}_{j})\) for all \(j \geqslant 0\), as claimed in property (T2). Indeed, if \(x \in E_{Q_{0}} \cap Q({\mathcal {T}}_{j})\) for some \(j \geqslant 0\), then (4.8) holds, and \(Q({\mathcal {T}}_{j})\) is contained in \(\leqslant K\) elements of \(\mathbf {Top}\). This means that if \(x \in Q({\mathcal {T}}_{j}) \, \setminus \, \cup \mathbf {Leaves}({\mathcal {T}}_{j})\), then x is contained in \(\leqslant K - 1\) distinct leaves, and hence satisfies (4.9). But this would imply \(x \notin E_{Q_{0}}\). Hence \(x \in \mathbf {Leaves}({\mathcal {T}}_{j})\), 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 \(\mu (E_{Q_{0}}) \geqslant \tfrac{1}{2}\mu (Q_{0})\), we infer that
Recalling that \(N = KM\), this yields
Now, we discard all light trees with the property \(\mu (E_{Q_{0}} \cap Q({\mathcal {T}}_{j})) < \tfrac{1}{4}\mu (Q({\mathcal {T}}_{j}))\). Then, by the uniform upper bound (4.6), we have
Hence, the heavy trees with
satisfy
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 \({\mathcal {T}}_{0},{\mathcal {T}}_{1},\ldots \)
We have now proven Proposition 3.8 modulo Proposition 4.2, which concerns an individual heavy tree \({\mathcal {T}}_{j}\). Proving Proposition 4.2 will occupy the rest of the paper.
5 A criterion for positive width
Let \(E \subset {\mathbb {R}}^{d}\) be a closed n-regular set, write \(\mu := {\mathcal {H}}^{n}|_{E}\), and let \({\mathcal {D}}\) 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 \({\mathbb {R}}^{n}\), but only in the case \(n = d - 1\). I start here with the higher co-dimensional generalisation.
Definition 5.1
(Measure on the affine Grassmannian) Fix \(0< m < d\), and let \({\mathcal {A}} := {\mathcal {A}}(d,m)\) be the collection of all affine planes of dimension m. Define a measure \(\lambda := \lambda _{d,m}\) on \({\mathcal {A}}\) via the relation
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 G(d, n).
Definition 5.2
(Width) For \(Q \in {\mathcal {D}}\) and a plane \(W \in {\mathcal {A}}(d,d - n)\), we define
where we recall that \(B_{Q} = B(c_{Q},C\ell (Q))\) is a ball centred at some point \(c_{Q} \in Q \subset E\) containing Q. Then, we also define
Finally, if \({\mathcal {F}}\subset {\mathcal {D}}\) is an arbitrary collection of dyadic cubes, we set
The \(\mu (Q)\)-normalisation in (5.3) is the right one, because for \(V \in G(d,n)\) fixed, it is only possible that \(\mathrm {width}_{Q}(E,\pi _{V}^{-1}\{w\}) \ne 0\) if \(w \in \pi _{V}(B_{Q}) \subset V\), and \({\mathcal {H}}^{n}(\pi _{V}(B_{Q})) \sim \mu (Q)\). 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 \(C \geqslant 1\), depending only on the 1-regularity constant of E, such that
where \({\mathcal {D}}(Q_{0}) := \{Q \in {\mathcal {D}} : Q \subset Q_{0}\}\).
Proof
Fix \(Q_{0} \in {\mathcal {D}}\). By definitions,
The main tool in the proof is Eilenberg’s inequality
where \(A \subset {\mathbb {R}}^{d}\) is Borel, see [23, Theorem 7.7]. In particular, we infer from (5.8) that
for all \(V \in G(d,n)\) and for \({\mathcal {H}}^{n}\) a.e. \(w \in V\). We continue our estimate of (5.7) for a fixed plane \(V \in G(d,n)\), and for any \(w \in V\) such that \(q := q_{V,w} < \infty \). If \(q \in \{0,1\}\), then
so these pairs (V, w) contribute nothing to the integral in (5.7). So, assume that \(q \geqslant 2\), and enumerate the points in \(B_{Q_{0}} \cap E \cap \pi _{V}^{-1}\{w\}\) as
We will next need to construct a “spanning graph” whose vertices are the points \(x_{1},\ldots ,x_{q}\), and whose edges “\({\mathcal {E}}\)” are a (relatively small) subset of the \(\sim q^{2}\) segments connecting the vertices. More precisely, we need the following properties from \({\mathcal {E}}\):
-
(E1)
\({\text {card}}{\mathcal {E}} \lesssim _{d} q\).
-
(E2)
For every \(1 \leqslant i < j \leqslant q\), there is a connected union of edges in \({\mathcal {E}}\) which connects \(x_{i}\) to \(x_{j}\) inside \({\bar{B}}(x_{i},2|x_{i} - x_{j}|)\).
Property (E2) sounds like quasiconvexity, but is weaker: there are no restrictions on the length of the connecting \({\mathcal {E}}\)-path, as long as it is contained in \(B(x_{i},2|x_{i} - x_{j}|)\). Let us then find the edges with the properties (E1)–(E2). Let \(\xi _{1},\ldots ,\xi _{p} \subset S^{d - 1}\) be a maximal \(\tfrac{1}{4}\)-separated set on \(S^{d - 1}\), with \(p \sim _{d} 1\), and let
be a directed open cone around the half-line \(\{r\xi _{j} : r > 0\}\). By the net property of \(\xi _{1},\ldots ,\xi _{p}\),
We claim that the following holds: if \(y \in x + C_{j}\), then
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 \(y \in S^{d - 1}\), then the set \(C_{y} := \{re : e \in B(y,1) \cap S^{d - 1} \text { and } 0 < r \leqslant 1\}\) is contained in B(y, 1). Consequently,
We are then prepared to define the edge set \({\mathcal {E}}\). Fix one of the points \(x_{i}\), \(1 \leqslant i \leqslant q\). For every of \(1 \leqslant j \leqslant p\), 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 \(j \in \{1,\ldots ,p\}\) by (5.9). Thus, for every \(x_{i}\), one draws \(\sim _{d} 1\) edges. Let \({\mathcal {E}}\) be the collection of all edges so obtained. Then \({\text {card}}{\mathcal {E}} \sim _{d} q\), so requirement (E1) is met.
To prove (E2), fix \(s_{0} := x_{i}\) and \(t := x_{j}\) with \(1 \leqslant i < j \leqslant q\). The plan is to find, recursively, a collection of segments \(I_{j} := [s_{j - 1},s_{j}] \in {\mathcal {E}}\), \(1 \leqslant j \leqslant k\), whose union is connected, contains \(\{s_{0},t\}\) (indeed \(s_{k} = t\)) and is contained in
By (5.9), there is a half-cone \(C_{j_{1}}\) with \(t \in s_{0} + C_{j_{1}}\). Let \(I_{1} = [s_{0},s_{1}] \in {\mathcal {E}}\) be the edge connecting \(s_{0}\) to one of the nearest points \(s_{1} \in \{x_{1},\ldots ,x_{q}\} \cap (s_{0} + C_{j_{1}})\). Evidently \(|s_{0} - s_{1}| \leqslant |s_{0} - t|\), since \(t \in \{x_{1},\ldots ,x_{q}\} \cap (s_{0} + C_{j_{1}})\) 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
In particular,
Also, we see from (5.11) that \(\partial I_{1} = \{s_{0},s_{1}\} \subset {\bar{B}}(t,|s_{0} - t|)\), and hence \(I_{1} \subset {\bar{B}}(t,|s_{0} - t|)\) 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 \(t \in s_{1} + C_{j_{2}}\) (unless \(s_{1} = t\) and we are done already), and we let \(I_{2} = [s_{1},s_{2}] \in {\mathcal {E}}\) be the edge connecting \(s_{1}\) to the nearest point \(s_{2} \in \{x_{1},\ldots ,x_{q}\} \cap (s_{1} + C_{j_{2}})\). Then \(|s_{1} - s_{2}| \leqslant |s_{1} - t|\) (otherwise we chose t over \(s_{2}\)), so
From the inclusions above, we infer that \(I_{2} \subset {\bar{B}}(t,|s_{0} - t|)\), and also
We proceed inductively, finding further segments \([s_{i},s_{i + 1}] \in {\mathcal {E}}\), which are contained in \({\bar{B}}(t,|s_{0} - t|)\), and with the property that \(|s_{j + 1} - t|< |s_{j} - t|< \cdots < |s_{0} - t|\). Since the points \(s_{j}\) are drawn from the finite set \(\{x_{1},\ldots ,x_{q}\}\), these strict inequalities eventually force \(s_{k} = t\) for some \(k \geqslant 1\), and at that point the proof of property (E2) is complete.
Let us then use the edges \({\mathcal {E}}\) constructed above to estimate the integrand in (5.7). I claim that
To see this, fix \(Q \in {\mathcal {D}}(Q_{0})\), and let \(x_{i},x_{j} \in B_{Q} \cap E \cap \pi _{V}^{-1}\{w\} \subset \{x_{1},\ldots ,x_{q}\}\) be points such that
According to property (E2) of the edge family \({\mathcal {E}}\), there exists a connected union of segments in \({\mathcal {E}}\) which is contained in
and which contains \(\{x_{i},x_{j}\}\). Since the union is connected, the total length of the segments involved exceeds \(|x_{i} - x_{j}|\):
Swapping the order of summation proves (5.13). To complete the proof of the proposition, fix \(I \in {\mathcal {E}}\), and consider the inner sum in (5.13). Note that the inclusion \(I \subset 4B_{Q}\) is only possible if \(\ell (Q) > rsim |I|\). On the other hand, for a fixed side-length \(2^{-j} > rsim |I|\), there are \(\lesssim 1\) cubes \(Q \in {\mathcal {D}}(Q_{0})\) with \(\ell (Q) = 2^{-j}\) and \(I \subset 4B_{Q}\). Putting these observations together,
From this, (5.13), and the cardinality estimate \({\text {card}}{\mathcal {E}} \lesssim _{d} q\) 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. \(\square \)
Recall that our objective, in Proposition 4.2, is to prove that each heavy tree \({\mathcal {T}}_{j}\) satisfies \(\mathrm {width}({\mathcal {T}}_{j}) > rsim \mu (Q({\mathcal {T}}_{j}))\) if the parameter \(M \geqslant 1\) was chosen large enough. To accomplish this, we start by recording a technical criterion which guarantees that a general tree \({\mathcal {T}}\subset {\mathcal {D}}\) satisfies \(\mathrm {width}({\mathcal {T}}) > rsim \mu (Q({\mathcal {T}}))\). Afterwards, the criterion will need to be verified for heavy trees.
Proposition 5.14
For every \(c,\delta > 0\) and \(C_{0} \geqslant 1\) there exists \(N \geqslant 1\) such that the following holds. Assume that the n-regularity constant of E is at most \(C_{0}\). Let \({\mathcal {T}} \subset {\mathcal {D}}\) be a tree with top cube \(Q_{0} := Q({\mathcal {T}})\). Assume that there is a subset \({\mathcal {G}} \subset \mathbf {Leaves}({\mathcal {T}})\) with the following properties.
-
All the cubes in \({\mathcal {G}}\) have PBP with common plane \(V_{0} \in G(d,n)\) and constant \(\delta \):
$$\begin{aligned} {\mathcal {H}}^{n}(\pi _{V}(E \cap B_{Q})) \geqslant \delta \mu (Q), \qquad Q \in {\mathcal {G}}, \, V \in B(V_{0},\delta ). \end{aligned}$$(5.15) -
Write \(f_{V} := \sum _{Q \in {\mathcal {G}}} {\mathbf {1}}_{\pi _{V}(B_{Q})}\) for \(V \in B(V_{0},\delta )\). Assume that there is a subset \(S_{G} \subset B(V_{0},\delta )\) such that the “high multiplicity” sets \(H_{V} := \left\{ x \in V : {\mathcal {M}}f_{V}(x) \geqslant N\right\} \) satisfy
$$\begin{aligned} \int _{H_{V}} f_{V}(x) \, dx \geqslant cN^{-1}\mu (Q_{0}), \qquad V \in S_{G}. \end{aligned}$$(5.16)
Here \({\mathcal {M}}f_{V}\) 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 \({\mathcal {G}}\) had approximately the same generation in \({\mathcal {D}}\). In our application, this cannot be assumed, unfortunately, and we will need another auxiliary result to deal with the issue:
Lemma 5.17
Fix \(M,d,\gamma \geqslant 1\) and \(c > 0\). Then, the following holds if \(A = A_{d} \geqslant 1\) is large enough, depending only on d (as in “\({\mathbb {R}}^{d}\)”), and
Let \({\mathcal {B}}\) be a collection of balls contained in \(B(0,1) \subset {\mathbb {R}}^{d}\), and associate to every \(B \in {\mathcal {B}}\) a weight \(w_{B} \geqslant 0\). Set
and write \(H_{N} := \{{\mathcal {M}}f \geqslant N\}\), where \({\mathcal {M}}f\) is the Hardy–Littlewood maximal function of f. Assume that
Then, there exists a collection \({\mathcal {R}}_{\mathrm {heavy}}\) of disjoint cubes such that the “sub-functions”
satisfy the following properties:
The lemma is easy in the case where the balls in \({\mathcal {B}}\) have common radius, say r. Then one can take \({\mathcal {R}}_{\mathrm {heavy}}\) to be a suitable collection of disjoint cubes of side-length \(\sim r\). In the application to Proposition 5.14, this case corresponds to the situation where \(\ell (Q) \sim \ell (Q')\) for all \(Q,Q' \in {\mathcal {G}}\). 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 \(\mathrm {width}(Q)\) and \(\mathrm {width}({\mathcal {T}})\) from (5.3)–(5.4) and integrating (5.19) over \(V \in S_{G}\).
To prove (5.19), we assume, to avoid a rescaling argument, that \(\ell (Q_{0}) = 1\). Then, we begin by re-interpreting (5.16) in such a way that we may apply Lemma 5.17. Namely, we identify \(V \in S_{G}\) with \({\mathbb {R}}^{n}\), and consider the collection of balls
More precisely, let \({\mathcal {B}}\) be an index set for the balls \(\pi _{V}(B_{Q})\) such that if some ball \(B = \pi _{V}(B_{Q})\) arises from multiple distinct cubes \(Q \in {\mathcal {G}}\), then B has equally many indices in \({\mathcal {B}}\).
Note that the balls in \({\mathcal {B}}\) are all contained in
since \(B_{Q} \subset B_{Q'}\) whenever \(Q,Q' \in {\mathcal {D}}\) and \(Q \subset Q'\). We then define \(f := \sum _{B \in {\mathcal {B}}} {\mathbf {1}}_{B}\) and \(H_{N} := \{x \in V : {\mathcal {M}}f(x) \geqslant N\}\). It follows from (5.16), and the assumption \(\ell (Q_{0}) = 1\), that
In other words, the hypotheses of Lemma 5.17 are met with \(\gamma = 1\). We fix \(M := C\delta ^{-1}\), where \(C \geqslant 1\) is a large constant to be specified soon, depending only on the n-regularity constant of E. We then assume that \(N > AM^{3}/c\), in accordance with (5.18). Lemma 5.17 now provides us with a collection \({\mathcal {R}} = {\mathcal {R}}_{\mathrm {heavy}}\) of disjoint cubes in \({\mathbb {R}}^{n} \cong V\) such that
In this proof we abbreviate \(|\cdot | := {\mathcal {H}}^{n}|_{V}\). We recall that
where \(T(R) := \pi _{V}^{-1}(R)\). Therefore, the conditions in (5.20) are equivalent to
where the implicit constants depend on the n-regularity constant of \(\mu \). We now make a slight refinement to the set \({\mathcal {G}}\): for \(R \in {\mathcal {R}}\) fixed, we apply the 5r-covering theorem to the balls \(\{2B_{Q} : Q \in {\mathcal {G}} \text { and } B_{Q} \subset T(R)\}\). As a result, we obtain a sub-collection \({\mathcal {G}}_{R} \subset {\mathcal {G}}\) with the properties
and
In particular, by (5.21),
and
by (5.21). We also write \({\mathcal {B}}_{R} := \{\pi _{V}(B_{Q}) : Q \in {\mathcal {G}}_{R}\}\), \(R \in {\mathcal {R}}\), so \({\mathcal {B}}_{R} \subset {\mathcal {B}}\) is a collection of balls contained in R satisfying
Just like \({\mathcal {B}}\), the set \({\mathcal {B}}_{R}\) should also, to be precise, be defined as a set of indices, accounting for the possibility that \(B = \pi _{V}(B_{Q})\) arises from multiple cubes \(Q \in {\mathcal {G}}_{R}\). Next, recall a key assumption of the proposition, namely that all the cubes in \({\mathcal {G}}\) have PBP with common ball \(B(V_{0},\delta ) \subset G(d,n)\). In particular, for our fixed plane \(V \in S_{G} \subset B(V_{0},\delta )\), we have
Since the balls \(B_{Q}\), \(Q \in {\mathcal {G}}\), 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 \(B_{Q} \cap E\) by \(B_{Q} \cap E \cap B_{0}\). For \(B = \pi _{V}(B_{Q})\) with \(Q \in {\mathcal {G}}\), write \(E_{B} := \pi _{V}(B_{Q} \cap E \cap B_{0})\), so (5.26) implies that \(|E_{B}| > rsim \delta |B|\). Then, for \(R \in {\mathcal {R}}\) fixed, we infer from (5.25) that
We now choose the constant \(C \geqslant 1\) so large that
Then, if we consider the “set of multiplicity \(\leqslant 1\)”,
we may infer from (5.27) that
Consequently, if \(P_{R} := R \, \setminus \, L_{R}\) is the “positive multiplicity set”, we have
Fix \(w \in P_{R} \subset R\), and write
(If the sum happens to equal \(\infty \), pick \(m \geqslant 2\) arbitrary; eventually one will have to let \(m \rightarrow \infty \) in this case). Unraveling the definitions, the \((d - n)\)-plane \(W := W_{w} := \pi _{V}^{-1}\{w\}\) contains m points of \(E \cap B_{0}\) inside m distinct balls \(B_{Q}\), with \(Q \in {\mathcal {G}}_{R}\). Let \(P \subset E \cap W\) be the set of these m points, and define the following set \({\mathcal {E}}\) of edges connecting (some) pairs of points in P: for every point \(p \in P\), pick exactly one of the points \(q \in P \, \setminus \, \{p\}\) at minimal distance from p, and add the edge (p, q) to \({\mathcal {E}}\). Note that \({\text {card}}{\mathcal {E}} = m\), since \({\mathcal {E}}\) contains precisely one edge of the form (p, q) for every \(p \in P\). We have now used the assumption \(m \geqslant 2\): otherwise we could not have drawn any edges in the preceding manner! Note that the edges in the graph \((P,{\mathcal {E}})\) are directed: \((p,q) \in {\mathcal {E}}\) does not imply \((q,p) \in {\mathcal {E}}\).
Now that the edge set \({\mathcal {E}}\) has been constructed, define the following relation between edges \(I \in {\mathcal {E}}\) and the cubes \(Q \in {\mathcal {T}}\): write \(I \prec Q\) if \(I \subset B_{Q}\), and \(|I| \geqslant \rho \ell (Q)\). Slightly abusing notation, here I also refers to the segment [p, q], for an edge \((p,q) \in {\mathcal {E}}\). The choice of the constant \(\rho > 0\) will become apparent soon, and it will only depend on the n-regularity constant of E. We now claim that
We already know that \({\text {card}}{\mathcal {E}} = m\), so it remains to prove the first inequality. Fix \(I = (p,q) \in {\mathcal {E}}\), with \(p,q \in P\). Then, by the definition of P, the points p and q are contained in two balls \(B_{p} := B_{Q_{p}}\) and \(B_{q} := B_{Q_{q}}\), respectively, with \(Q_{p},Q_{q} \in {\mathcal {G}}_{R}\) and \(Q_{p} \ne Q_{q}\). In particular, we recall from (5.22) that \(2B_{p} \cap 2B_{q} = \emptyset \). Hence \(p \notin 2B_{q}\), and \(|I| > rsim \ell (Q_{q})\). On the other hand, \(p,q \in B_{0}\), so \(|I| \lesssim \ell (Q_{0})\). Let \(Q' \supset Q_{q}\) be the smallest cube in the ancestry of \(Q_{q}\) such that \(p,q \in B_{Q'}\). Then \(Q_{q} \subsetneq Q' \subset Q_{0}\), hence \(Q' \in {\mathcal {T}}\), and
Since \(p,q \in B_{Q'}\), by convexity also \(I \subset B_{Q'}\). If the constant “\(\rho \)” in the definition of “\(\prec \)” was chosen appropriately, we infer from \(I \subset B_{Q'}\) and (5.30) that \(I \prec Q'\). This proves the lower bound in (5.29).
Next, we claim that
Indeed, fix \(Q \in {\mathcal {T}}\) and assume that there is at least one edge \(I \in {\mathcal {E}}\) such that \(I \prec Q\). Then \(I \subset B_{Q} \cap W\), and both endpoints of I lie in E, so \({\text {diam}}(E \cap B_{Q} \cap W) \geqslant |I|\). Thus, (5.31) boils down to showing that \({\text {card}}\{I \in {\mathcal {E}} : I \prec Q\} \lesssim _{d} 1\). Let \(P_{Q} := \{p \in P : (p,q) \in {\mathcal {E}} \text { and } (p,q) \prec Q \text { for some } q \in P \, \setminus \, \{p\}\}\). Then
since \({\mathcal {E}}\) contains precisely one edge of the form (p, q) for all \(p \in P\), i.e. the map \(I = (p,q) \mapsto p\) is injective \(\{I \in {\mathcal {E}} : I \prec Q\} \rightarrow P_{Q}\). So, it remains to argue that \({\text {card}}P_{Q} \lesssim _{d} 1\). Otherwise, if \({\text {card}}P_{Q} \gg _{d} 1\), there exist two distinct points \(p_{1},p_{2} \in P_{Q}\) with \(|p_{1} - p_{2}| < \rho \ell (Q)\). However, if \(q \in P\) is such that \(I := (p_{1},q) \prec Q\), then \(|I| \geqslant \rho \ell (Q)\), and since \((p_{1},q) \in {\mathcal {E}}\), the point q must be one of the nearest neighbours of p in \(P \, \setminus \, \{p\}\). This is not true, however, since \(|p_{1} - p_{2}| < |p_{1} - q|\). 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 \(w \in R\) next gives
Finally, summing the result over the (disjoint) cubes \(R \in {\mathcal {R}}\), and using (5.23), we find that
This completes the proof of (5.19), and the proof of the proposition. \(\square \)
6 From big \(\beta \) numbers to heavy cones
Proposition 5.14 contains criteria for showing that \(\mathrm {width}({\mathcal {T}}) > rsim \mu (Q({\mathcal {T}}_{j}))\). To prove Proposition 4.2, these criteria need to be verified for the heavy trees \({\mathcal {T}}_{j}\). The selling points (T1)–(T4) of a heavy tree \({\mathcal {T}}_{j}\) were that all of its leaves are contained in M cubes in \({\mathcal {T}}_{j}\) with non-negligible \(\beta \)-number (see (T3)), and the total \(\mu \) measure of the leaves is at least \(\tfrac{1}{4}\mu (Q({\mathcal {T}}_{j}))\) (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 \({\mathcal {T}}_{j}\), 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 \(V_{0} \in G(d,n)\), \(\alpha > 0\), and \(x \in {\mathbb {R}}^{d}\). We write
For \(0< r< R < \infty \), we also define the truncated cones
Note the non-standard notation: \(X(x,V,\alpha )\) is a cone with axis \(V^{\perp } \in G(d,d - n)\)! The next proposition extracts “conical” information from many big \(\beta \)-numbers:
Proposition 6.2
Let \(\alpha ,d,\epsilon ,\theta > 0\) and \(C_{0},H \geqslant 1\). Then, there exists \(M \geqslant 1\), depending only on the previous parameters, such that the following holds. Let \(E_{0} \subset {\mathbb {R}}^{d}\) be a n-regular set with regularity constant at most \(C_{0}\), and let \(E \subset E_{0} \cap B(0,1)\) be a subset of measure \({\mathcal {H}}^{n}(E) \geqslant \theta > 0\) with the following property: for every \(x \in E\), there exist M distinct dyadic scales \(0< r < 1\) such that
Then, there exists a subset \(G \subset E\) of measure \({\mathcal {H}}^{1}(G) \geqslant \theta /2\) such that for all \(x \in G\),
The key point of Proposition 6.2 is that information about the \(\beta \)-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 \(\alpha ,d,\theta > 0\) and \(C_{0},H \geqslant 1\). Then, there exist constants \(\tau > 0\) and \(L \geqslant 1\), depending only on the previous parameters, such that the following holds. Let \(E_{0} \subset {\mathbb {R}}^{d}\) be an n-regular set with regularity constant at most \(C_{0}\), and let \(B \subset E_{0} \cap B(0,1)\) be a subset with \({\mathcal {H}}^{n}(B) \geqslant \theta \) satisfying the following: there exists \(V \in G(d,n)\) such that for every \(x \in B\),
Then, there exists a subset \(B' \subset B\) with \({\mathcal {H}}^{n}(B') \geqslant \tau \) which is contained on an L-Lipschitz graph over V. In fact, one can take \(L \sim 2^{H}/\alpha \).
We may then prove Proposition 6.2.
Proof of Proposition 6.2
It suffices to show that the subset \(B \subset E\) such that (6.3) fails has measure \({\mathcal {H}}^{n}(B) < \theta /2\) if \(M \geqslant 1\) was chosen large enough. Assume to the contrary that \({\mathcal {H}}^{n}(B) \geqslant \theta /2\). By definition, for every \(x \in B\), there exists an plane \(V_{x} \in G(d,n)\) such that
We observe that the dependence of \(V_{x}\) on \(x \in B\) can be removed, at the cost of making B and \(\alpha \) slightly smaller. Indeed, choose an \(\tfrac{\alpha }{2}\)-net \(V_{1},\ldots ,V_{k} \subset G(d,n)\) with \(k \sim _{\alpha ,d,n} 1\), and note that for every \(x \in B\), there exists \(1 \leqslant j \leqslant k\) such that
By the pigeonhole principle, there is a subset \(B' \subset B\) of measure \({\mathcal {H}}^{n}(B') > rsim _{\alpha ,d,n} {\mathcal {H}}^{n}(B) \geqslant \theta /2\) such that the choice of \(V := V_{j}\) is common for \(x \in B'\). It follows that (6.5) holds for this V, for all \(x \in B'\), with \(\tfrac{\alpha }{2}\) in place of \(\alpha \). We replace B by \(B'\) without altering notation, that is, we assume that (6.5) holds for all \(x \in B\), and for some fixed \(V \in G(d,n)\).
Now Proposition 6.4 can be applied to the set B, and the plane V. The conclusion is that there is a further subset \(B' \subset B\) of measure
which is contained in \(\Gamma \cap B(0,1)\), where \(\Gamma \subset {\mathbb {R}}^{d}\) is an L-Lipschitz graph over V for some \(L \sim 2^{H}/\alpha \sim _{\alpha ,H} 1\). We will derive a contradiction, using that \(B' \subset E\) and, consequently,
for all \(x \in B'\), and for M distinct dyadic scales \(0< r < 1\) (which may depend on \(x \in B'\)). For technical convenience, we prefer to work with a lattice \({\mathcal {D}}\) of dyadic cubes on \(E_{0}\). As usual, we define
Then, reducing “M” by a constant factor if necessary, it follows from (6.7) that every \(x \in B'\) is contained in \(\geqslant M\) distinct cubes \(Q \in {\mathcal {D}}\) of side-length \(0 < \ell (Q) \leqslant 1\) satisfying \(\beta _{E_{0}}(Q) \geqslant \epsilon \). Moreover, since \(B' \subset E \subset E_{0} \cap B(0,1)\), we may assume that \(B_{Q} \subset B(0,C)\) for all the cubes \(Q \in {\mathcal {D}}\), for some \(C \sim _{C_{0}} 1\).
The main tool is that since \(\Gamma \) is an n-dimensional L-Lipschitz graph in \({\mathbb {R}}^{d}\), 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 \(\Gamma _{\mathrm {bad}}\) of points \(x \in \Gamma \cap B(0,1)\) for which
for \(\geqslant M/2\) distinct dyadic scales \(0< r < 1\) has measure \({\mathcal {H}}^{n}(\Gamma _{\mathrm {bad}}) \ll 1\), and in particular \({\mathcal {H}}^{n}(\Gamma _{\mathrm {bad}}) \leqslant {\mathcal {H}}^{n}(B')/2\), assuming that \(M \geqslant 1\) is large enough, depending only on \(L,c,C_{0},d,H,\epsilon \), and \(\theta \). In (6.8), \(c > 0\) is a constant so small that
In particular, c only depends on the n-regularity constant of E. Further, in (6.8), the quantity \(\beta _{\Gamma ,\infty }(B(x,r))\) is the \(L^{\infty }\)-type \(\beta \)-number
As pointed out after Definition 1.4, the WGL holds for the \(L^{\infty }\)-type \(\beta \)-numbers if and only if it does for the \(L^{1}\)-type \(\beta \)-numbers \(\beta _{\Gamma }(B(x,r))\) (Dorronsoro’s strong geometric lemma holds for the latter, hence implies the WGL for the former).
We then focus attention on \(B'' := B' \, \setminus \, \Gamma _{\mathrm {bad}} \subset \Gamma \cap B(0,1)\), which still satisfies
recalling (6.6). Comparing (6.7) and (6.8), we find that every point \(x \in B''\) has the following property: there exist M/2 cubes \(Q \in {\mathcal {D}}\) such that \(x \in Q\),
Consider now a cube \(Q \in {\mathcal {D}}\) containing at least one point \(x \in B''\) such that (6.11) holds. In particular, recalling the choice of \(c > 0\) from (6.9), the intersection
is contained in a slab \(T \subset {\mathbb {R}}^{d}\) (a neighbourhood of an n-plane) of width \(\leqslant cc^{-1}\epsilon \ell (Q)/100 = \epsilon \ell (Q)/100\). Since \(\beta _{E_{0}}(Q) \geqslant \epsilon \), however, we have
In other words, for every \(Q \in {\mathcal {D}}\) containing some \(x \in B''\) such that (6.11) holds, there exists a subset \(E_{Q} \subset E_{0} \cap B_{Q} \subset B(0,C)\)
-
of measure \({\mathcal {H}}^{n}(E_{Q}) > rsim \epsilon {\mathcal {H}}^{n}(Q)\) which is contained
-
in the \(\sim \ell (Q)\)-neighbourhood of \(\Gamma \), yet
-
outside the \(\sim \epsilon \ell (Q)\)-neighbourhood of \(\Gamma \).
The collection of such cubes in \({\mathcal {D}}\) will be denoted \({\mathcal {G}}\). As observed above (6.11), we have
On the other hand, the sets \(E_{Q}\) have bounded overlap in the sense
since \(y \in {\mathbb {R}}^{d}\) can only lie in the sets \(E_{Q}\) associated to cubes \(Q \in {\mathcal {D}}\) with \(\ell (Q) \sim _{\epsilon } {\text {dist}}(y,\Gamma )\). Combining (6.12)–(6.13), we find that
We have shown that \({\mathcal {H}}^{n}(B'') \lesssim _{\epsilon } M^{-1}\). This inequality contradicts (6.10) if \(M \geqslant 1\) is large enough, depending on \(\alpha ,d,\epsilon ,C_{0},\theta \), and H. The proof of Proposition 6.2 is complete. \(\square \)
7 Heavy trees have positive width
We are equipped to prove Proposition 4.2. Fix a heavy tree \({\mathcal {T}}:= {\mathcal {T}}_{j}\), and recall from the heavy tree property (T3) that if \(Q \in \mathbf {Leaves}({\mathcal {T}})\), then
Moreover, by (T1)–(T2), the total measure of \(\mathbf {Leaves}({\mathcal {T}})\) is
Based on this information, we seek to verify the hypotheses of Proposition 5.14, which will eventually guarantee that \(\mathrm {width}({\mathcal {T}}) > rsim 1\) and finish the proof of Proposition 4.2. We split the argument into three parts.
7.1 Part I: finding heavy cones
Abbreviate \(Q_{0} := Q({\mathcal {T}})\) and \({\mathcal {L}} := \mathbf {Leaves}({\mathcal {T}})\). To avoid a rescaling argument later on, we assume with no loss of generality that
For every \(Q \in {\mathcal {L}}\), the PBP condition implies the existence of a plane \(V_{Q} \in G(d,n)\) such that
We would prefer that all the planes \(V_{Q}\) are the same, and this can be arranged with little cost. Namely, pick a \(\tfrac{\delta }{2}\)-net \(\{V_{1},\ldots ,V_{m}\} \subset G(d,n)\) with \(m \sim _{\delta ,d,n} 1\), and note that for all \(Q \in {\mathcal {L}}\), there is some \(V_{j}\) such that \(S_{j} := B(V_{j},\tfrac{\delta }{2}) \subset B(V_{Q},\delta ) =: S_{Q}\). Therefore, by the pigeonhole principle, there is a fixed index \(1 \leqslant j \leqslant m\) with the property
Let \({\mathcal {L}}_{G}\) be the good leaves satisfying \(S_{j} \subset S_{Q}\) for this j, and write \(S := S_{j}\) and \(V_{0} := V_{j}\). We have just argued that \(\mu (\cup {\mathcal {L}}_{G}) \sim _{\delta ,d,n} 1\), and (7.2) holds for all \(Q \in {\mathcal {L}}_{G}\), for all
From this point on, I cease recording the dependence of the “\(\lesssim \)” notation on the n-regularity and PBP constants \(C_{0}\) and \(\delta \).
For technical purposes, let us prune the set of good leaves a little further. Namely, apply the 5r-covering theorem to the balls \(10B_{Q}\), \(Q \in {\mathcal {L}}_{G}\). As a result, we obtain a sub-collection of the good leaves, still denoted \({\mathcal {L}}_{G}\), with the separation property
and such that the lower bound \(\mu (\cup {\mathcal {L}}_{G}) \sim 1\) 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 \(S = B_{G(d,n)}(V_{0},\tfrac{\delta }{2})\) is “small enough”, in a manner depending only on d.
For every \(Q \in {\mathcal {L}}_{G}\), pick an n-dimensional disc \(D_{Q} \subset B_{Q}\) which is parallel to the plane \(V_{0}\) and which satisfies
Such discs are pairwise disjoint by the separation property (7.3). We will also use frequently that the restrictions \(\pi _{V}|_{D_{Q}} :D_{Q} \rightarrow V\) are bilipschitz for all \(Q \in {\mathcal {L}}_{G}\) and \(V \in S = B_{G(d,n)}(V_{0},\tfrac{\delta }{2})\) if \(\delta > 0\) is small enough, as we assume. Therefore, the projections \(\pi _{V}(D_{Q}) \subset V\) are n-regular ellipsoids which contain, and are contained in, n-dimensional balls of radius \(\sim \mathrm {rad}(D_{Q})\).
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 \(\pi _{V\sharp }({\mathcal {H}}^{n}|_{E})\) is large at \(x \in V \in S\),
-
(b)
The Hardy–Littlewood maximal function of \(\pi _{V\sharp }({\mathcal {H}}^{n}|_{E_{D}})\) is large at \(x \in V \in S\).
Statement (b) contains much more information! Statement (a) could e.g. be true because a single cube \(Q \in {\mathcal {L}}_{G}\) satisfies \(\pi _{V}(Q) = \{x\}\). But since \(\pi _{V}|_{D_{Q}}\) is bilipschitz for all \(Q \in {\mathcal {L}}_{G}\) and \(V \in S\), statement (b) forces \(\pi _{V}^{-1}\{x\}\) to intersect many distinct balls \(B_{Q} \supset D_{Q}\). Recalling Proposition 5.14, this is helpful for finding a lower bound for \(\mathrm {width}({\mathcal {T}})\).
Let us verify that \(E_{+}\) is n-regular, with n-regularity constant \(\lesssim 1\). We leave checking the lower bound to the reader. To check the upper bound, fix \(x \in E_{+}\) and a radius \(r > 0\). Since E itself is n-regular, it suffices to show that
Write
and
For every \(Q \in {\mathcal {L}}^{\leqslant }_{G}\) we have \(Q \subset B(x,C'r)\) for some constant \(C' \sim 1\), so
Here we used that the leaves \({\mathcal {L}}\) consist of disjoint cubes. To finish the proof of (7.4), we claim that \({\text {card}}{\mathcal {L}}_{G}^{>} \leqslant 1\). Assume to the contrary that \(D_{Q},D_{Q'} \in {\mathcal {L}}_{G}^{>}\) with \(Q \ne Q'\). Then certainly \(2B_{Q} \cap B(x,r) \ne \emptyset \ne 2B_{Q'} \cap B(x,r)\), and both \(B_{Q},B_{Q'}\) have diameters \(\geqslant r\). This forces \(10B_{Q} \cap 10B_{Q'} \ne \emptyset \), violating the separation condition (7.3). This completes the proof of (7.4).
Let \(\mu _{+} := {\mathcal {H}}^{n}|_{E \cup E_{D}} = \mu + \sum _{Q \in {\mathcal {L}}_{G}} {\mathcal {H}}^{n}|_{D_{Q}}\), and define the associated \(\beta \)-numbers
We next claim that for every \(x \in E_{D}\) there exist \( > rsim M\) distinct dyadic radii \(0 < r \lesssim 1\) such that \(\beta _{+}(B(x,r)) > rsim \epsilon \). This follows easily by recalling that if \(x \in D_{Q}\) with \(Q \in {\mathcal {L}}_{Q} \subset {\mathcal {L}}\), then
by the definition of good leaves, but let us be careful: let \(x \in D_{Q}\), and let \(Q' \in {\mathcal {T}}\) be one of the ancestors of Q with
Since \(x \in D_{Q} \subset B_{Q} \subset B_{Q'}\), we have \(B_{Q'} \subset B(x,r)\) for some (dyadic) \(r \sim \mathrm {rad}(B_{Q'}) \lesssim 1\). Then, if \(V \in {\mathcal {A}}(d,n)\) is arbitrary, we simply have
which proves that \(\beta _{+}(B(x,r)) > rsim \epsilon \). A fixed radius “r” can only be associated to \(\lesssim 1\) cubes \(Q'\) in the ancestry of Q, so \( > rsim M\) of them arise in the manner above. The claim follows.
We note that
We aim to apply Proposition 6.2 to the set \(E_{D}\), but we will perform a final pruning before doing so. Let \(c > 0\) be a small constant to be determined soon, and let \({\mathcal {L}}_{G,\mathrm {light}} \subset {\mathcal {L}}_{G}\) consist of the good leaves with the following property: there exists a point \(x_{Q} \in D_{Q}\) and a radius \(0 < r_{Q} \leqslant 1\) such that
Evidently \(D_{Q} \subset B(x_{Q},r_{Q}/5)\) if \(c > 0\) is small enough, since if \(D_{Q} \not \subset B(x_{Q},r_{Q}/5)\), then
We also observe that since \(x_{Q} \in D_{Q} \subset B_{Q} \subset B_{Q_{0}}\), and \(r_{Q} \leqslant 1 = \ell (Q_{0})\), we have \(B(x_{Q},r_{Q}) \subset 2B_{Q_{0}}\) for all \(Q \in {\mathcal {L}}_{G,\mathrm {light}}\). Now, use the 5r covering theorem to find a subset \({\mathcal {L}}' \subset {\mathcal {L}}_{G,\mathrm {light}}\) such that the associated balls \(B(x_{Q},r_{Q}/5)\) are disjoint, and
It follows from (7.6), and the n-regularity of \(\mu _{+}\), that
Comparing this upper bound with (7.5), we find that if \(c > 0\) was chosen small enough, depending only on the PBP and n-regularity constants of E, then
where \({\mathcal {L}}_{G,\mathrm {heavy}} = {\mathcal {L}}_{G} \, \setminus \, {\mathcal {L}}_{G,\mathrm {light}}\). Let \(E_{D,\mathrm {dense}}\) be the union of the discs \(D_{Q}\) with \(Q \in {\mathcal {L}}_{G,\mathrm {heavy}}\). We summarise the properties of \(E_{D,\mathrm {dense}} \subset E_{D} \subset E_{+}\):
-
(1)
\(\mu _{+}(E_{D,\mathrm {dense}}) \sim 1\),
-
(2)
If \(x \in E_{D,\mathrm {dense}}\), there are \( > rsim M\) dyadic scales \(0 < r \lesssim 1\) such that \(\beta _{+}(B(x,r)) > rsim \epsilon \),
-
(3)
If \(x \in E_{D,\mathrm {dense}}\), then \(\mu _{+}(E_{D} \cap B(x,r)) > rsim r\) for all \(0 < r \leqslant 1\).
We then apply Proposition 6.2 to the set \(E_{D,\mathrm {dense}}\) with a “multiplicity” parameter \(H \geqslant 1\) 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 \(\alpha \) and \(\theta \) in the statement of the proposition are set to be such that \(\alpha \sim _{d,\delta } 1\) (specifics to follow later), and \(\theta \sim 1\) is so small that \({\mathcal {H}}^{n}(E_{D,\mathrm {dense}}) \geqslant \theta \), which is possible by (1) above. As a good first approximation of how to choose \(\alpha \), recall from Lemma 2.4 that if \(x \in {\mathbb {R}}^{d}\) and \(|\pi _{V_{0}}(x)| \leqslant \alpha |x|\), where \(\alpha = \alpha (d,\delta ) > 0\) is small enough, then there exists a plane \(V \in B_{G(d,n)}(V_{0},\tfrac{\delta }{2}) = S\) such that \(\pi _{V}(x) = 0\). In symbols, the previous statement is equivalent to
In fact, in the case \(n = d - 1\), this would be a suitable definition for \(\alpha \), and the reader may think that \(\alpha \) is at least so small that (7.7) holds. In the case \(n < d - 1\), additional technicalities force us to pick \(\alpha \) slightly smaller.
Proposition 6.2 then states that if \(M \geqslant 1\) is chosen large enough, in a manner depending only on \(\alpha ,H,d,\delta ,\epsilon ,\theta \), and the n-regularity constant of E, the following holds: there exists a subset \(G \subset E_{D,\mathrm {dense}}\) of measure
with the property
(The upper bound in (7.8) follows from \(G \subset E_{D}\) and \({\text {diam}}(E_{D}) \lesssim \ell (Q_{0}) = 1\)). We next upgrade (7.9) to a measure estimate, using the definition of \(E_{D,\mathrm {dense}}\). Namely, recall from (3) above that if \(y \in E_{D,\mathrm {dense}}\), then \(\mu _{+}(E_{D} \cap B(y,r)) > rsim r^{n}\) for all \(0 < r \leqslant 1\). By definitions and a few applications of the triangle inequality,
and hence
for all those scales \(2^{-j}\) such that \(X(x,V_{0},\tfrac{\alpha }{2},2^{-j - 1},2^{-j})\) contains some \(y \in E_{D,\mathrm {dense}}\). (Here we used that \(\alpha \sim _{d,\delta } 1\).) For \(x \in G\), 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 \(2^{-i} \in \{2^{-j - 1},2^{-j},2^{-j + 1}\}\) satisfies \({\mathcal {H}}^{n}(E_{D} \cap X(x,V_{0},\alpha ,2^{-i - 1},2^{-i})) \geqslant c2^{-in}\). Here \(c \sim 1\) 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:
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 \(V \in S = B_{G(d,n)}(V_{0},\tfrac{\delta }{2})\), write
interpreted as a function on \({\mathbb {R}}^{n}\), and let \({\mathcal {M}} f_{V}\) stand for the centred Hardy–Littlewood maximal function of \(f_{V}\). We will prove the following claim:
Claim 7.12
For every \(x \in G\), there exists a subset \(S_{x} \subset S\) of measure \(\gamma _{d,n}(S_{x}) > rsim 1/\sqrt{H}\) 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
By (7.11), there exist H distinct indices \(j \geqslant 0\) such that \({\mathcal {H}}^{n}(E_{j,x}) \geqslant c2^{-jn}\). 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 \(\pi _{V}^{-1}\{\pi _{V}(x)\}\), \(V \in S\), or then \(E_{j,x}\) meets fairly many of the planes \(\pi _{V}^{-1}\{\pi _{V}(x)\}\), \(V \in S\), for every one of the H indices “j”.
Case 1. Fix \(x \in G\), assume with no loss of generality that \(x = 0\). This has the notational benefit that \(\pi _{V}^{-1}\{\pi _{V}(x)\} = V^{\perp }\) for \(V \in G(d,n)\). Assume that there exists at least one index \(j \geqslant 0\) such that \({\mathcal {H}}^{n}(E_{j,x}) \geqslant c2^{-jn}\), and
Fix such an index \(j \geqslant 0\), and abbreviate \(E_{j,0} := E_{0}\). Then (7.15) will imply that most of the (non-negligible) \({\mathcal {H}}^{n}\) mass of \(E_{0} \subset X(0,V_{0},\alpha )\) is contained in narrow slabs around \((d - n)\)-planes with “high density”. As in the classical proof of the Besicovitch–Federer projection theorem, the case \(n < d - 1\) 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 Sect. 2, we define
and we write \(\gamma _{W,n + 1,n}\) for the \({\mathcal {O}}(d)\)-invariant probability measure on G(W, n). The metric on G(W, n) is inherited from G(d, n). Recall the Fubini formula established in Lemma 2.2:
for \(B \subset G(d,n)\) Borel. We will need to find a Borel set \({\mathcal {W}} \subset G(d,n + 1)\), in fact a ball, which may depend on j and x, with the following properties:
-
(W1)
\(\gamma _{d,n + 1}({\mathcal {W}}) \sim _{d,\delta } 1\),
-
(W2)
For every \(W \in {\mathcal {W}}\), the set \(S \cap G(W,n)\) contains a ball \(S_{W} = B_{G(W,n)}(V_{W},\tfrac{\delta }{4})\),
-
(W3)
There exists a subset \(E_{{\mathcal {W}},0} \subset E_{0}\) of measure \({\mathcal {H}}^{n}(E_{{\mathcal {W}},0}) \geqslant c2^{-jn}\) with the property
$$\begin{aligned} E_{{\mathcal {W}},0} \subset \bigcup _{V \in S_{W}} V^{\perp }, \qquad W \in {\mathcal {W}}. \end{aligned}$$
The “c” appearing in property (W3) may be a constant multiple (depending on \(\delta ,d\)) of the constant in \({\mathcal {H}}^{n}(E_{0}) \geqslant c2^{-jn}\). Finding \({\mathcal {W}}\) with the properties (W1)–(W3) is easy if \(n = d - 1\), so let us discuss this case first to get some intuition. Simply take \({\mathcal {W}} := G(d,d) = \{{\mathbb {R}}^{d}\}\). Note that in this case \(G(W,n) \equiv G(d,n)\). Evidently (W1)–(W2) are satisfied, even with \(S_{W} := S\). Also, (W3) is satisfied with \(E_{{\mathcal {W}},0} := E_{0}\) by (7.7), which implies that \(E_{0} \subset X(0,V_{0},\alpha ) \subset \bigcup _{V \in S} V^{\perp }\).
We then treat the general case. In the process, we also finally fix the angular parameter \(\alpha \sim _{d,\delta } 1\). Recall that \(E_{0} \subset X(0,V_{0},\alpha ,2^{-j - 1},2^{-j})\), that is, \(|\pi _{V_{0}}(z)| \leqslant \alpha |z|\) and \(|z| \sim 2^{-j}\) for all \(z \in E_{0}\). Start by choosing a point \(z_{0} \in E_{0}\) such that
where \(0 < \rho \leqslant \min \{\tfrac{1}{10},\alpha ,\delta \}\) is a parameter to be chosen momentarily (we will have \(\rho \sim _{\delta ,d} 1\)). We then define
so at least the measure estimate in (W3) is satisfied by (7.17). Write \(W_{0} := {\text {span}}(V_{0},z_{0}) \in G(d,n + 1)\) (evidently \(z_{0} \notin V_{0}\) since \(|\pi _{V_{0}}(z_{0})| < |z_{0}|\)), and set \({\mathcal {W}} := B(W_{0},\rho )\). Then \(\gamma _{d,n + 1}({\mathcal {W}}) \sim _{d,\delta } 1\), so property (W1) is satisfied.
We next verify (W2). Let \(W \in {\mathcal {W}}\), that is, \(d(W,W_{0}) \leqslant \rho \). Then, since \(V_{0} \subset W_{0}\), Lemma 2.1 implies that there exists a plane \(V_{W} \in G(W,n)\) with \(d(V_{W},V_{0}) \lesssim \rho \). In particular, \(V_{W} \in B_{G(d,n)}(V_{0},\tfrac{\delta }{4})\) if \(\rho \) is chosen small enough, and consequently
This completes the proof of (W2).
To prove (W3), we need to check that if \(W \in {\mathcal {W}}\) and \(z \in E_{{\mathcal {W}},0}\), then there exists a plane \(V \in S_{W}\) with \(\pi _{V}(z) = 0\). This will be accomplished by an application of Lemma 2.4 inside \(W \cong {\mathbb {R}}^{n + 1}\). First, since \(z \in E_{{\mathcal {W}},0} \subset E_{0}\), \(V_{W} \subset W\), and \(d(V_{W},V_{0}) \lesssim \rho \leqslant \alpha \), we have
Second,
using that \(z_{0} \in W_{0}\), and \(z \in B(z_{0},\rho 2^{-j}) \subset B(z_{0},|z_{0}|/2)\), and \(d(W,W_{0}) \leqslant \rho \). Combining (7.18)–(7.19), and setting \(z_{W} := \pi _{W}(z) \in W\), we find that
Finally, the estimate (7.20) allows us to apply Lemma 2.4 to the point \(z_{W} \in W\) in the space \(G(W,n) \cong G(n + 1,n)\). The conclusion is that if \(\alpha \) is small enough, depending only on \(\delta ,n\), then there exists a plane \(V \in B_{G(W,n)}(V_{W},\tfrac{\delta }{4}) = S_{W}\) such that \(\pi _{V}(z_{W}) = 0\). But now \(V \subset W\), and \(\pi _{W}(z - z_{W}) = 0\), so also \(\pi _{V}(z) = \pi _{V}(z_{W}) + \pi _{V}(z - z_{W}) = 0\). 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 \(\gamma _{d,n}(\{V \in S : V^{\perp } \cap E_{0} \ne \emptyset \}) \leqslant 1/\sqrt{H}\). Combined with the Fubini formula (7.16), this implies that the set of planes \(W \in G(d,n + 1)\) such that
has \(\gamma _{d,n + 1}\)-measure at most \(C^{-1}\), for \(C \geqslant 1\). Choose \(C \sim _{\delta } 1\) here so large that the planes \(W \in G(d,n + 1)\) in question have total measure \(\leqslant \tfrac{1}{2}\gamma _{d,n + 1}({\mathcal {W}})\). After discarding these “bad” planes from \({\mathcal {W}}\), we may assume that the opposite of (7.21) holds for all \(W \in {\mathcal {W}}\):
Fix \(W \in {\mathcal {W}}\), so (7.22) holds, and abbreviate \(\gamma _{W,n + 1,n} =: \gamma _{n + 1,n}\). Then, let \({\mathcal {S}}\) be a system of dyadic cubes on the (n-regular) ball \(S_{W} \subset G(W,n)\), with top cube \(S_{W}\). Then, cover the set
by a disjoint collection \({\mathcal {Q}} \subset {\mathcal {S}}\) of these cubes such that
For \(Q \in {\mathcal {Q}}\), write \({\mathcal {C}}(Q) := \cup \{V^{\perp } : V \in Q\}\), generalising the notation \({\mathcal {C}}(S)\) introduced in (7.7). Since \({\bar{S}}_{W}\) is covered by the cubes \(Q \in {\mathcal {Q}}\), the set \(E_{{\mathcal {W}},0} \subset E_{0} \cap \bigcup _{V \in S_{W}} V^{\perp }\) is covered by the cones \({\mathcal {C}}(Q)\), \(Q \in {\mathcal {Q}}\). Now, let \({\mathcal {Q}}_{\mathrm {light}}\) be the cubes \(Q \in {\mathcal {Q}}\) satisfying
Then,
Recalling from (W3) that \({\mathcal {H}}^{n}(E_{{\mathcal {W}},0}) \geqslant c2^{-jn}\), and that \(E_{{\mathcal {W}},0}\) is covered by the union of the cones \({\mathcal {C}}(Q)\), \(Q \in {\mathcal {Q}}\), we infer that there is a subset \({\bar{E}}_{{\mathcal {W}},0} \subset E_{{\mathcal {W}}}\) of measure \({\mathcal {H}}^{n}({\bar{E}}_{{\mathcal {W}},0}) \geqslant \tfrac{c}{2} \cdot 2^{-jn}\) which is covered by the union of the cones \({\mathcal {C}}(Q)\), \(Q \in {\mathcal {Q}} \, \setminus \, {\mathcal {Q}}_{\mathrm {light}}\). Every cube \(Q \in {\mathcal {Q}} \, \setminus \, {\mathcal {Q}}_{\mathrm {light}}\) satisfies the inequality reverse to (7.23), and is consequently contained in some maximal cube in \({\mathcal {S}}\) with this property. Let \({\mathcal {Q}}_{\mathrm {heavy}}\) be the collection of such maximal (hence disjoint) cubes. Then, since \(Q \subset Q'\) implies \({\mathcal {C}}(Q) \subset {\mathcal {C}}(Q')\), we see that \({\bar{E}}_{{\mathcal {W}},0}\) is also covered by the union of the cones \({\mathcal {C}}(Q)\), \(Q \in {\mathcal {Q}}_{\mathrm {heavy}}\), and consequently
We moreover claim that the union of the heavy cubes, denoted \(H_{W}\), satisfies
Indeed, if \(S_{W} \in {\mathcal {Q}}_{\mathrm {heavy}}\), there is nothing to prove, since \(\gamma _{n,n + 1}(S_{W}) \sim _{\delta ,d} 1\). If, on the other hand, \(S_{W} \notin {\mathcal {Q}}_{\mathrm {heavy}}\), then the parent \({\hat{Q}}\) of every cube \(Q \in {\mathcal {Q}}_{\mathrm {heavy}}\) satisfies (7.23), by the maximality of Q. Of course (7.24) remains valid if we replace “Q” by “\({\hat{Q}}\)”. Putting these pieces together, we find that
This completes the proof of (7.25).
We are now ready to prove Claim 7.12 in Case 1, that is, define the set \(S_{x} = S_{0} \subset S\) such that (7.13) holds for all \(V \in S_{0}\). Define
Then, by the Fubini formula (7.16), and the uniform lower bound (7.25), we have
as required by Claim 7.12. It remains to establish the lower bound (7.13), namely that if \(V \in S_{0} (= S_{x})\), then \({\mathcal {M}}f_{V}(\pi _{V}(x)) = {\mathcal {M}}f_{V}(0) > rsim \sqrt{H}\). Fix \(V \in S_{0}\), let first \(W \in {\mathcal {W}}\) be such that \(V \in H_{W}\), and then let \(Q \in {\mathcal {Q}}_{W,\mathrm {heavy}} = {\mathcal {Q}}_{\mathrm {heavy}}\) be the unique cube with \(V \in Q\) (we do not claim, however, that the choice of W would be unique). By definitions, especially recalling that \(E_{{\mathcal {W}},0} \subset E_{0} \subset E_{D} \cap {\bar{B}}(2^{-j}) \, \setminus B(2^{-j - 1})\), we have
where of course \({\mathcal {C}}(Q,r,R) := {\mathcal {C}}(Q) \cap {\bar{B}}(R) \, \setminus \, B(r)\), and we recall that \({\mathcal {C}}(Q) = \{V^{\perp } : V \in Q\}\). Note that \({\mathcal {C}}(Q,2^{-j - 1},2^{-j}) \subset T = T_{V}\), where \(T \subset {\mathbb {R}}^{d}\) is a slab of the form
of width \(\sim _{d} 2^{-j}\ell (Q)\) around the plane \(V^{\perp } \in G(d,d - n)\). Indeed, if \(x \in {\mathcal {C}}(Q,2^{-j - 1},2^{-j})\), then \(\pi _{V'}(x) = 0\) for some \(V' \in Q\). Then \(d(V,V') \lesssim _{d} \ell (Q)\), and \(|\pi _{V}(x)| \leqslant d(V,V') \cdot |x| \lesssim 2^{-j}\ell (Q)\), which means that \(x \in T\) if the constant \(C \geqslant 1\) is chosen appropriately.
Write \(B_{V} := B(0,C2^{-j}\ell (Q)) \subset V\). With this notation, recalling that \(D_{Q} \subset B_{Q}\), and using that the projections \(\pi _{V}|_{D_{Q}} :D_{Q} \rightarrow V\) are bilipschitz for \(Q \in {\mathcal {L}}_{G}\) and \(V \in S_{0} \subset S\), we infer that
In final estimate, we used that \(\gamma _{n + 1,n}(Q) \sim \ell (Q)^{n}\). This is the whole point of the integralgeometric argument: without splitting G(d, n) into a “product” of \(G(d,n + 1)\) and G(W, n), we could have, more easily, reached the penultimate estimate with “\(\gamma _{d,n}(Q)\)” in place of “\(\gamma _{n + 1,n}(Q)\)”. But \(\gamma _{d,n}(Q) \sim \ell (Q)^{n(d - n)} \ll \ell (Q)^{n}\) if \(n < d - 1\), and the final estimate would have failed. We have now proved Claim 7.12 in Case 1.
Case 2. Again, fix \(x \in G\), assume with no loss of generality that \(x = 0\), and let \(j_{1},\ldots ,j_{H} \geqslant 0\) be distinct scale indices such that \({\mathcal {H}}^{n}(E_{j_{i},0}) \geqslant c2^{-j_{i}n}\) for all \(1 \leqslant i \leqslant H\), recall the notation from (7.14). This time, we assume that
where \({\bar{S}}_{0,i} := \{V \in S : V^{\perp } \cap E_{j_{i},0} \ne \emptyset \}\). It follows from (7.28) that
Let
Then, it follows by splitting the integration in (7.29) to \(S \, \setminus \, S_{0}\) and \(S_{0}\), that
Recalling that \(\gamma _{d,n}(S) \leqslant \tfrac{1}{2}\) (that is, \(S = B_{G(d,n)}(V_{0},\tfrac{\delta }{2})\) is a fairly small ball), we find that \(\gamma _{d,n}(S_{0}) > rsim 1/\sqrt{H}\), as required by Claim 7.12. It remains to check that \({\mathcal {M}}f_{V}(\pi _{V}(x)) = {\mathcal {M}}f_{V}(0) > rsim \sqrt{H}\) whenever \(V \in S_{0}\).
Fixing \(V \in S_{0}\), it follows by definition that there are \(\geqslant \sqrt{H}\) indices \(i \in \{1,\ldots ,H\}\) with the property that \(V \in {\bar{S}}_{0,i}\), which meant by definition that
For each of these indices i, the plane \(V^{\perp }\) intersects at least one of the discs \(D_{Q}\) with \(Q \in {\mathcal {L}}_{G}\), whose union is \(E_{D}\). Moreover, since the sets \(E_{j,0} \subset {\bar{B}}(2^{-j}) \, \setminus \, {\bar{B}}(2^{-j - 1})\) are disjoint for distinct indices \(j \geqslant 0\), we conclude that \(V^{\perp }\) meets \(\geqslant \sqrt{H}\) distinct discs \(D_{Q}\). Consequently, recalling also that \(D_{Q} \subset B_{Q}\) for all \(Q \in {\mathcal {L}}_{G}\),
A similar lower bound for \({\mathcal {M}}f_{V}\) follows easily from the special structure of \(f_{V}\): whenever \(V \in S_{0} \subset S\) and \(f_{V}(0) \geqslant \sqrt{H}\), we may pick the \(h := \sqrt{H}\) largest balls \(B_{1},\ldots ,B_{h}\) of the form \(\pi _{V}(B_{Q}) \subset V\), \(Q \in {\mathcal {L}}_{G}\), which contain 0. Writing \(r := \min \{\mathrm {rad}(B_{k}) : 1 \leqslant k \leqslant h\}\),
as claimed. This completes the proof of (7.13), and Claim 7.12, in Case 2.
7.3 Part III: conclusion
We then proceed with the proof of Proposition 4.2. Recall from (7.8) that \({\mathcal {H}}^{n}(G) \sim 1\). In Claim 7.12, we showed that to every \(x \in G\) we may associate a set of planes \(S_{x} \subset S\) of measure \(\gamma _{d,n}(S_{x}) > rsim 1/\sqrt{H}\) such that \({\mathcal {M}}f_{V}(\pi _{V}(x)) > rsim \sqrt{H}\) holds for all \(V \in S_{x}\). Writing \(G_{V} := \{x \in G : V \in S_{x}\}\) for \(V \in S\), it follows that
Recalling from (7.8) that \({\mathcal {H}}^{n}(G_{V}) \leqslant {\mathcal {H}}^{n}(G) \lesssim 1\) for all \(V \in S\), we infer that the subset
has measure \(\gamma _{d,n}(S_{G}) > rsim 1/\sqrt{H}\). The plan is now to verify that the hypotheses of Proposition 5.14 are valid for the subset \(S_{G} \subset S\), and with parameter \(N \sim \sqrt{H}\) (this “N” has nothing to do with \(N = KM\)). Consider \(V \in S_{G}\). By definition, \({\mathcal {H}}^{n}(G_{V}) > rsim 1/\sqrt{H}\), and
Write \(H_{V} := \pi _{V}(G_{V})\). Then, (7.30) is equivalent to
Moreover, recalling that \(G_{V} \subset G \subset E_{D}\) is covered by the discs \(D_{Q}\), \(Q \in {\mathcal {L}}_{G}\), and using the inequality (based on \(D_{Q} \subset B_{Q}\) and the bilipschitz property of \(\pi _{V}|_{D_{Q}} :D_{Q} \rightarrow V\))
we find that
Now, (7.32) says that the hypothesis (5.16) of Proposition 5.14 is satisfied for the set of leaves \({\mathcal {G}} := {\mathcal {L}}_{G}\), the set of planes \(S_{G} \subset S\), and with the constant “\(H'\)” in place of “N”. Moreover, by their definition below (7.2), all the cubes \(Q \in {\mathcal {L}}_{G}\) satisfy the PBP condition with common plane \(V_{0}\):
Consequently, Proposition 5.14 states that if the parameter \(H'\) is chosen large enough, depending only on \(C_{0}\) and \(\delta \), then
As explained above (7.8), choosing \(H' = \sqrt{H}\) this big means forces us to choose the parameter \(M \geqslant 1\) large enough in a manner depending on
So, in fact \(M \sim _{C_{0}d,\delta ,\epsilon } 1\), as claimed in Proposition 4.2. Since the lower bound for \(\mathrm {width}({\mathcal {T}})\) 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 Sect. 4, we have now proved Proposition 3.8. As we recorded in Lemma 3.4, this implies that n-regular sets \(E \subset {\mathbb {R}}^{d}\) having PBP satisfy the WGL, and then the BPLG property follows from Theorem 1.5. This completes the proof of Theorem 1.6.
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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.
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T.O. is supported by the Academy of Finland via the project Quantitative rectifiability in Euclidean and non-Euclidean spaces, Grant Nos. 309365, 314172, and via the project Incidences on Fractals, Grant No. 321896.
Appendix A: A variant of the Lebesgue differentiation theorem
Appendix A: A variant of the Lebesgue differentiation theorem
Here we prove Lemma 5.17, which we restate below for the reader’s convenience:
Lemma A.1
Fix \(M,d,\gamma \geqslant 1\) and \(c > 0\). Then, the following holds if \(A = A_{d} \geqslant 1\) is large enough, depending only on d (as in “\({\mathbb {R}}^{d}\)”), and
Let \({\mathcal {B}}\) be a collection of balls contained in \([0,1)^{d} \subset {\mathbb {R}}^{d}\), and associate to every \(B \in {\mathcal {B}}\) a weight \(w_{B} \geqslant 0\). Set
and write \(H_{N} := \{{\mathcal {M}}f \geqslant N\}\), where \({\mathcal {M}}f\) is the Hardy–Littlewood maximal function of f. Assume that
Then, there exists a collection \({\mathcal {R}}_{\mathrm {heavy}}\) of disjoint cubes such that the “sub-functions”
satisfy the following properties:
Remark A.5
Comparing with (A.3), the first property in (A.4) states a non-negligible fraction of the \(L^{1}\)-mass of f is preserved in the functions \(f_{R}\), \(R \in {\mathcal {R}}_{\mathrm {heavy}}\). In conjunction with (A.2), the second property in (A.4) states that the functions \(f_{R}\) can be arranged to have arbitrarily high \(L^{1}\)-density in R, at the cost of choosing the parameter N large.
Remark A.6
While proving Lemma A.1, we will apply the well-known inequalities
valid for \(f \in L^{1}({\mathbb {R}}^{d})\), every \(\lambda > 0\), and a certain constant \(C = C_{d} \geqslant 1\). The first inequality in (A.7) is stated in [32, (6)], but we provide the short details. Let \(C = C_{d} \geqslant 1\) be a constant to be specified in a moment. Write \(\Omega _{h} := \{{\mathcal {M}}f > h\}\) for \(h > 0\). For every \(x \in \Omega _{C\lambda }\), choose a radius \(r_{x} > 0\) such that, denoting \(B_{x} := B(x,r_{x})\), we have
This is possible, since \(f \in L^{1}({\mathbb {R}}^{d})\). For example, one can take \(r_{x} > 0\) 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 \(f \in L^{1}({\mathbb {R}}^{d})\). We then apply the 5r-covering lemma to the balls \(\tfrac{1}{5}B_{x}\) to obtain a countable sub-sequence \(\{B_{i}\}_{i \in {\mathbb {N}}} \subset \{B_{x}\}_{x \in \Omega _{C\lambda }}\) with the properties that (i) the balls \(\tfrac{1}{5}B_{i}\) are disjoint, and (ii) the balls \(B_{i}\) cover \(\bigcup \{\tfrac{1}{5}B_{x} : x \in \Omega _{C\lambda }\} \supset \Omega _{C\lambda }\). We observe that if \(C = C_{d} \geqslant 1\) is large enough, it follows from (A.8) that \(\tfrac{1}{5}B_{i} \subset \Omega _{\lambda }\) for all \(i \in {\mathbb {N}}\). Consequently,
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 \(f \notin L^{1}([0,1)^{d})\), there is nothing to prove: then \({\mathcal {R}}_{\mathrm {heavy}} := \{[0,1)^{d}\}\) satisfies the conclusions (A.4). So, assume that \(f \in L^{1}([0,1)^{d})\), and hence \(f \in L^{1}({\mathbb {R}}^{d})\), since \({\text {spt}}f \subset [0,1)^{d}\). Let \(C = C_{d} \geqslant 1\) be the constant from (A.7). Choosing \(N/(2C)< \lambda < N/C\), and combining the inequalities (A.7) with the main assumption (A.3), we find that
With this in mind, we replace N by N/(2C), and we re-define \(H_{N}\) to be the set \(H_{N} := \{x : f(x) \geqslant N\}\). As we just argued, the hypothesis (A.3) remains valid with the new notation, possibly with slightly worse constants.
Fix \(N \geqslant 1\) and abbreviate
It would be helpful if the elements in \({\mathcal {B}}\) 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 \({\mathcal {D}}_{1},{\mathcal {D}}_{2},\ldots ,{\mathcal {D}}_{d + 1}\) with the following property: every cube \(Q \subset [0,1)^{d}\), and consequently every ball \(B \subset [0,1)^{d}\), is contained in a dyadic cube \(R \in {\mathcal {D}}_{1} \cup \cdots \cup {\mathcal {D}}_{d + 1}\) with \(|R| \leqslant C_{d}|Q|\) (resp. \(|R| \leqslant C_{d}|B|\)). The constant “\(d + 1\)” is not crucial—any dimensional constant would do. The fact that \(d + 1\) systems in \({\mathbb {R}}^{d}\) suffice was shown by Mei [24], but such “adjacent” dyadic systems can even be produced in metric spaces, see [19].
In particular, for every \(B \in {\mathcal {B}}\), we may assign an index \(i = i_{B} \in \{1,\ldots ,d + 1\}\), possibly in a non-unique way, such that \(B \subset Q'\) for some \(Q' \in {\mathcal {D}}_{i}\) with \(|Q'| \leqslant C_{d}|B|\). We let \({\mathcal {B}}_{i}\) be the set of balls in \({\mathcal {B}}\) with fixed index \(i \in \{1,\ldots ,d + 1\}\), and we write
We claim that there exists \(i \in \{1,\ldots ,d + 1\}\) such that if \(H_{N/(d + 1)}^{i} := \{x : f_{i}(x) \geqslant N/(d + 1)\}\), then
Indeed, one notes that if \(x \in H_{N}\) is fixed, then \(f_{1}(x) + \cdots + f_{d + 1}(x) = f(x) \geqslant N\), and hence there exists \(i = i_{x} \in \{1,\ldots ,d + 1\}\) such that \(f_{i}(x) \geqslant f(x)/(d + 1) \geqslant N/(d + 1)\). In particular \(x \in H_{N/(d + 1)}^{i}\). Then \({\mathbf {1}}_{H_{N/(d + 1)}^{i}}(x)f_{i}(x) \geqslant f(x)/(d + 1)\) for this particular i, and
This implies (A.9). We now fix \(i \in \{1,\ldots ,d + 1\}\) satisfying (A.9). Then \(f_{i}\) satisfies the hypothesis (A.3) with the slightly worse constants “\(\theta /(d + 1)^{2}\)” and “\(N/(d + 1)\)”. Also, it evidently suffices to prove the claimed lower bounds in (A.4) for “\(f_{i}\)” and its “sub-functions”
in place of f and the “sub-functions” \(f_{R}\). Let us summarise the findings: by passing from \({\mathcal {B}}\) to \({\mathcal {B}}_{i}\) and from f to \(f_{i}\) if necessary, we may assume that every ball in the original collection “\({\mathcal {B}}\)” is contained in an element “R” of some dyadic system “\({\mathcal {D}}\)” with \(|R| \leqslant C_{d}|B|\). We make this a priori assumption in the sequel.
For every dyadic cube \(R \in {\mathcal {D}}\), we define the weight
Here the relation \(B \sim R\) means that \(B \subset R\), and \(|R| \leqslant C_{d}|B|\). By the previous arrangements, for every \(B \in {\mathcal {B}}\) there exist \(\sim _{d} 1\) dyadic cubes \(R \in {\mathcal {D}}\) such that \(B \sim R\). It is worth pointing out that
because if \(x \in B \in {\mathcal {B}}\), then \(B \sim R\) for some \(R \in {\mathcal {D}}\). It follows that \(x \in R\), and \(w_{B}\) is one of the terms in the sum defining \({\mathfrak {w}}_{R}\).
We now begin the proof in earnest. If \(\Vert f\Vert _{1} > M\) there is nothing to prove: then we simply declare \({\mathcal {R}}_{\mathrm {heavy}} := \{[0,1)^{d}\}\), and (A.4) is satisfied. So, we may assume that
We will next perform \(k \in {\mathbb {N}}\) successive stopping time constructions, for some \(1 \leqslant k \leqslant \gamma + 1\), which will generate a families \({\mathcal {R}}_{1},{\mathcal {R}}_{2},\ldots ,{\mathcal {R}}_{k} \subset {\mathcal {D}}\) of disjoint dyadic cubes. The cubes in \({\mathcal {R}}_{k + 1}\) will be contained in the union of the cubes in \({\mathcal {R}}_{k}\). A subset of one of these families will turn out to be the family “\({\mathcal {R}}_{\mathrm {heavy}}\)” whose existence is claimed.
Let \({\mathcal {R}}_{1} \subset {\mathcal {D}}\) 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
Indeed, if \(x \in H_{N}\), then
It then follows from the definition of the coefficients \({\mathfrak {w}}_{R}\) (and the fact that every \(B \in {\mathcal {B}}\) is contained in some \(R \in {\mathcal {D}}\)) that there exist dyadic cubes \(R \in {\mathcal {D}}\) containing x such that (A.11) holds, and in particular \(x \in R\) for some \(R \in {\mathcal {R}}_{1}\).
Next, we calculate that
since the cubes in \({\mathcal {R}}_{1}\) are disjoint. Moreover, by (A.10),
so
for some constant \(A = A_{d} \geqslant 1\). 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 \(N > (2A)^{\gamma + 1}3^{(\gamma + 1)^{2}}M^{\gamma + 2}/c\). Next, we claim that if \(x \in R \in {\mathcal {R}}_{1}\), then
The second inequality follows directly from the definition of the maximal cubes \(R \in {\mathcal {R}}_{1}\). Regarding the first inequality, note that if \(B \in {\mathcal {B}}\) is a ball satisfying \(x \in B \cap R\) and \(B \not \subset R\), then \(B \subset R'\) for some strict ancestor \(R' \in {\mathcal {D}}\) of R. Then the coefficient \(w_{B}\) appears in the sum defining \({\mathfrak {w}}_{R'}\) for this ancestor \(R' \supsetneq R\). As a corollary of (A.15), and recalling that \(f(x) \geqslant N\) for all \(x \in H_{N}\), 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 \({\mathcal {R}}_{2}\) will be generated. The case distinction is based on examining the following “heavy” cubes in \({\mathcal {R}}_{1}\):
Case 1 Assume first that
Then
In this case, we set \({\mathcal {R}}_{\mathrm {heavy}} := {\mathcal {R}}_{1,\mathrm {heavy}}\), 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 \({\mathcal {R}}_{1}\). Therefore,
where \({\mathcal {R}}_{1,\mathrm {light}} = {\mathcal {R}}_{1} \, \setminus \, {\mathcal {R}}_{1,\mathrm {heavy}}\).
We now proceed to define the next generation stopping cubes \({\mathcal {R}}_{2}\). Fix \(R_{0} \in {\mathcal {R}}_{1,\mathrm {light}}\), and consider the maximal dyadic sub-cubes \(R \subset R_{0}\) with the property
Again, the left hand side of (A.19) is constant on R, so the stopping condition is well-posed. The cubes so obtained are denoted \({\mathcal {R}}_{2}(R_{0})\), and we set
We claim that the (fairly large) part of \(H_{N}\) covered by cubes in \({\mathcal {R}}_{1,\mathrm {light}}\) is remains covered by the cubes in \({\mathcal {R}}_{2}\). Indeed, fix \(x \in R_{0} \cap H_{N}\), where \(R_{0} \in {\mathcal {R}}_{1,\mathrm {light}} \subset {\mathcal {R}}_{1}\). Then
by definitions of \({\mathcal {R}}_{1}\) and \(N_{1}\), so
and hence x is contained in some (maximal) dyadic cube \(R \subset R_{0}\) satisfying (A.19).
Arguing as in (A.15), we infer the following: if \(x \in R \in {\mathcal {R}}_{2}\), then
Indeed, the first inequality follows exactly as in (A.15). To see the second inequality, split the cubes \(R' \supsetneq R\) into the ranges \(R \subsetneq R' \subset R_{0}\) and \(R_{0} \subsetneq R \subset [0,1)^{d}\), where \(R_{0} \in {\mathcal {R}}_{1}\). Then, use the definitions of the stopping cubes \({\mathcal {R}}_{1}\) and \({\mathcal {R}}_{2}\). As a corollary of (A.21), we infer an analogue of (A.16) for \(R \in {\mathcal {R}}_{2}\):
We next estimate the total volume of the cubes in \({\mathcal {R}}_{2}\). Fix \(R_{0} \in {\mathcal {R}}_{1,\mathrm {light}}\), and first estimate
Of course, this computation was just a repetition of (A.13). Also the next estimate can be carried out in the same way as the estimate just below (A.13):
Combining the previous two displays, the stopping cubes in \({\mathcal {R}}_{2}(R_{0})\) have total volume \(\leqslant AM|R_{0}|/N_{2}\) for every \(R_{0} \in {\mathcal {R}}_{1,\mathrm {light}}\). Therefore,
recalling (A.14). Since \(N \gg \max \{A,M\}\), this means that the total volume of the stopping cubes tends to zero rapidly as their generation increases.
We are now prepared to make another case distinction, this time based on the heavy sub-cubes in \({\mathcal {R}}_{2}\):
Case 2.1
Assume first that
Then,
In this case, we declare \({\mathcal {R}}_{\mathrm {heavy}} := {\mathcal {R}}_{2,\mathrm {heavy}}\), and we see that (A.4) is satisfied.
Case 2.2 Assume then that (A.24) fails. Since the part of \(H_{N}\) contained in the \({\mathcal {R}}_{1,\mathrm {light}}\)-cubes is also contained in the \({\mathcal {R}}_{2}\)-cubes (as established right below (A.20)), we deduce from (A.18) that
Here of course \({\mathcal {R}}_{2,\mathrm {light}} := {\mathcal {R}}_{2} \, \setminus \, {\mathcal {R}}_{2,\mathrm {heavy}}\). So, we find ourselves in a situation analogous to (A.18), except that the integral of \(f{\mathbf {1}}_{H_{N}}\) over the light cubes has decreased by half.
Repeating the construction above, we proceed to define—inductively—new collections of stopping cubes. The stopping cubes \({\mathcal {R}}_{k}\) are contained in the the union of the stopping cubes \({\mathcal {R}}_{k - 1,\mathrm {light}}\), and they are defined as the maximal sub-cubes “R” of \(R_{0} \in {\mathcal {R}}_{k - 1,\mathrm {light}}\) satisfying
Repeating the argument under (A.20), this definition ensures that the part of \(H_{N}\) covered by the cubes in \({\mathcal {R}}_{k - 1,\mathrm {light}}\) remains covered by the union of the cubes in \({\mathcal {R}}_{k}\). Moreover, induction shows that
The general analogue of the inequality (A.22) is
and the total volume of the cubes in \({\mathcal {R}}_{k}\) satisfies
in analogy with (A.23). Once the cubes in \({\mathcal {R}}_{k}\) have been constructed, we split into two cases, depending on whether
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
and the proof of (A.4) concludes if \(k \leqslant \gamma + 1\). So, the only remaining task is to show that the first case must occur for some \(k \leqslant \gamma + 1\). Indeed, if the second case of (A.29) occurs for any \(k \geqslant 1\), we have
Recalling that \(N_{k} = \lfloor N/2^{k}\rfloor \geqslant N/3^{k}\), hence \(N_{1}\ldots N_{k} \geqslant N^{k}3^{-k^{2}}\), this yields
Assuming that \(N > 3^{(\gamma + 1)^{2}}(2A)^{\gamma + 1}M^{\gamma + 2}/c\) (in agreement with (A.2)), the inequality above cannot hold for \(k = \gamma + 1\). Thus, the “heavy” case of (A.29) occurs latest at step \(k = \gamma + 1\). The proof of the lemma is complete. \(\square \)
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Orponen, T. Plenty of big projections imply big pieces of Lipschitz graphs. Invent. math. 226, 653–709 (2021). https://doi.org/10.1007/s00222-021-01055-z
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DOI: https://doi.org/10.1007/s00222-021-01055-z