1. Introduction

HAUSDORFF DIMENSION OF UNIFORMLY NON FLAT SETS WITH TOPOLOGY

Guy David

Résumé. Soient d un entier et E un ensemble fermé non vide dans ℝ^{n}. On suppose que E est localement uniformément non plat, au sens où pour x ∈ E et r > 0 assez petit, E ∩ B(x, r) ne reste jamais ε_{0}r-proche d’un d-plan affine. Si de plus E vérifie localement uniformément certaines propriétés de non-dégénérescence topologique de dimension d, comme la condition B de Semmes, alors sa dimension de Hausdorff est strictement plus grande que d. Notre démonstration est basée sur les propriétés de rectifiabilité uniforme des ensembles quasiminimaux (“restricted sets”) d’Almgren.

Abstract. Let d be an integer, and let E be a nonempty closed subset of ℝ^{n}. Assume that E is locally uniformly non flat, in the sense that for x ∈ E and r > 0 small, E∩B(x, r) never stays ε_{0}r-close to an affine d-plane. Also suppose that E satisfies locally uniformly some appropriate d-dimensional topological nondegeneracy condition, like Semmes’ Condition B. Then the Hausdorff dimension of E is strictly larger than d. We see this as an application of uniform rectifiability results on Almgren quasiminimal (restricted) sets.

AMS classification: 49Q20, 28A75.

Mots clé / Key words: Hausdorff dimension, quasiminimal sets, restricted sets, flatness.

1. Introduction

Let E be a closed subset in Euclidean space ℝ^{n}, and let an integer dimension d < n be given. Suppose that at all scales and locations, E is both uniformly non flat (relative to that dimension) and satisfies some d-dimensional topological nondegeneracy condition. We want to show that the Hausdorff dimension of E is (strictly) larger than d.

A simple instance of this is when d = 1, E is connected, and its P. Jones numbers are such that β_{E}(x, r) ≥ ε for x ∈ E and 0 < r ≤ 1. Then C. Bishop and P. Jones [BJ] showed that E has Hausdorff dimension at least d^{∗}, where d^{∗} > 1 depends only on n and ε.

Here we shall keep the same condition to measure the uniform nonflatness of E. Call P the set of affine planes of dimension d and set

(1.1)  β_{E}(x, r) = inf_{P ∈P} { sup {r^{−1} dist (y, P) ; y ∈ E ∩ B(x, r)}}

for x ∈ ℝ^{n} and r > 0. We require the existence of ε_{0} > 0 and r_{0} > 0 such that

(1.2)  β_{E}(x, r) ≥ ε_{0} for all x ∈ E and 0 < r ≤ r_{0}.

Note that using L^{p} variants of β_{E}(x, r) would only make (1.2) harder to check, and would thus lead to weaker theorems below. We could also prove the results below with a bilateral version of β_{E}(x, r). That is, we could replace (1.2) with the weaker requirement that for every choice of x ∈ E and 0 < r ≤ r_{0}, we cannot find a plane P such that dist (y, P) ≤ ε_{0}r for every y ∈ E ∩ B(x, r) and dist (z, E) ≤ ε_{0}r for every z ∈ P ∩ B(x, r). The proof would be the same (just replace the weak geometric lemma by its bilateral version near (6.10)). But this remark is essentially useless, because with our topological nondegeneracy condition, the weaker analogue of (1.2) implies (1.2) (with a smaller ε_{0}).

For the topological nondegeneracy condition, there seems to be no obvious best choice, so we shall present two main ones and a few consequences, and hope that they will turn out to be the right ones. The general idea, as in [Se2], [DS2], and Section 12 of [DS3], is to require some property of topological nature (i.e., for instance, invariant under deformations) that would automatically imply that E is at least d-dimensional. Connectedness (in dimension d = 1), as in the result of Bishop and Jones, would be an example. We want to require something like this at all scales and locations, too.

For our first topological condition, we shall consider deformations of E in balls B(x, r). These deformations will be given by one-parameter families of Lipschitz mappings {φ_{t}}, 0 ≤ t ≤ 1, defined on ℝ^{n}, with the following properties:

(1.3)  φ_{t}(\overline{B}(x, r)) ⊂ \overline{B}(x, r) for each t ∈ [0, 1];

(1.4)  for each y ∈ ℝ^{n}, φ_{t}(y) is a continuous function of t ∈ [0, 1];

(1.5)  φ_{t}(y) = y for t = 0 and for y ∈ ℝ^{n} \ B(x, r);

(1.6)  dist (φ_{t}(y), E) ≤ α_{0}r for t ∈ [0, 1] and y ∈ E ∩ B(x, r),

where the constant α_{0} ∈ (0, 1) is given in advance. Thus we only allow ourselves to move E ∩ B(x, r) in a thin tube around E. This is important, because it may force φ_{1}(E) to keep some of the topological properties of E.

Minor modifications of this are possible. For instance, we may require that t → φ_{t} be a continuous function of t, valued in Lipschitz functions. We could also have decided to define φ_{t} only on [E ∩ B(x, r)] ∪ ∂B(x, r), but it would then have been easy to extend it to ℝ^{n}. Note that in (1.3) we do not require any bound on the Lipschitz constant of φ_{t}, and probably we could even replace the Lipschitz requirement with simple continuity.

For our first topological condition, we shall require the existence of two small positive constants η_{0} and δ_{0} such that

(1.7)  H^{d}[B(x, (1 − η_{0})r) ∩ φ_{1}(E)] ≥ δ_{0}r^{d}

when (1.3)-(1.6) hold. Let us already give the corresponding statement.

2. Condition B and other topological nondegeneracy conditions

Theorem 1.8. For each choice of positive constants ε_{0}, r_{0}, α_{0}, η_{0}, and δ_{0}, we can find d^{∗} > d such that Hdim (E) ≥ d^{∗} whenever E is a nonempty closed set in ℝ^{n} such that (1.2) holds and

(1.9)

(1.7) holds for all choices of x ∈ E, 0 < r ≤ r_{0}, and of Lipschitz deformations {φ_{t}} with the properties (1.3), (1.4), (1.5), and (1.6).

Here Hdim (E) denotes the Hausdorff dimension of E. Thus our conclusion is that the Hausdorff measure H^{m}(E) is infinite for every m < d^{∗}. See for instance [Fa] for details about Hdim and H^{d}.

We shall give in Section 2 a few additional comments about our topological condition (1.7), and in particular check that it is implied by S. Semmes’ Condition B (a uniform separation property in codimension 1). However it is a little unpleasant that in (1.7) we require a lower bound on H^{d}[B(x, (1 − η_{0})r) ∩ φ_{1}(E)], rather than the more natural H^{d}(φ_{1}(E ∩ B(x, r))). Since we fear that the difference may cause trouble in potential applications, we shall give a slightly more general, but also somewhat more technical, version of Theorem 1.8, where we use approximations of E by small d-dimensional skeletons. See Theorem 3.8 below.

Both theorems are easy to localize. That is, if E satisfies the hypotheses of these theorems for all balls B(x, r) contained in an open set Ω, then E∩B(y, ρ) also has Hausdorff dimension at least d^{∗} when y ∈ E ∩ Ω and ρ > 0. This will be clear from the proof.

Our proof will be a little indirect; the general idea is the following. We shall proceed by contradiction, assume that E is not too large, and construct a functional and a minimizer F for this functional. By definition of the functional and the topological condition, F will have a big intersection with a small modification E_{ρ} of E. We shall also check that F is a quasiminimal set (in the terminology of F. Almgren [Al], a restricted set), then use the local uniform rectifiability estimates from [DS3] to show that F is very flat in most balls; a contradiction with (1.2) will ensue. All this will be in the spirit of Section 12 of [DS3]. See the beginning of Section 4 for a slightly more precise description of the argument.

Even though the author likes the indirect argument below, he is aware that there could be much simpler, direct proofs of Theorems 1.8 and 3.8.

Potential applications for the results of this paper could be lower bounds on the dimension of limit sets of some groups, as was the case with the original paper of Bishop and Jones. But the author knows very little about the subject.

The author wishes to thank Leonid Potyagailo, who asked the question and told him about the potential applications to limit sets, A. Ancona for help with the topology in Section 2, and P. Jones for interesting discussions.

2. Condition B and other topological nondegeneracy conditions

The following condition was introduced by S. Semmes [Se1], in connection with L^{2}- boundedness of singular integral operators on subsets of ℝ^{n}. This is a topological condition of codimension 1.

Definition 2.1. Let E be a closed set in ℝ^{n}. We say that E satisfies Condition B locally if there are constants α_{1} > 0 and r_{0} > 0 such that, for all choices of x ∈ E and 0 < r ≤ r_{0}, we can find two balls B_{1} and B_{2} contained in B(x, r) \ E, of radius α_{1}r, and such that B_{1} and B_{2} lie in different connected components of B(x, r) \ E.

We added “locally” because in [Se1] and related works, the property is assumed to hold for all r < diam (E). Also, we weakened the property a little, because normally one requires the balls B_{1} and B_{2} to be contained in different connected components of ℝ^{n} \ E.

Lemma 2.2. If the nonempty closed set E satisfies Condition B locally, then it satisfies the hypothesis (1.9) in Theorem 1.8, with d = n − 1. Hence, if E also satisfies (1.2) with d = n − 1, then Hdim (E) ≥ d^{∗} > n − 1, for some d^{∗} that depends only on n, d, α_{1}, and ε_{0}.

So let E satisfy Condition B locally, and let us prove (1.9); the rest of the lemma will follow from Theorem 1.8. We shall keep the same r_{0} and take

(2.3)  α_{0} = η_{0} = α_{1}/2 and δ_{0} = 2^{−n+1}c_{n−1}α_{1}^{n−1},

where c_{n−1} denotes the Lebesgue measure of the unit ball in ℝ^{n−1}.

Let x ∈ E and 0 < r ≤ r_{0} be given, and let us apply Condition B. We get two balls B_{1} = B(x_{1}, α_{1}r) and B_{2} = B(x_{2}, α_{1}r), with the properties stated in Definition 2.1. Set B′_{j} = B(x_{j}, α_{1}r/2) for j = 1, 2.

Next let {φ_{t}} be a family of Lipschitz functions, with the properties (1.3)-(1.6). Note that for y ∈ E and 0 ≤ t ≤ 1, dist (φ_{t}(y), E) ≤ α_{0}r = α_{1}r/2, by (1.6) or (1.5). Thus

(2.4)  φ_{t}(E) never meets B′_{1} ∪ B′_{2}.

Note that φ_{t}(∂B(x, r)) = ∂B(x, r) does not meet B′_{1} ∪ B′_{2} either, by (1.5) and because B_{j} ⊂ B(x, r). We claim that

(2.5)  B′_{1} and B′_{2} lie in different connected components of ℝ^{n} \ φ_{1}(E ∪ ∂B(x, r)).

This comes directly from 4.3 on Chapter XVII of [Du], page 360 (for instance). The point is that if y_{1} ∈ B′_{1} and y_{2} ∈ B′_{2}, then y_{1} and y_{2} lie in different components of ℝ^{n} \ [E ∪ ∂B(x, r)] by definition of B_{1} and B_{2}, and this stays the same with φ_{t}(E) for all t, because φ_{t}(E) never passes through y_{1} or y_{2}. The proof uses Borsuk’s theorem on extensions of maps into spheres, and Brouwer’s theorem that says that the identity map on the sphere is not nullhomotopic.

Call L the line through the centers x_{1} and x_{2}, and π the orthogonal projection onto the hyperplane P through x_{1} and perpendicular to L. Let us check that

(2.6)  π[φ_{1}(E) ∩ B(x, r − η_{0}r)] contains P ∩ B′_{1}.

Let ξ ∈ P ∩ B′_{1} be given, and call L_{ξ} the line through ξ parallel to L. Then L_{ξ} meets B′_{1} and B′_{2}, because π(B′_{1}) = π(B′_{2}) = P ∩ B′_{1}. Since B′_{1} and B′_{2} lie in different components of ℝ^{n} \φ_{1}(E ∪∂B(x, r)), we can find a point z_{ξ} in φ_{1}(E ∪∂B(x, r)))∩L_{ξ}, somewhere between the intersections of L_{ξ} with B′_{1} and B′_{2}. Note that z_{ξ} lies in the convex hull of B′_{1} ∪ B′_{2}, which itself is contained in B(x, r − α_{1}r/2) = B(x, r − η_{0}r). So z_{ξ} ∈ φ_{1}(E) ∩ B(x, r − η_{0}r), and since π(z_{ξ}) = ξ, we see that ξ ∈ π[φ_{1}(E) ∩ B(x, r − η_{0}r)], as needed for (2.6).

Since π is Lipschitz with constant 1, (2.6) says that

(2.7)  \begin{aligned}H^{d}(φ_{1}(E) ∩ B(x, r − η_{0}r)) & ≥ H^{d}(π[φ_{1}(E) ∩ B(x, r − η_{0}r)]) \\ & ≥ H^{d}(P ∩ B(x_{1}, α_{1}r/2)) = 2^{−n+1}c_{n−1} α_{1}^{n−1}r^{d}, \end{aligned}

which is exactly (1.7) with δ_{0} = 2^{−n+1}c_{n−1} α_{1}^{n−1}.

This completes our proof of (1.9), and Lemma 2.2 follows. □

It is reasonably easy to come up with generalizations of Condition B in higher codimensions, but it is a little harder to guess which one is the most natural. We shall give such a condition here, more as an example of what can be done than a final answer to a natural problem. The following definition is a local version of Definition 3 on page 106 of [Da].

Definition 2.8. Let E be a closed set in ℝ^{n}. We say that E ∈ SS(d, n) locally if there are constants α_{1} > 0 and r_{0} > 0 such that, for all choices of x ∈ E and 0 < r ≤ r_{0}, we can find an affine subspace W of dimension n − d and an Euclidean sphere Σ ⊂ W ∩ B(x, r/2) of dimension n − d − 1, such that

(2.9)  dist (Σ, E) ≥ 2α_{1}r

and Σ is linked with E in B(x, r), in the sense that there is no one-parameter family {f_{t}} of continuous functions on ℝ^{n} such that

(2.10)  f_{t}(\overline{B}(x, r)) ⊂ \overline{B}(x, r) for each t ∈ [0, 1],

(2.11)  for each y ∈ ℝ^{n}, f_{t}(y) is a continuous function of t ∈ [0, 1],

(2.12)  f_{t}(y) = y for t = 0 and for y ∈ ℝ^{n} \ B(x, r),

(2.13)  dist (f_{t}(y), Σ) ≥ α_{1}r for t ∈ [0, 1] and y ∈ E ∩ B(x, r),

and

(2.14)  f_{1}(E ∩ B(x, r)) ⊂ ∂B(x, r).

In other words, there is no way to deform E ∩ B(x, r) inside of \overline{B}(x, r) so that it lands on ∂B(x, r), without crossing a small neighborhood of Σ.

When d = n − 1, Σ is composed of two points, and our condition looks a lot like Condition B, except that instead of saying that the two centers of the B_{j} in Condition B lie in different components of ℝ^{n} \ E, we say that we cannot deform E past the two points without crossing them. But here the conditions are equivalent, by Borsuk’s and Brouwer’s theorems.

In higher codimensions, one could come up with other ways to say that E is linked with Σ, and it is not clear that they are all equivalent. The linking condition in Definition 2.8 is probably not the most natural one (as the case of codimension 1 suggests), but at least we can use it easily. Also, there are probably a few algebraic ways to make sure that E and Σ are linked and E ∈ SS(d, n) locally, but this is a slightly different subject. See Remark 2.24 for a little more about this.

Lemma 2.15. If E is a closed set and E ∈ SS(d, n) locally, then we can find α_{0}, δ_{0}, and η_{0} such that (1.7) holds for all choices of x ∈ E, 0 < r ≤ r_{0}, and all families {φ_{t}} of Lipschitz deformations such that (1.3), (1.4), (1.5), and (1.6) hold.

In other words, (1.9) holds. The constants α_{0}, δ_{0}, and η_{0} depend only on the constants in SS(d, n); in fact, we can keep the same r_{0} and take

(2.16)  α_{0} = α_{1}, η_{0} = 1/2, and δ_{0} = 2^{−d} c_{d} α_{1}^{d}.

Now Theorem 2.9 says that if E is closed, nonempty, and E ∈ SS(d, n) and (1.2) holds, then the Hausdorff dimension of E is at least d^{∗}, where d^{∗} > d depends only on n, d, α_{1}, and ε_{0}.

Let us prove the lemma. Choose constants as in (2.16). As before, we start with x ∈ E and 0 < r ≤ r_{0}, Definition 2.8 gives a sphere Σ, and we want to use it to prove (1.7) for all families {φ_{t}} of Lipschitz functions that satisfy (1.3)-(1.6).

So we give ourselves such a family, assume that (1.7) fails, and try to reach a contradiction. The idea will be to use {φ_{t}} to find deformations {f_{t}} that satisfy (2.10)-(2.14).

Note that {φ_{t}} automatically satisfies (2.10)-(2.12), by (1.3)-(1.5). Also,

(2.17)  dist (φ_{t}(E), Σ) ≥ dist (E, Σ) − α_{0}r ≥ α_{0}r

by (1.6), (2.9), and because α_{0} = α_{1}. In particular, (2.13) holds for {φ_{t}}. But φ_{1} does not necessarily satisfy (2.14); we only know that (1.7) fails, i.e., that

(2.18)  H^{d}[φ_{1}(E) ∩ B(x, r/2)] < δ_{0}r^{d} = 2^{−d} c_{d} α_{1}^{d} r^{d},

by (2.16). We have to deform φ_{1}(E) some more, to send it to ∂B(x, r).

Call x_{0} the center of Σ, W^{⊥} the affine d-plane through x_{0} perpendicular to W, and π the orthogonal projection onto W^{⊥}. Then (2.18) says that

(2.19)  H^{d}[φ_{1}(E) ∩ B(x, r/2)] < H^{d}(W^{⊥} ∩ B(x_{0}, α_{1}r/2)).

Hence π(φ_{1}(E) ∩ B(x, r/2)) does not contain W^{⊥} ∩ B(x_{0}, α_{1}r/2).

Let z ∈ W^{⊥} ∩B(x_{0}, α_{1}r/2) be such that D = π^{−1}(z)∩B(x, r/2) does not meet φ_{1}(E). Set U = {y ∈ B(x, r) ; dist (y, Σ) < α_{1}r}, and call V the convex hull of U. Thus V is an α_{1}r-neighborhood of the ball of W enclosed in Σ.

Also set D_{ε} = {y ∈ V ; dist (y, D) < ε}. Since φ_{1}(E) is closed and does not meet D, we can choose ε < α_{1}r/2 so small that D_{ε} does not meet φ_{1}(E).

Now we can find a continuous deformation {ψ_{t}}, 0 ≤ t ≤ 1, such that ψ_{0}(y) = y everywhere, ψ_{t}(y) = y when y ∈ ℝ^{n} \ V,

(2.20)  ψ_{t}(\overline{V} \ [U ∪ D_{ε}]) ⊂ \overline{V} \ [U ∪ D_{ε}] for all t,

and

(2.21)  ψ_{1}(V \ [U ∪ D_{ε}]) ⊂ ∂V.

Let us just give an idea of the construction. We first define ψ_{1}, and then we shall take ψ_{t}(y) = ty + (1 − t)ψ_{1}(y). We keep ψ_{1}(y) = y on D. For y ∈ V \ D_{ε}, we call y^{∗} the point of D such that y^{∗} − y is parallel to W^{⊥}, and then let ψ_{1}(y) be the point of the half line from y^{∗} through y that lies in ∂V. In other words, we move y straight away from D and parallel to W^{⊥} until we hit ∂V. For the rest of V, i.e., on V ∩ D_{ε}, we interpolate nicely between the two definitions. The properties (2.20) and (2.21) are easy to check; the main point for (2.20) is that if y ∈ V \ [U ∪ D_{ε}] then all the points of the half-line from y^{∗} through y that lie after y also lie out of U ∪ D_{ε}.

We complete our deformation {ψ_{t}} by another continuous family {h_{t}}, 0 ≤ t ≤ 1, such that h_{t}(y) = y for t = 0 and for y ∈ ℝ^{n} \ B(x, r), h_{t}(\overline{B}(x, r)) ⊂ \overline{B}(x, r) for all t,

(2.22)  h_{t}(B(x, r) \ V) ⊂ \overline{B}(x, r) \ V for all t, and h_{1}(\overline{B}(x, r) \ V) ⊂ ∂B(x, r).

This one is very easy to construct, since V is essentially a smaller ball inside B(x, r); we just need to push the points out to ∂B(x, r).

Our family {f_{t}} is obtained by composing the three families above. That is, we set f_{t}(y) = φ_{3t}(y) for 0 ≤ t ≤ 1/3, then f_{t}(y) = ψ_{3t−1}(φ_{1}(y)) for 1/3 < t ≤ 2/3, and finally f_{t}(y) = h_{3t−2}(ψ_{1}(φ_{1}(y))) for t > 2/3. The properties (2.10)-(2.12) are straightforward. Let us check (2.13).

When t ≤ 1/3, (2.13) follows from (2.17). We also get that φ_{1}(E) ⊂ ℝ^{n} \ [U ∪ D_{ε}], by definition of U and D_{ε}. Then (2.20) says that f_{t}(E) ⊂ ℝ^{n} \ [U ∪ D_{ε}] for 1/3 < t ≤ 2/3, which proves (2.13) in that case. Also note that

(2.23)  f_{2/3}(E) = ψ_{1}(φ_{1}(E)) ⊂ ℝ^{n} \ V,

by (2.21). Finally, for t > 2/3, f_{t}(E) = h_{3t−2}(ψ_{1}(φ_{1}(E))) stays out of V by (2.22), and hence (2.13) holds.

Our last condition (2.14) holds because f_{1}(E ∩ B(x, r)) = h_{1}(ψ_{1}(φ_{1}(E ∩ B(x, r))) ⊂ h_{1}(\overline{B}(x, r) \ V) ⊂ ∂B(x, r), by (2.23) and (2.22).

By Definition 2.8, such a family {f_{t}} cannot exist; this gives the desired contradiction and proves Lemma 2.15. □

3. Our main technical topological condition

Remark 2.24. It would be easy to prove the analogue of Lemma 2.15 with a somewhat weaker definition of SS(d, n). First, the Euclidean sphere Σ could be replaced with a bilipschitz image of such a sphere, for instance. In [Da] this was hard to do because one wanted to know that E has “big projection”, and for this one really wanted to show that the straight projection π(E ∩ B(x, r/2)) contains W^{⊥} ∩ B(x_{0}, α_{1}r/2). The argument was the same as above (for φ_{t}(y) ≡ y); see below (2.19)). Here we just want to prove (1.7) (or contradict (2.18)), and for this a bilipschitz image of a sphere is enough. Since the point is merely to get (1.7), one could probably replace spheres by other objects too.

In fact, we could get (1.7) from various other linking conditions, like the existence of a homotopically nontrivial mapping from E ∩ B(x, r) to a sphere 𝕊^{d}. We do not want to try to give a list here, because we would surely miss the most relevant example. But at least we should say that (1.7) has a stability property that should makes it pleasant to check. That is, if we choose δ_{0} small enough, depending on n, d, and α_{0}, then either (1.7) holds (for all deformations {φ_{t}} in B(x, t)), or else we can even find {φ_{t}} such that B(x, (1 − η_{0})r) ∩ φ_{1}(E) has Hausdorff dimension at most d − 1. We can get this by a projection on skeletons, a little bit like when we will construct f and π in Section 5. The reader in kindly invited to consult Subsection 12.2 of [DS3] (called Stability of Sets) for this; more ideas on topological conditions that may lead to (1.7) can be found in Subsection 12.3 of [DS3] (Topological Interpretations).

3. Our main technical topological condition

For our main technical topological nondegeneracy condition, we need to introduce dyadic cubes and skeletons. Let ρ > 0 be a small number; the reader may think of ρ as a small power of 2, but since we often work at one scale at a time, this does not really matter. Our set of dyadic cubes of size ρ is

(3.1)  Q(ρ) = {Q = ρ [0, 1]^{n} + ρ ℓ; ℓ ∈ ℤ^{n}}.

For each Q ∈ Q(ρ) and each integer m ∈ [0, n], we denote by S_{m}(Q) the m-dimensional skeleton of Q. Thus S_{0}(Q) is the set of all vertices of Q, S_{n}(Q) is the cube Q itself, and in general S_{m}(Q) is the union of all the m-dimensional faces of Q. It will be convenient to replace the closed set E with

(3.2)  E_{ρ} = ⋃_{Q∈Q(ρ) ; Q∩E≠∅} S_{d}(Q),

where d is the same fixed integer as in the introduction. One of the points is that E_{ρ} has locally finite H^{d}-measure, while we expect E to be much larger. And the difference is not so large, because we can deform E onto a subset of E_{ρ}, at least if Hdim (E) < d + 1. [See Lemma 3.14.]

Our second topological nondegeneracy condition T ND is the following. We assume that for each choice of a big constant C_{1} > 0, there are (small) positive constants r_{0}, α < 1, η, and δ_{1} such for every choice of

(3.3)  x_{1} ∈ E, 0 < r_{1} ≤ r_{0}, and 0 < ρ ≤ \frac{α r_{1}}{2\sqrt{n}}

(the reader should not pay too much attention to the constant 2\sqrt{n} here, the point is merely to make sure that points of E_{ρ} lie within \frac{α r}{2} of E) such that

(3.4)  H^{d}(E_{ρ} ∩ B(x_{1}, r_{1})) ≤ C_{1}r_{1}^{d},

there is a ball B(x, r) centered on E and contained in B(x_{1}, r_{1}) such that, for each oneparameter family {φ_{t}}_{0≤t≤1} of Lipschitz functions on ℝ^{n} that satisfy (1.3), (1.4), (1.5), and for which

(3.5)  dist (φ_{t}(y), E) ≤ αr_{1} for t ∈ [0, 1] and y ∈ E_{ρ} ∩ B(x, r),

we have that

(3.6)  H^{d}(φ_{1}(E_{ρ} ∩ B(x, r))) ≥ δ_{1}r_{1}^{d} + H^{d}(E_{ρ} ∩ A_{ηr_{1}}(x, r)),

where we set

(3.7)  A_{ηr_{1}}(x, r) = B(x, r) \ B(x, r − ηr_{1}).

When this condition is satisfied, we shall also say that E ∈ T ND.

We still have an unpleasant boundary term in (3.6) (namely, H^{d}(E_{ρ} ∩ A_{ηr_{1}}(x, r))), but hopefully it will be easier to control in applications than the one in Theorem 1.8, in particular because we get some limited control from the extra condition (3.4).

Also, we get some extra flexibility by allowing our nondegeneracy condition (3.6) to hold only on some balls B(x, r).

Note that we did not specify officially that r ≥ C^{−1}r_{1}, but since we can apply (3.6) with φ_{t}(y) ≡ y, we get that E_{ρ} ∩ B(x, r) ⊄ A_{ηr_{1}}(x, r), hence r ≥ ηr_{1}. This is also why we put δ_{1}r_{1}^{d} (and not δ_{1}r^{d}) in the right-hand side.

Incidentally, the statement below would still be true if we replaced the ball B(x, r) with a cube, or some other object that is bilipschitz-equivalent to a ball (and modified the annulus A_{ηr_{1}}(x, r) accordingly), but hopefully we shall not need to know this.

Theorem 3.8. If the nonempty closed set E ⊂ ℝ^{n} satisfies (1.2) (for some choice of r_{0} > 0 and ε_{0} > 0) and T ND, then Hdim (E) ≥ d^{∗} for some d^{∗} > d.

As we shall see at the beginning of Section 7, we shall only need to apply T ND with one value of C_{1}, which depends only on n and d, and then d^{∗} will depend only on n, d, ε_{0}, and the values of α, η, and δ_{1} that we get by applying T ND for this choice of C_{1}.

In the rest of this section, we want to show that Theorem 1.8 follows from Theorem 3.8. Hopefully, this will also convince the reader that T ND is not too difficult to prove. The later sections will be devoted to the proof of Theorem 3.8.

So let E satisfy the hypotheses of Theorem 1.8, and suppose in addition that

(3.9)  Hdim (E) < d + 1

(otherwise, there is nothing to prove). We want to show that E ∈ T ND, so we give ourselves a large constant C_{1}, and we try to find positive constants so that the condition in T ND holds. For future reference, let us announce already that we shall keep the same value for r_{0} and choose

(3.10)  α = \frac{1}{2} Min (η_{0}, α_{0}), η = 2^{−d−1}10^{−1}C_{1}^{−1}δ_{0}, and δ_{1} = 2^{−d−1}δ_{0}.

Let x_{1}, r_{1}, and ρ be as in (3.3), and assume that H^{d}(E_{ρ} ∩ B(x_{1}, r_{1})) ≤ C_{1}r_{1}^{d}, as in (3.4). We take x = x_{1} and choose r such that

(3.11)  r_{1}/2 ≤ r ≤ 2r_{1}/3

and the mass of E_{ρ} in the thin annulus A_{ηr_{1}}(x, r) = B(x, r) \ B(x, r − ηr_{1}) is very small. More precisely, note that we can find more than (10η)^{−1} choices or r such that (3.11) holds and the A_{ηr_{1}}(x, r) are disjoint, so we can pick r such that

(3.12) H^{d}(E_{ρ} ∩ A_{ηr_{1}}(x, r)) ≤ 10η H^{d}(E_{ρ} ∩ B(x_{1}, r_{1})) ≤ 10η C_{1}r_{1}^{d} ≤ 2^{−d−1}δ_{0} r_{1}^{d} ≤ δ_{1} r_{1}^{d},

by (3.4) and by our choice of η and δ_{1} in (3.10).

So we picked x ∈ E and r such that B(x, r) ⊂ B(x_{1}, r_{1}). What we need to do now is take a family {φ_{t}} of Lipschitz functions on ℝ^{n} that satisfies (1.3), (1.4), (1.5), and (3.5), and show that it satisfies (3.6). By (3.12), it will be enough to check that

(3.13)  H^{d}(φ_{1}(E_{ρ} ∩ B(x, r))) ≥ 2δ_{1}r_{1}^{d}.

To do so, we want to use {φ_{t}} to construct a deformation {f_{t}} of E, and then apply the hypotheses of Theorem 1.8 to {f_{t}}. So it is a good idea to first deform E into a subset of E_{ρ}.

Call Q_{1}(ρ) the set of cubes Q ∈ Q(ρ) such that Q meets B(x, r), Q_{2}(ρ) the set of cubes Q ∈ Q(ρ) that meet a cube of Q_{1}(ρ), and set D_{j} = ⋃_{Q∈Q_{j}(ρ)} Q for j = 1, 2.

Lemma 3.14. There is a Lipschitz function ψ : ℝ^{n} → ℝ^{n} such that

(3.15)  ψ(x) = x for x ∈ ℝ^{n} \ D_{2},

(3.16)  ψ(Q) ⊂ Q for Q ∈ Q(ρ),

and

(3.17)  ψ(E) ∩ Q ⊂ S_{d}(Q) for Q ∈ Q_{1}(ρ).

The construction of ψ is a standard Federer-Flemming argument. We shall obtain ψ as a composition of various projection mappings onto skeletons of smaller and smaller dimensions. Let us give a rapid proof here; the reader may also look at Proposition 3.1 in [DS3], where a variant of the lemma is proved with more details.

We start with a “projection” π_{1} that will send E ∩ D_{2} into faces of dimension n − 1. Let us first define π_{1} on Q when Q is a cube of Q_{2}(ρ). Choose a point x_{Q} ∈ int(Q) \ E (which is possible by (3.9)), and decide already that the restriction of π_{1} to [Q ∩ E] ∪ ∂Q will be the radial projection on ∂Q centered at c_{Q}. That is, π_{1}(y) is the intersection of ∂Q with the half line through y that emanates from c_{Q}. Then extend π_{1} to Q, so that π_{1}(Q) ⊂ Q and π_{1} is Lipschitz on Q. The Lipschitz constant may be enormous, if c_{Q} is very close to E, but this does not matter. Notice that we get a coherent definition of π_{1} on D_{2}. Indeed, if Q and Q′ ∈ Q_{2}(ρ) meet, then both definitions yield π_{1}(y) = y on Q ∩ Q′, because Q ∩ Q′ is contained in ∂Q and in ∂Q′. We take π_{1}(y) = y out of D_{2}. This is still coherent with our definition of π_{1} on D_{2}.

So we constructed a first Lipschitz function π_{1} on ℝ^{n} such that π_{1}(Q) ⊂ Q for Q ∈ Q(ρ) and for which π_{1}(E) is contained in the union of E ∩ (ℝ^{n} \ D_{2}) and of the (n − 1)- dimensional faces of the cubes Q, Q ∈ Q_{2}(ρ). Note also that Hdim (π_{1}(E)) < d + 1, by (3.9) and because π_{1} is Lipschitz.

If d = n − 1, we can take ψ = π_{1}. Otherwise, we have to continue the construction. We want to send points of the (n − 1)-dimensional faces of the cubes Q, Q ∈ Q_{1}(ρ), to the boundary of these faces. Notice that when we talk about the boundary ∂F and the interior int(F) of an m-dimensional face F, we shall really see F as a m-dimensional object. Thus ∂F is an union of faces of dimension m − 1.

For each (n − 1)-dimensional face F contained in D_{1}, we choose a point c_{F} ∈ int(F) \ π_{1}(E). Such a point exists because Hdim (π_{1}(E)) < d + 1 ≤ n − 1.

We define π_{2} first on ∂F ∪ (π_{1}(E) ∩ F), where we demand that π_{2}(y) coincides with the radial projection on ∂F with center c_{F}. Then we extend this to a Lipschitz function defined on F, and such that π_{2}(F) ⊂ F. We do this on each (n − 1)−face F contained in D_{1}, and this gives a definition of π_{2} on ⋃_{Q_{1}(ρ)} S_{n−1}(Q). This definition is coherent, because π_{2} is the identity on the intersections of (n − 1)−faces of cubes Q ∈ Q_{1}(ρ).

Then we define π_{2} on the (n−1)−faces F′ of cubes Q ∈ Q_{2}(ρ) such that F′ ⊄ D_{1}. We set π_{2}(y) = y on F′; notice that this is coherent with our definition on the faces F ⊂ D_{1}, because F ∩ F′ ⊂ ∂F for such a combination of faces. This gives a definition of π_{2} on all the faces of dimension n − 1 in D_{2}. We extend this to the whole D_{2}, making sure that π_{2}(Q) ⊂ Q for Q ∈ Q_{2}(ρ) and π_{2}(y) = y on ∂D_{2}. Then we set π_{2}(y) = y out of D_{2}, and we get a Lipschitz map π_{2} on ℝ^{n}.

Now π_{2} ∘ π_{1}(E) is contained in ℝ^{n} \ D_{2}, plus the interior of the hyperfaces F of cubes Q ∈ Q_{2}(ρ) such that F ⊄ D_{1} (i.e., that are not contained in a cube of Q_{1}(ρ)), plus the faces of dimension n − 2 of cubes Q ∈ Q_{1}(ρ). This comes from our earlier description of π_{1}(E), and the fact the we moved all the points of π_{1}(E) that lied on a (n − 1)−face F ⊂ D_{1} to a boundary ∂F. Also, Hdim (π_{2} ∘ π_{1}(E)) ≤ Hdim (π_{1}(E)) < d + 1, because π_{2} is Lipschiz.

If d = n − 2, we can take ψ = π_{2} ∘ π_{1} ; see the short justification below. Otherwise, we still need to project. We look first at faces of dimension n − 2 of cubes Q ∈ Q_{1}(ρ). For each such face F, we pick a point c_{F} ∈ int(F) \ π_{2} ∘ π_{1}(E). Such a point exists, because Hdim (π_{2} ∘ π_{1}(E)) < d + 1 ≤ n − 2. We decide that on ∂F ∪ [π_{2} ∘ π_{1}(E)], π_{3} coincides with the radial projection on ∂F with center c_{F}. On the other faces of dimension n − 2, we set π_{3}(y) = y (which is coherent).

Then we extend π_{3} to the faces of dimension n − 1. When we do this, we make sure that π_{3}(F) ⊂ F for each such face, that π_{3}(y) = y on the faces that do not touch D_{1}, and also that the interior of F is sent to the interior of F. Then we extend π_{3} to D_{2}, still making sure that π_{3}(Q) ⊂ Q for Q ∈ Q(ρ). Finally we set π_{3}(y) = y out of D_{2}.

Now π_{3} ∘ π_{2} ∘ π_{1}(E) is contained in ℝ^{n} \ D_{2}, plus the interior of the hyperfaces F of cubes Q ∈ Q_{2}(ρ) such that F ⊄ D_{1}, plus the faces of dimension n − 3 of cubes Q ∈ Q_{1}(ρ). Indeed, if z ∈ π_{2} ∘ π_{1}(E), then either z ∈ ℝ^{n} \ D_{2} and π_{3}(z) = z, or z lies in the interior of some hyperface F of a cube Q ∈ Q_{2}(ρ), with F ⊄ D_{1}, and then π_{3}(z) also lies in the interior of F, or else z lies in a face of dimension n − 2 of a cube Q ∈ Q_{1}(ρ), and then π_{3}(z) lies on the boundary of that face.

We continue this procedure until we get to d-dimensional faces. Each time, we first construct π_{k} on the faces F of dimension n − d + 1 of cubes Q ∈ Q_{1}(ρ). There we chose a center c_{F} ∈ int(F) \ π_{k−1} ∘ … π_{1}(E), and decide that for y ∈ ∂F ∪ [π_{k−1} ∘ … π_{1}(E) ∩ F], π_{k}(y) is the radial projection on ∂F of y (with center c_{F}). We set π_{k}(y) = y on the other faces of dimension n−d+1. Then we extend π_{k} to faces F′ of larger and larger dimensions, making sure that π_{k}(F′) ⊂ F′ for all F′, and that π_{3}(y) = y on the faces that do not touch D_{1}. When we get to hyperfaces, we also make sure that π_{k}(int(F′)) ⊂ int(F′) for the hyperfaces F′ that are contained in D_{2}, but not in D_{1}.

Eventually we get to k = n − d, and then π_{k} ∘ … π_{1}(E) is contained in the union of the d-dimensional faces of cubes Q ∈ Q_{1}(ρ), plus the interior of some n − 1-dimensional faces that are not contained in a cube of Q_{1}(ρ), plus ℝ^{n} \ D_{2}. Thus (3.17) holds (for ψ = π_{k} ∘ … π_{1}), and the other properties (3.15) and (3.16) are true by construction. □

Variants of Lemma 3.14 are often used, where we also require that for Q ∈ Q(ρ), H^{d}(ψ(E ∩ Q)) ≤ CH^{d}(E ∩ Q). Such a control is obtained by selecting more carefully the points c_{F} in the various faces, with a nice Fubini argument. But for the moment we do not need such refinements.

Now we use Lemma 3.14 to construct a deformation {ψ_{t}} of E onto E_{ρ}. Set ψ_{t}(y) = ty + (1 − t)ψ(y) for 0 ≤ t ≤ 1 ; this gives a one-parameter family of Lipschitz functions ψ_{t}, which all satisfy (3.15) and (3.16). In particular,

(3.18)  |ψ_{t}(y) − y| ≤ \sqrt{n} ρ for y ∈ ℝ^{n} and 0 ≤ t ≤ 1.

We may now return to our be a one-parameter family of deformations {φ_{t}}_{0≤t≤1}, as in the definition of T ND. Define new Lipschitz mappings f_{t}, 0 ≤ t ≤ 1, on ℝ^{n} by

(3.19)  \{ \begin{matrix}f_{t}(y) & = ψ_{2t}(y)  for 0 ≤ t ≤ 1/2, \\ f_{t}(y) & = φ_{2t−1}(ψ(y)) for 1/2 ≤ t ≤ 1. \end{matrix}

We want to check that {f_{t}} satisfies the conditions (1.3)-(1.6), relative to the slightly larger ball B = B(x, (1 + η_{0})r). First, each f_{t} is clearly Lipschitz. Next, let us check that f_{t}(\overline{B}) ⊂ \overline{B}, as in (1.3). Observe that

(3.20)  B(x, r) ⊂ D_{1} ⊂ D_{2} ⊂ B(x, r + 2\sqrt{n} ρ) ⊂ B(x, r + αr_{1}) ⊂ B

by defintion of D_{1} and D_{2}, (3.3), (3.11), and because α ≤ \frac{η_{0}}{2} by (3.10).

If y ∈ D_{2}, then ψ_{t}(y) ∈ D_{2} for all t, by (3.16), and so (3.20) says that f_{s}(y) ∈ B for s ≤ 1/2. If in addition ψ(y) ∈ \overline{B}(x, r), then φ_{t}(ψ(y)) ∈ \overline{B}(x, r), by (1.3) for φ_{t}, and f_{s}(y) = φ_{2s−1}(ψ(y)) ∈ \overline{B}(x, r) ⊂ B for s ≥ 1/2, by (3.20). If instead ψ(y) ∉ \overline{B}(x, r), then (1.5) says that φ_{t}(ψ(y)) = ψ(y), so f_{s}(y) = ψ(y) for s ≥ 1/2. Thus f_{s}(y) ∈ D_{2} ⊂ B, as for s = 1/2 above. So f_{t}(D_{2}) ⊂ B, and we are left with \overline{B} \ D_{2}. But

(3.21)  ψ_{t}(y) = y and f_{t}(y) = y for all t when y ∈ ℝ^{n} \ D_{2},

by (3.15) (for ψ_{t}), and then (1.5) (because ψ_{t}(y) = y lies out of B(x, r), by (3.20)). Thus f_{t}(\overline{B} \ D_{2}) ⊂ \overline{B} too, and f_{t} satisfies (1.3).

Note that (1.4) (the continuity of f_{t}(y) in t) holds because ψ_{1} = ψ and φ_{0}(y) = y. Also, f_{0}(y) = y because ψ_{0}(y) = y, and f_{t}(y) = y for y ∉ B, by (3.20) and (3.21). Which takes care of (1.5).

We still need to check (1.6). So let y ∈ E ∩ B and 0 ≤ s ≤ 1 be given; we want to show that

(3.22)  dist (f_{s}(y), E) ≤ α_{0}(1 + η_{0})r.

First,

(3.23)  dist (ψ_{t}(y), E) ≤ |ψ(y) − y| ≤ \sqrt{n} ρ ≤ \frac{1}{2} α r_{1} ≤ αr ≤ \frac{1}{2} α_{0}r

by (3.18), (3.3), (3.11), and (3.10). So (3.22) already holds for 0 ≤ s ≤ 1/2.

For s > 1/2 we shall distinguish between cases again. Observe that

(3.24)  ψ(y) ∉ B(x, r) when y ∈ ℝ^{n} \ D_{1},

because y lies in a cube Q ∈ Q(ρ) \ Q_{1}(ρ), and then (3.16) says that ψ(y) lies in Q, which does not meet B(x, r) because B(x, r) ⊂ int(D_{1}) (by (3.20) and because B(x, r) is open).

If y ∈ E\D_{1} and s ≥ 1/2, (3.24) says that ψ(y) ∉ B(x, r), then f_{s}(y) = φ_{2s−1}(ψ(y)) = ψ(y), by (1.5) for {φ_{t}}. Then (3.22) follows from (3.23).

We are left with the case when y ∈ D_{1} (and s ≥ 1/2). Let us first check that

(3.25)  ψ(y) ∈ E_{ρ} when y ∈ E ∩ D_{1}.

Let Q be a cube of Q_{1}(ρ) that contains x. Then ψ(y) ∈ S_{d}(Q) by (3.17). Since Q touches E (because x ∈ E), S_{d}(Q) ⊂ E_{ρ} and (3.25) follows.

If in addition ψ(y) ∈ B(x, r), we can apply (3.5) for {φ_{t}} and get that f_{s}(y) = φ_{2s−1}(ψ(y)) lies within αr_{1} of E. So (3.22) holds in that case, again because αr_{1} ≤ α_{0}r. If ψ(y) ∉ B(x, r), then f_{s}(y) = ψ(y) and (3.22) holds because of (3.23), just like when s ≤ 1/2.

So {f_{t}} satisfies (1.3)-(1.6). Since E satisfies the hypotheses of Theorem 1.8, the analogue of (1.7) for B(x, (1 + η_{0})r) holds (because (1 + η_{0})r ≤ r_{0} by (3.3) and (3.11)). That is,

(3.26) H^{d}[B(x, r) ∩ f_{1}(E)] ≥ H^{d}[B(x, (1 − η_{0})(1 + η_{0})r) ∩ f_{1}(E)] ≥ δ_{0}((1 + η_{0})r)^{d}.

4. A functional on unions of faces

On the other hand, we have already seen that f_{1}(y) lies out of B(x, r) when y ∈ ℝ^{n}\D_{1} (by (3.24) and (1.5) for {φ_{t}}). So

(3.27)  B(x, r) ∩ f_{1}(E) ⊂ f_{1}(E ∩ D_{1}) = (φ_{1} ∘ ψ)(E ∩ D_{1}) ⊂ φ_{1}(E_{ρ}),

where the last inclusion comes from (3.25). Hence

(3.28)  H^{d}(B(x, r) ∩ φ_{1}(E_{ρ})) ≥ δ_{0}r^{d} ≥ 2^{−d}r_{1}^{d}δ_{0} ≥ 2δ_{1}r_{1}^{d},

by (3.26), (3.11), and (3.10). And H^{d}(φ_{1}(B(x, r) ∩ E_{ρ})) is even larger, since φ_{1}(y) lies out of B(x, r) when y ∉ B(x, r).

This completes our proof of (3.13); as was said before (3.13), we get that E satisfies T ND (as soon as it satisfies (3.9) and the topological condition in Theorem 1.8). Thus Theorem 1.8 follows from Theorem 3.8. □

4. A functional on unions of faces

Let E be a closed set that satisfies the hypotheses of Theorem 3.8. We want to prove that its Hausdorff dimension is at least d^{∗} > d, and the main point will be to establish the following lemma.

Lemma 4.1. For every large C_{1} > 0, we can find α_{2} > 0 and r_{0} > 0 such that

(4.2)  H^{d}(E_{ρ} ∩ B(x_{1}, r_{1})) > C_{1}r_{1}^{d}

for all choices of x_{1} ∈ E, r_{1} ∈ (0, r_{0}), and ρ > 0 such that ρ ≤ α_{2}r_{1}.

We shall see in Section 7 that Theorem 3.8 is relatively easy and straightforward once we prove the lemma. And in fact it would be enough to prove the lemma for some (large) C_{1} that can be computed from the dimension.

Our strategy for proving the lemma (which will keep us busy in Sections 4-6) will be to assume that (4.2) fails (so that (3.4) holds) and produce a contradiction. For this we shall use (3.4) and T ND to construct an Ahlfors-regular, uniformly rectifiable set F^{∗} with a big intersection with E_{ρ}. The contradiction will then come from the fact that F^{∗} is fairly flat at most scales and locations (by uniform rectifiability), while our assumption (1.2) says that E (and hence E_{ρ}, at least at scales larger than ρ) is never flat.

To find F^{∗}, we shall minimize a functional like J(F) = H^{d}(F ∩ E_{ρ}) + MH^{d}(F \ E_{ρ}), where M is a large constant, and among sets F that are a finite union F^{∗} of d-dimensional faces of cubes, plus a set of lower dimension F \ F^{∗}, and that can be obtained from E_{ρ} by deformations like the {φ_{t}} in the definition of T ND. The point of taking finite unions of faces is that the existence of a minimizer F will be essentilly trivial.

Our hypothesis T ND will be used to find a ball B(x, r) ⊂ B(x_{1}, r_{1}) where to define the functional, and then prove that H^{d}(F) is never too small (by (3.6)). Then, if M is large enough (compared to δ_{1}^{−1}), we will get that H^{d}(F^{∗} ∩ E_{ρ}) = H^{d}(F ∩ E_{ρ}) is not too small either (because H^{d}(F \ E_{ρ}) is very small).

To show that F^{∗} is Ahlfors-regular and uniformly rectifiable, we shall prove that F is a quasiminimal set (with the terminology of Almgren, a restricted set) in some open set, and then use the main result in [DS3]. The quasiminimality constant, and then the Ahlfors-regularity and uniform rectifiability constants will depend on M, hence essentially all the constants at hand (except α_{2}), but we shall still be able to get a contradiction at some scale larger than ρ, if ρ/r_{1} (or equivalently α_{2}) is small enough.

A good part of the argument below is quite similar to parts of [DS3], Section 12, but with small differences that seem to make it hard to import the results of [DS3] directly. So we shall repeat some of the arguments here, mainly for the convenience of the reader.

We start with the preparation of an appropriate domain and the definition of a functional. Since we are given C_{1} > 0, we get other constants r_{0}, α, η and δ_{1} from T ND. Let

(4.3)  x_{1} ∈ E, 0 < r_{1} < r_{0}, and 0 < ρ ≤ α_{2} r_{1}

be given, as in the statement of Lemma 4.1, and where α_{2} will be chosen near the end of the argument. We suppose that (4.2) fails (or equivalently that (3.4) holds) and we want a contradiction.

By T ND, we can find x ∈ E and r > 0 such that B(x, r) ⊂ B(x_{1}, r_{1}) and (3.6) holds for all the appropriate deformations {φ_{t}}.

We want to define a domain V ⊂ B(x, r) where things will really happen. Since we shall work with lots of slightly different security balls slightly smaller than B(x, r), it will be convenient to set

(4.4)  B_{j} = B(x, r − ηr_{1} + \frac{jηr_{1}}{10} ) for 0 ≤ j ≤ 10.

Recall that ηr_{1} ≤ r (because (3.6) holds for the identity mapping, see the comment a little below (3.7)). So there is no difficulty with the definitions in (4.4). Also observe that B_{0} ⊂ … ⊂ B_{10}, and that the annulus from (3.7) is

(4.5)  A_{ηr_{1}}(x, r) = B(x, r) \ B_{0} = B_{10} \ B_{0}.

Things will be easier if the domain V is composed of (multiples) of dyadic cubes. First choose a number L such that

(4.6)  \frac{1}{200} Min (η, α) r_{1} ≤ \sqrt{n} L ≤ \frac{1}{100} Min (η, α) r_{1},

and such that ρ^{−1}L is a power of 2. Call Q(L) the set of “dyadic cubes” of sidelength L, as in (3.1). Set

(4.7)  Q_{D} = {Q ∈ Q(L) ; Q ∩ B_{6} ≠ ∅},

(4.8)  Q_{V} = \{Q ∈ Q_{D}; dist (Q, E) ≤ \frac{αr_{1}}{2} \},

(4.9)  D = ⋃_{Q∈Q_{D}} Q, and V = ⋃_{Q∈Q_{V}} Q.

Notice that diam (Q) = \sqrt{n} L ≤ 100^{−1}Min (η, α) r_{1} for Q ∈ Q(L), by (4.6), so
(4.10)  B_{6} ⊂ D ⊂ B_{7} ⊂ B(x, r),

and also

(4.11)  dist (y, E) ≤ \frac{αr_{1}}{2} + \sqrt{n} L < αr_{1} for y ∈ V.

Let us already decide that we shall take α_{2} < (200\sqrt{n})^{−1}Min (η, α), to make sure that in the following construction,

(4.12)  ρ ≤ L

by (4.6)). Note that each cube of Q(ρ) is neatly contained in a cube of Q(L), because ρ^{−1}L is a power of 2. We shall work for a long time with a given ρ, but it will be important to make sure that our various constants do not depend on ρ (or α_{2}), because our final contradiction will require ρ to be small enough compared to r_{1}.

Consider the class F_{0} of closed subsets of D that are a (finite) union of d-dimensional faces of cubes Q ∈ Q(ρ), plus perhaps a set of Hausdorff dimension ≤ d − 1. For F ∈ F_{0}, denote by F^{∗} the union of the faces of dimension d (of cubes in Q(ρ)) that are contained in F. Thus

(4.13)  F^{∗} ⊂ F and Hdim(F \ F^{∗}) ≤ d − 1.

We are mostly interested in the subclass F of the sets F ∈ F_{0} that can be written

(4.14)  F = φ_{1}(E_{ρ} ∩ D),

where {φ_{t}}, 0 ≤ t ≤ 1, is a family of Lipschitz mapping on ℝ^{n} such that

(4.15)  φ_{t}(D) ⊂ D for all t,

(4.16)  φ_{t}(y) is a continuous function of t for every y ∈ ℝ^{n},

(4.17)  φ_{t}(y) = y for t = 0 and for y ∈ ℝ^{n} \ D,

(4.18)  dist (φ_{t}(y), E) < αr_{1} for y ∈ E_{ρ} ∩ D and 0 ≤ t ≤ 1,

and even

(4.19)  φ_{1}(y) ∈ V for y ∈ E_{ρ} ∩ D.

Note that even though the set F^{∗} \ F has a lower Hausdorff dimension, it may play an important role in the description above. That is, (4.14) gives a representation of F as a deformation of E_{ρ} ∩ D, but we do not necessarily have a similar representation of F^{∗}, because for instance we cannot always deform F onto F^{∗}.

Let us check that F is not empty, and that

(4.20)  E_{ρ} ∩ D ∈ F.

The point is that we can take the trivial deformation φ_{t}(y) = y in (4.15)-(4.19). Of course only (4.18) and (4.19) need to be checked. But

(4.21)  dist (y, E) ≤ \sqrt{n} ρ ≤ \sqrt{n} L ≤ 100^{−1}αr_{1} for y ∈ E_{ρ},

by definition of E_{ρ}, (4.12) and (4.6). Hence

(4.22)  E_{ρ} ∩ D = E_{ρ} ∩ V,

by (4.8). So (4.18) and (4.19) hold, φ_{t}(y) = y satisfies the required conditions, and E_{ρ} ∩ D ∈ F (because it is a union of d-dimensional faces of cubes from Q(ρ).

Let us use T ND. Each family {φ_{t}} as in (4.15)-(4.19) satisfies (1.3), (1.4), and (1.5), because D ⊂ B(x, r) (by (4.10)). Let us check that it also satisfies (3.5), i.e., that dist (φ_{t}(y), E) ≤ αr_{1} for y ∈ E_{ρ} ∩ B(x, r) and t ∈ [0, 1]. If y lies out of D, then φ_{t}(y) = y, hence φ_{t}(y) ∈ E_{ρ} and the result follows from (4.21). Otherwise, we can simply apply (4.18).

So (3.5) holds, we can apply T ND, and (3.6) says that

(4.23)  H^{d}(φ_{1}(E_{ρ} ∩ B(x, r))) ≥ δ_{1}r_{1}^{d} + H^{d}(E_{ρ} ∩ A_{ηr_{1}}(x, r)).

Set H = E_{ρ} ∩ [B(x, r) \ D] for a few lines. Then φ_{1}(E_{ρ} ∩ B(x, r)) = φ_{1}(E_{ρ} ∩ D) ∪ φ_{1}(H) (because D ⊂ B(x, r), see (4.10)), and this union is disjoint because φ_{1}(D) ⊂ D and φ_{1}(y) = y ∉ D when y lies out of D. Moreover, φ_{1}(H) = H (because φ_{1}(y) = y on H), so the left-hand side of (4.23) is H^{d}(φ_{1}(E_{ρ} ∩ D)) + H^{d}(H). Also,

(4.24)  H = E_{ρ} ∩ [B(x, r) \ D] ⊂ E_{ρ} ∩ [B(x, r) \ B_{6}] ⊂ E_{ρ} ∩ A_{ηr_{1}}(x, r)

by (4.10) and (4.5) (or the definition (3.7)). So we can substract H^{d}(H) from both sides of (4.23), and we get that

(4.25)  H^{d}(φ_{1}(E_{ρ} ∩ D)) ≥ δ_{1}r_{1}^{d} + H^{d}(E_{ρ} ∩ A_{ηr_{1}}(x, r) ∩ D).

This holds for every family {φ_{t}} such that (4.15)-(4.19) hold. When we restrict to families where F = φ_{1}(E_{ρ} ∩ D) ∈ F_{0} as in (4.14), we get that

(4.26)  H^{d}(F) = H^{d}(φ_{1}(E_{ρ} ∩ D)) ≥ δ_{1}r_{1}^{d} + H^{d}(E_{ρ} ∩ A_{ηr_{1}}(x, r) ∩ D) for F ∈ F.
Set M = \frac{2C_{1}}{δ_{1}} , with C_{1} as in (4.2) and (3.4), and define a functional on F by
(4.27)  J(F) = H^{d}(F ∩ E_{ρ}) + MH^{d}(F \ E_{ρ}) for F ∈ F.

5. Almgren quasiminimality of F

Note that J(F) = H^{d}(F^{∗} ∩ E_{ρ}) + MH^{d}(F^{∗} \ E_{ρ}), where F^{∗} still denotes the finite union of d-faces that satisfies (4.13). Since F ≠ ∅ (by (4.20)) and there is only a finite number of possible sets F^{∗}, we can find a minimizer F, i.e., a set F ∈ F such that

(4.28)  J(F) = inf {J(F′) ; F′ ∈ F}.
Recall from (4.20) that F′ = E_{ρ} ∩ D ∈ F. Hence
(4.29)  J(F) ≤ J(E_{ρ} ∩ D) = H^{d}(E_{ρ} ∩ D) ≤ H^{d}(E_{ρ} ∩ B(x, r)) ≤ C_{1}r_{1}^{d},

by (4.10) and because we assumed that (4.2) does not hold (to get a contradiction). Then

(4.30)  H^{d}(F \ E_{ρ}) ≤ M^{−1}J(F) ≤ \frac{δ_{1}}{2C_{1}} C_{1}r_{1}^{d} = \frac{δ_{1} r_{1}^{d}}{2}

by (4.27), (4.29), and our definition of M. Now (4.26) yields

(4.31)  H^{d}(F ∩ E_{ρ}) ≥ H^{d}(F) − H^{d}(F \ E_{ρ}) ≥ δ_{1}r_{1}^{d}/2 + H^{d}(E_{ρ} ∩ A_{ηr_{1}}(x, r) ∩ D).

Thus F has a fairly big intersection with E_{ρ}. It will be useful to have a lower bound on the size of the part of F ∩ E_{ρ} that lies in B_{0} = B(x, r − ηr_{1}) (recall the notation (4.4)), because it will be easier to show that F^{∗} is regular there. Note that F ∩ E_{ρ} is the disjoint union of F ∩E_{ρ} ∩B_{0} and F ∩E_{ρ} \B_{0}. Recall that F ⊂ D ⊂ B(x, r) by definition of F_{0} and (4.10), so F ∩E_{ρ} \B_{0} ⊂ E_{ρ} ∩A_{ηr_{1}}(x, r)∩D, by (4.5). So we may substract H^{d}(F ∩E_{ρ} \B_{0}) from both sides of (4.31) and we get that

(4.32)  H^{d}(F ∩ E_{ρ} ∩ B_{0}) ≥ δ_{1}r_{1}^{d}/2.

Our next goal is to show that F^{∗} is locally uniformly rectifiable in B_{0} because F is a minimizer for some nice functional. This will be used to produce lots of balls where F (and hence E_{ρ}) are very flat, and a contradiction with (1.2) will ensue.

5. Almgren quasiminimality of F

Let F be the minimizer for J introduced in the last section, and set

(5.1)  B_{2} = B(x, r − \frac{9η r_{1}}{10} )

(as in (4.4)). The goal of this section is to show that

(5.2)  S = F ∩ B_{2} is a (B_{2}, k, δ)-quasiminimizer for H^{d},

where k is a constant that depends on C_{1} and δ_{1} (in fact, k = CM = 2CC_{1}δ_{1}^{−1} for some geometric constant C),

(5.3)  δ = 4^{−n−1}L,

and the notion of quasiminimizer for H^{d} is the same as in [DS3] and [Al], and will be explained soon. See [DS3], pages 7 and 8, for the official definition that we shall use.

What does (5.2) mean? First, S should be closed in B_{2} and non empty (Condition (1.1) in [DS3]). This comes from (4.14) and (4.32) respectively. Next, H^{d}(S) should be locally finite (Condition (1.2) in [DS3]), which is obvious because H^{d}(F) ≤ J(F) < +∞. And then S should satisfy (1.8) in [DS3].

For this, we give ourselves a Lipschitz function ψ : ℝ^{n} → ℝ^{n} and that satisfies the conditions (1.5)-(1.7) in [DS3]. In particular, (1.5) and (1.6) says that if we set

(5.4)  W = {z ∈ ℝ^{n}; ψ(z) ≠ z},

then

(5.5)  diam (W ∪ ψ(W)) < δ

and

(5.6)  W ∪ ψ(W) ⊂⊂ B_{2},

i.e., dist (W ∪ ψ(W), ℝ^{n} \ B_{2}) > 0. The condition (1.7) in [DS3] says that there is a homotopy from the identity to f, with conditions like (5.5) and (5.6), but we shall not need it. We need to show that under these conditions,

(5.7)  H^{d}(S ∩ W) ≤ kH^{d}(ψ(S ∩ W)).

Our strategy for proving this is to use ψ to construct a competitor F_{1} ∈ F, and then deduce (5.7) from the minimality of J(F). Of course we cannot take F_{1} = ψ(F) directly, because it probably does not lie in F, so we shall have to modify it in two ways.

First, it could be that ψ(F) gets out of V, which would be bad because of our constraint (4.19) on deformations. To fix this problem, we shall construct a Lipschitz retraction onto V, and use it to push back ψ(F) inside V.

But also, ψ(F) is not necessarily a finite union of d-faces, plus a set of lower dimension, so we shall need to project it onto d-dimensional skeletons (a little like in Lemma 3.14).

We start with the Lipschitz retraction.

Lemma 5.8. If V is a finite union of (closed) dyadic cubes of sidelength L, there is 4^{n}−Lipschitz function h defined on

(5.9)  V_{+} = \{y ∈ ℝ^{n}; dist (y, V) ≤ \frac{L}{4}\},

and such that

(5.10)  h(V_{+}) ⊂ V,

(5.11)  h(y) = y for y ∈ V,

and

(5.12)  |h(y) − y| ≤ \sqrt{n} L for y ∈ V_{+}.

The statement is not exactly the same as in Lemma 12.2 in [DS3], but the proof will be the same. We shall get h as a composition of a few mappings g_{j}.

Set g_{1}(y) = y on V. Call Q_{+} the collection of cubes Q ∈ Q(L) that touch V but are not contained in it. Thus

(5.13)  V_{+} \ V ⊂ ⋃_{Q∈Q_{+}} Q ∩ V_{+}.

Call c_{Q} the center of Q ∈ Q_{+}. Then dist (c_{Q}, V_{+}) ≥ dist (c_{Q}, V) − L/4 ≥ L/4. We define g_{1} on Q ∩ V_{+} to be the radial projection on ∂Q with center c_{Q} (as defined in the proof of Lemma 3.14). Note that g_{1}(y) = y on ∂Q, so our different definitions coincide (on intersections of cubes). Observe also that g_{1} is 4−Lipschitz on V_{+}, because each c_{Q} if far from V_{+}. Finally, we claim that

(5.14)  dist (g_{1}(y), V) ≤ dist (y, V) ≤ 4^{−1}L for y ∈ V_{+} .

This is trivial when y ∈ V. Then let y ∈ V_{+} \ V be given, let Q ∈ Q_{+} contain y, and let z be a point of V such that |z − y| = dist (y, V). Also call Q_{z} a cube of V that contains z. Then Q_{z} touches Q (because otherwise it would be too far), and even z ∈ Q ∩ Q_{z} (by inspection). Note that c_{Q} lies further from T = Q ∩ Q_{z} than y. Then dist (g_{1}(y), Q) ≤ dist (g_{1}(y), T) ≤ dist (y, T) ≤ |z − y| = dist (y, V), as needed for (5.14). Now (5.14) yields

(5.15)  g_{1}(V_{+}) ⊂ V ∪ ⋃_{Q∈Q_{+}}[S_{n−1}(Q) ∩ V_{+}].

We set g_{2}(y) = y on V. For each (d − 1)-face T of a cube Q ∈ Q_{+} such that T is not already contained in V, we decide that for y ∈ T ∩ V_{+}, g_{2}(y) is the radial projection of y on ∂T (centered at the center of T). This definition is coherent, because T only meets different faces or V along ∂T, where g_{2} is the identity. Note that we do not need to extend g_{2}, because it is already defined on g_{1}(V_{+}), by (5.15). Also, g_{2} is 4−Lipschitz, again because y ∈ V_{+} is never close to the center of T.

As before, dist (g_{2}(y), V) ≤ dist (y, V) ≤ 4^{−1}L for y ∈ T ∩ V_{+}, and hence g_{2}(y) ∈ S_{n−2}(T) ∩ V_{+}. Thus

(5.16)  g_{2} ∘ g_{1}(V_{+}) ⊂ V ∪ ⋃_{Q∈Q_{+}}[S_{n−2}(Q) ∩ V_{+}].

We can define g_{3} on V and on the sets T ∩ V_{+}, where T is a (n − 2)−dimensional face of some cube Q ∈ Q_{+} and is not contained in V, as we did before.

After a few iterations, we get a 4^{n}−Lipschitz mapping h = g_{1} ∘ … g_{n} that sends V_{+} to V, plus the intersection of V_{+} with a set of vertices. But only the vertices of V can lie in V^{+}, so h(V_{+}) ⊂ V, as in (5.10). We have that h(y) = y for y ∈ V because all the g_{j} fix points of V, and our last condition (5.12) holds because g_{j}(Q) ⊂ Q for each Q ∈ Q(L) and each j, so that if we start from y in some cube Q ∈ Q(L), h(y) lies in the same Q. □

We may now return to the Lipschitz mapping ψ of the beginning of this section. Recall from (4.14) and (4.19) that

(5.17)  F = φ_{1}(E_{ρ} ∩ D) ⊂ V,

for some family {φ_{t}} that satisfies (4.15)-(4.19). Notice that if ψ(y) ≠ y, then y ∈ W (by (5.4)), and then |ψ(y) − y| ≤ diam (W ∪ ψ(W)) < δ by (5.5). Hence

(5.18)  |ψ(y) − y| ≤ δ = 4^{−n−1}L for y ∈ ℝ^{n},

by (5.3). By (5.17) and (5.18), ψ(y) ∈ V_{+} (and hence h(ψ(y)) is defined) for y ∈ F. Set

(5.19)  F_{1} = ψ(F) and F_{2} = h(F_{1}).

Thus F_{2} ⊂ V. We still need to project it back on d-dimensional skeletons. For this part of the construction, we shall work on cubes of sidelength ρ. Some additional notation will be useful. Call S_{d} ⊂ ℝ^{n} the union of all the d-dimensional faces of cubes Q ∈ Q(ρ). With the notations of the beginning of Section 3,

(5.20)  S_{d} = ⋃_{Q∈Q(ρ)} S_{d}(Q).

For A ⊂ ℝ^{n}, call N (A) the union of all the cubes Q ∈ Q(ρ) which meet a cube of Q(ρ) which meets A.

Let W be as in (5.4), and set

(5.21)  X = h(ψ(F ∩ W)).

This is the part that we may need to project back onto S_{d}. Indeed

(5.22)  h(ψ(y)) = ψ(y) = y for y ∈ F \ W,

by (5.4), (5.17), and (5.11), so h(ψ(F \ W)) = F \ W. Except for a set of lower dimension, this set is contained in F^{∗}, which itself is contained in S_{d} by definition of F_{0}.

We may now apply Lemma 11.14 on page 90 of [DS3] to X. The notations are almost the same, except that ρ is called 2^{−j} there, and what we call N (A) here is denoted by N_{j}(A) there. We get the existence of a Lipschitz function f on ℝ^{n} such that

(5.23)  f(z) = z out of N (\overline{X}),

(5.24)  f(z) = z for z ∈ S_{d} ,

(5.25)  f(\overline{X}) ⊂ S_{d} ,

(5.26)  f(Q) ⊂ Q for Q ∈ Q(ρ),

and

(5.27)  H^{d}(f(\overline{X} ∩ Q \ S_{d})) ≤ CH^{d}(\overline{X} ∩ Q \ S_{d}) for every Q ∈ Q(ρ).

The proof of Lemma 11.14 in [DS3] is similar to the proof of Lemma 3.14 above, except that we also need (5.27) now. As was alluded to above, this can be obtained by chosing the centers c_{F} in the various faces more carefully, so that their average distance to the points of the current image of \overline{X} is not too small. □

Set F_{3} = f(F_{2}) = f(h(ψ(F))), and let us already check that
(5.28)  Hdim(F_{3} \ S_{d}) ≤ d − 1.

We already know from (5.22) that h(ψ(F^{∗} \ W)) = F^{∗} \ W ⊂ S_{d}, because F^{∗} ⊂ S_{d} by definition of F_{0}. Then f(h(ψ(F^{∗} \ W))) ⊂ S_{d} too, by (5.24). Since f(h(ψ(F^{∗} ∩ W))) ⊂ f(X) ⊂ S_{d} by definition (5.21) of X and by (5.25), we are left with f(h(ψ(F \F^{∗}))), which has Hausdorff dimension at most d − 1 by (4.13) and because f ∘ h ∘ ψ is Lipschitz.

Now (5.28) does not exactly say that F_{3} ∈ F_{0}, because F_{3} ∩ S_{d} is not necessarily an exact union of full d−faces. To fix this, we need to compose with yet another function π.

Call A the set of d-dimensional faces of cubes in Q(ρ) that meet F_{3} on a set of dimension > d−1 but are not contained in F_{3}. For T ∈ A, choose an origin c_{T} ∈ int(T)\F_{3}. Such a point exists by definition of A, and dist (c_{T} , F_{3}) > 0 because F_{3} is closed. For z ∈ F_{3} ∩ T, we let π(z) be the radial projection on ∂T with center c_{T}. Then we extend this definition to T, so that π is Lipschitz on T, π(T) ⊂ T, and π(z) = z on ∂T. We do this for each T ∈ A, then set π(z) = z on the rest of S_{d}, and then extend π to ℝ^{n} so that it stays Lipschitz,

(5.29)  π(Q) ⊂ Q for every Q ∈ Q(ρ),

and that

(5.30)  π(z) = z on the cubes that do not touch a face T ∈ A.

The details are the same as in the previous similar constructions.

Our next middle-term goal is to show that

(5.31)  F_{4} = π(F_{3}) = π ∘ f ∘ h ∘ ψ(F)

lies in the set F of competitors for J.

By definition of π, π(F_{3}) is now an exact union of d−faces (precisely, the faces of S_{d} that were contained in F_{3}), plus perhaps a set of dimension ≤ d − 1 coming from the almost-empty faces, and the boundaries of faces T ∈ A where we used π to project. And the remaining piece π(F_{3} \ S_{d}) is still at most (d − 1)-dimensional, by (5.28). So we shall get that F_{4} ∈ F_{0} as soon as we check that F_{4} ⊂ D.

Recall that F_{2} = h(F_{1}) ⊂ V, by (5.19) and (5.10). Since the mappings f and π preserve cubes of Q(ρ) (by (5.26) and (5.29)), F_{4} = π(f(F_{2})) ⊂ V ⊂ D (by (4.8) and (4.9)), and

(5.32)  F_{4} ∈ F_{0}.

The verification that F_{4} ∈ F will take some time, because we need to construct a family of Lipschitz functions as in (4.14)-(4.19). The following function g, which we first define on V, will be useful. Set

(5.33)  g(z) = π ∘ f ∘ h ∘ ψ(z) for z ∈ V,

The definition makes sense, because if z ∈ V, then dist (ψ(z), V) < 4^{−n}L by (5.18), and then h(ψ(z)) is defined and lies in V, by (5.9) and (5.10). Then g(z) is well defined, because f and π are defined everywhere. Note that in addition

(5.34)  g(z) ∈ V ⊂ D for z ∈ V,

because f and π preserve cubes of Q(ρ), and V ⊂ D by definition (see (4.8) and (4.9)). We want to see how much g moves points. First observe that

(5.35)  h(ψ(z)) = ψ(z) = z for z ∈ V \ B_{2}.

by (5.4), (5.6), and (5.11). Even when z ∈ B_{2} we can still say something, since

(5.36) |h(ψ(z))−z| ≤ |h(ψ(z))−ψ(z)|+|ψ(z)−z| ≤ \sqrt{n} L+4^{−n−1}L ≤ 2\sqrt{n} L for z ∈ V,

by (5.12) and (5.18).

Next we look at f and the set N (\overline{X}) where it may move points. Recall from (5.21) that X = h(ψ(F ∩W)), which is defined because F ⊂ V (by (5.17)). Also recall from (5.4) that ψ(W) ⊂ B_{2}, so

(5.37)  dist (h(ψ(z)), B_{2}) ≤ |h(ψ(z)) − ψ(z)| ≤ \sqrt{n} L for z ∈ F ∩ W,

by (5.12) again. In other words, dist (y, B_{2}) ≤ \sqrt{n} L for y ∈ X.

By definition, every point of N (\overline{X}) lies within 2\sqrt{n} ρ of X, hence within 3\sqrt{n} L of B_{2} (by (4.12)). Since 3\sqrt{n} L < ηr_{1}/30 by (4.6), we get that

(5.38)  N (\overline{X}) ⊂ B_{3}.

Recall from (5.23) that f(z) = z out of N (\overline{X}). Then

(5.39)  f(h(ψ(z))) = h(ψ(z)) = ψ(z) = z for z ∈ V \ B_{3} ,

by (5.35), and (5.38).

Now we consider π. Recall from (5.30) that π(z) = z unless z ∈ Q for a cube Q ∈ Q(ρ) that touches a face T ∈ A. Let us check that

(5.40)  dist (T, B_{3}) ≤ \frac{ηr_{1}}{30} for T ∈ A.

Let T ∈ A be given, and first assume that T ⊂ F. Since T is not contained in F_{3} = f(h(ψ(F))), it must contain some point z ∈ F such that f(h(ψ(z))) ≠ z. Since z ∈ F ⊂ V, (5.39) says that z ∈ B_{3}. In this case (5.40) holds trivially.

If A is not contained in F, Hdim(A ∩ F) ≤ d − 1 (because F ∈ F_{0}), hence there are some points of F_{3} ∩ T that do not lie in F. Such points are of the form y = f(h(ψ(z))), with z ∈ F. Since y ∉ F, f(h(ψ(z))) ≠ z and (5.39) says that z ∈ B_{3}. Also,

(5.41)  \begin{aligned}|f(h(ψ(z))) − z| & ≤ |f(h(ψ(z))) − h(ψ(z))| + |h(ψ(z)) − z| \\ & ≤ \sqrt{n} ρ + 2\sqrt{n} L ≤ 3\sqrt{n} L ≤ \frac{ηr_{1}}{30} \end{aligned} for z ∈ V,

by (5.26), (5.36), (4.12), and (4.6). When we apply this with z ∈ F and y = f(h(ψ(z))) ∈ F_{3} ∩ T as above, we get that dist (T, B_{3}) ≤ |z − y| = |z − f(h(ψ(z)))| ≤ ηr_{1}/30, and (5.40) holds.

If z ∈ ℝ^{n} is such that π(z) ≠ z, (5.30) says that z ∈ Q for a cube Q ∈ Q(ρ) that touches a face T ∈ A; then dist (z, B_{3}) ≤ ηr_{1}/30 + 2\sqrt{n} ρ < ηr_{1}/10 by (5.40), (4.12), and (4.6). Thus

(5.42)  π(z) = z for z ∈ ℝ^{n} \ B_{4} ,

and hence (5.39) and (5.33) yield

(5.43)  g(z) = f(h(ψ(z))) = h(ψ(z)) = ψ(z) = z for z ∈ V \ B_{4}.

We are now ready to complete our definition of g.

Lemma 5.44. We can find a Lipschitz function g : ℝ^{n} → ℝ^{n} such that (5.33) holds,

(5.45)  g(z) = z for z ∈ ℝ^{n} \ B_{5},

and

(5.46)  g(z) ∈ \overline{B}_{5} ⊂ D for z ∈ \overline{B}_{5}.

Set H = [V ∩B_{5}]∪[∂B_{5}]. If we keep (5.33) on V ∩B_{5} and set g(z) = z on ∂B_{5}, we have a coherent definition of g on H, by (5.43). Moreover, g is Lipschitz on H. Indeed, since g is clearly Lipschitz on V (by (5.33)) and on \overline{B}_{5} \ B_{4} (because it is the identity there), it is enough to estimate g(y)−g(z) when y ∈ V ∩B_{4} and z ∈ ∂B_{5}. But then |g(y)−g(z)| ≤ 2r_{1} because g(y) ∈ V ⊂ B(x, r) (by (5.34) and (4.4)), while |y − z| ≥ ηr_{1}/10 (by (4.4) again). So g is Lipschitz on H.

Let us also check that g(H) ⊂ \overline{B}_{5}. When z ∈ [V \ B_{4}] ∪ ∂B_{5}, g(z) = z (by (5.43)), and so g(z) ∈ \overline{B}_{5}. Otherwise, notice that for z ∈ V,

(5.47)  \begin{aligned}|g(z) − z| & ≤ |g(z) − f(h(ψ(z)))| + |f(h(ψ(z))) − z| \\ & ≤ |π(f(h(ψ(z)))) − f(h(ψ(z)))| + 3\sqrt{n} L \\ & ≤ \sqrt{n} ρ + 3\sqrt{n} L < 4\sqrt{n} L ≤ \frac{ηr_{1}}{20} \end{aligned}

by (5.33), (5.41), (5.29), (4.12), and (4.6). In the remaining case when z ∈ V ∩ B_{4}, we still get that g(z) ∈ \overline{B}_{5}.

It is now easy to extend g from H to \overline{B}_{5}, so that it stays Lipschitz and with values in \overline{B}_{5}. [Otherwise, compose with a Lipschitz retraction on \overline{B}_{5}.] Finally extend g to the rest of ℝ^{n}, by setting g(z) = z out of \overline{B}_{5}. We still get a Lipschitz function, and the properties (5.33), (5.45), and (5.46) hold by construction. Lemma 5.44 follows. □

Recall that we want to use g to construct a family of Lipschitz functions as in (4.14)- (4.19) and show that F_{4} ∈ F. Denote by {φ_{t}} the family associated to F ∈ F and set

(5.48)  φ̃_{t}(y) = φ_{2t}(y) for 0 ≤ t ≤ 1/2

(and all y ∈ ℝ^{n}) and

(5.49)  φ̃_{t}(y) = (2 − 2t) φ_{1}(y) + (2t − 1) g(φ_{1}(y)) for 1/2 < t ≤ 1.

Each φ̃_{t} is Lipschitz, because g and the φ_{t} are. Next we need to check that φ̃_{t}(D) ⊂ D, as in (4.15). When t ≤ 1/2, this comes directly from (4.15) (for the φ_{t}). So we may restrict to t > 1/2.

Let y ∈ D be given. If φ_{1}(y) ∈ ℝ^{n} \ B_{5}, (5.45) says that g(φ_{1}(y)) = φ_{1}(y) and hence φ̃_{t}(y) = φ_{1}(y) ∈ D (by (5.49) and (4.15) for φ_{1}). Otherwise φ_{1}(y) ∈ B_{5}, (5.46) says that g(φ_{1}(y)) ∈ \overline{B}_{5}, and φ̃_{t}(y) ∈ \overline{B}_{5} ⊂ D by (5.49), because \overline{B}_{5} is convex, and by (4.10). So φ̃_{t}(D) ⊂ D in all cases.

The continuity in t of φ̃_{t}(y) (as in (4.16)) is clear. For the analogue of (4.17), we need to check that φ̃_{t}(y) = y when t = 0 and when y ∈ ℝ^{d} \D. When t = 0, this follows directly from (4.17). When t ≤ 1/2 and y ∈ ℝ^{d} \ D, this is just (4.17) again. When t > 1/2 and y ∈ ℝ^{d} \ D, we observe that φ_{1}(y) = y by (4.17), then g(φ_{1}(y)) = y by (5.45) and (4.10), so φ̃_{t}(y) = y by (5.49).

For (4.18), we need to check that dist (φ̃_{t}(y), E) < αr_{1} when y ∈ E_{ρ}∩D and 0 ≤ t ≤ 1. When t ≤ 1/2, this comes from (4.18) for the φ_{t}. For t > 1/2, first observe that

(5.50)  φ_{1}(y) ∈ V

by (4.19), so dist (φ_{1}(t), E) ≤ \frac{αr_{1}}{2} + \sqrt{n} L, by (4.11). Also, (5.47) says that |g(φ_{1}(y)) − φ_{1}(y)| ≤ 4\sqrt{n} L. Since by (5.49) φ̃_{t}(y) is a convex combination of φ_{1}(y) and g(φ_{1}(y)), we get that dist (φ̃_{t}(t), E) ≤ \frac{αr_{1}}{2} + 5\sqrt{n} L < αr_{1}, by (4.6) and as needed for the anagogue of (4.18).

We also need to check (4.19), i.e., that φ̃_{1}(y) ∈ V when y ∈ E_{ρ} ∩ D. This follows from (5.50) and (5.34). So our family {φ̃_{t}} satisfies (4.14)-(4.19). In addition, F_{4} = g(F) = g(φ_{1}(E_{ρ} ∩ D)) = φ̃_{1}(E_{ρ} ∩ D) by (5.31), (5.33), (4.14), and (5.49). Since we already know from (5.32) that F_{4} ∈ F_{0}, we get that F_{4} ∈ F (at last!).

Recall that F is a minimizer for J. Then J(F) ≤ J(F_{4}). Also recall that F_{4} = π(F_{3}), where π was designed to leave untouched all the d-dimensional faces of S_{d}, except the faces T ∈ A where F_{3} ∩ T was sent to ∂T. So F_{4} is contained in F_{3}, except perhaps for the image by π of the (d − 1)-dimensional part of F_{3} that did not live on S_{d} (see (5.28)), and a few (d − 1)-dimensional faces ∂T. Thus J(F_{4}) ≤ J(F_{3}), and altogether

(5.51)  J(F) ≤ J(F_{3}).

We want to analyse this carefully, so we cut F into the disjoint pieces F ∩ W and F \ W, where W is as in (5.4). Then

(5.52)  J(F) = J(F ∩ W) + J(F \ W),

where for convenience we still set

(5.53)  J(A) = H^{d}(A ∩ E_{ρ}) + MH^{d}(A \ E_{ρ})

as in (4.27), even when A ∉ F.

For the next few lines, it will be convenient to set

(5.54)  Φ = f ∘ h ∘ ψ.

Recall that F_{3} = Φ(F), so F_{3} = Φ(F ∩W)∪Φ(F \W). For y ∈ F \W, h(ψ(y)) = y by (5.22). If in addition y ∈ F^{∗} (the finite union of faces that almost cover F, see (4.13)), then y ∈ S_{d}, and (5.24) says that f(y) = y. Thus Φ(y) = y for y ∈ F^{∗} \ W, and Φ(F^{∗} \ W) = F^{∗} \ W. Since the small set Φ(F \ F^{∗}) does not contribute, J(F_{3}) ≤ J(Φ(F ∩ W)) + J(F \ W). So (5.51) and (5.52) yield

(5.55)  J(F ∩ W) ≤ J(Φ(F ∩ W))

and then

(5.56)  H^{d}(F ∩ W) ≤ MH^{d}(Φ(F ∩ W)),

because of the formula (5.53) for J.

In view of (5.7), we would prefer to have ψ(F ∩ W) instead of Φ(F ∩ W), so we want to show that h and f do not increase the measure of ψ(F ∩ W) too much. First, h is 4^{n}-Lipschitz (see Lemma 5.8), so

(5.57)  H^{d}(h ∘ ψ(F ∩ W)) ≤ 4^{nd}H^{d}(ψ(F ∩ W)).

6. Local uniform rectifiability and flatness

Recall from (5.21) that X = h ∘ ψ(F ∩ W), so the left-hand side of (5.57) is H^{d}(X). We cut X into X_{1} = X ∩S_{d} and X_{2} = X \S_{d}. By (5.24), f(z) = z on X_{1}, so H^{d}(f(X_{1})) = H^{d}(X_{1}). And

(5.58)  \begin{aligned}H^{d}(f(X_{2})) & ≤ ∑_{Q∈Q(ρ)} H^{d}(f(X_{2} ∩ Q)) ≤ C ∑_{Q∈Q(ρ)} H^{d}(f(\overline{X} ∩ Q \ S_{d})) \\ & ≤ C ∑_{Q∈Q(ρ)} H^{d}(\overline{X} ∩ Q \ S_{d}) ≤ C′H^{d}(\overline{X} \ S_{d}), \end{aligned}

by (5.27) and because the cubes Q have bounded overlap. Altogether,

(5.59) H^{d}(Φ(F ∩W)) = H^{d}(f(X)) ≤ H^{d}(f(X_{1}))+H^{d}(f(X_{2})) ≤ H^{d}(X_{1})+C′H^{d}(\overline{X} \S_{d})

by (5.54), (5.21), and (5.58). Next we check that

(5.60)  Hdim(\overline{X} \ [X ∪ S_{d}]) ≤ d − 1.

Indeed let z ∈ \overline{X} \ X be given. Recall from (5.21) that every point of X is of the form h(ψ(y)) for some y ∈ F ∩ W. Since W ⊂⊂ B_{2} and the domain of definition V_{+} of h is closed, we can also write z = h(ψ(y)), but with y ∈ F ∩ \overline{W}. In addition, y ∉ W, because z ∉ X. So ψ(y) = y. But y ∈ F ⊂ V by (5.17), so h(y) = y by (5.11). Altogether z = h(ψ(y)) = y ∈ F. If z ∈ F^{∗}, then it lies on S_{d}. So \overline{X} \ [X ∪ S_{d}] ⊂ F \ F^{∗}, and (5.60) follows from (4.13).

Because of (5.60), the right-hand side of (5.59) is at most H^{d}(X_{1}) + C′H^{d}(X \ S_{d}) = H^{d}(X_{1}) + C′H^{d}(X_{2}) ≤ C′H^{d}(X). Altogether,

(5.61)  \begin{aligned}H^{d}(F ∩ W) & ≤ MH^{d}(Φ(F ∩ W)) ≤ C′MH^{d}(X) \\ & = C′MH^{d}(h ∘ ψ(F ∩ W)) ≤ 4^{nd}C′MH^{d}(ψ(F ∩ W)) \end{aligned}

by (5.56), (5.59), (5.21), and (5.57).

Recall from (5.2) that S = F ∩ B_{2} and from (5.6) that W ⊂ B_{2}. So S ∩ W = F ∩ W, and (5.61) is just the same as (5.7).

This completes our proof of quasiminimality for F. Thus (5.2) holds (with k = 4^{nd}C′M), and we shall be able to apply the results of [DS3].

6. Local uniform rectifiability and flatness

In this section we use the local uniform rectifiability of F^{∗} that comes from (5.2) to show that F^{∗} is often flat, derive a contradiction with our assumption (1.2), and thus complete the proof of Lemma 4.1.

Let us first apply the main result of [DS3], which is stated on pages 10 and 11. Call S^{∗} the closed support in B_{2} of the restriction of H^{d} to S, as in (1.12) on page 9 of [DS3]. That is,

(6.1)  S^{∗} = {x ∈ B_{2} ; H^{d}(S ∩ B(x, r)) > 0 for all r > 0}.

Recall from (5.2) that S = F ∩ B_{2}. Since F is a finite union F^{∗} of d-faces, plus a set of Hausdorff dimension at most d − 1 (see the definition of F_{0} below (4.12)), we get that

(6.2)  S^{∗} = F^{∗} ∩ B_{2}.

Theorem 2.11 in [DS3] says that S^{∗} is locally Ahlfors-regular and uniformly rectifiable in B_{2}. More precisely, for each y ∈ S^{∗} ∩B_{1}, there is a closed, Ahlfors-regular and uniformly rectifiable set Z_{y} of dimension d such that

(6.3)  S^{∗} ∩ B(y, δ/2) ⊂ Z_{y} ⊂ S^{∗} ∩ B(y, δ).

Recall from (4.4) that B_{1} is a ball a little bit smaller than B_{2}, and from (5.3) and (4.6) that δ = 4^{−n−1}L ≤ ηr_{1}/100. In the statement of [DS3] one allows all balls B(y, R) centered on S^{∗}, with radius R < δ, and such that B(y, 3R) ⊂ B_{2}, but we shall only need the ones above.

We should also put stress on the fact that the Ahlfors-regularity and uniform rectifiability constants for the Z_{y} depend on our various constants, but not on ρ or α_{2}. The convention in this section will be that C denotes a constant that depends on n, d, α, η, δ_{1}, and even C_{1}, but in partiular not on ρ or α_{2}.

Let us not define uniform rectifiability here, and refer to [DS1] for all relevant information, but recall that Ahlfors-regular means that there is a constant C such that

(6.4)  C^{−1}t^{d} ≤ H^{d}(Z_{y} ∩ B(z, t)) ≤ Ct^{d} for z ∈ Z_{y} and 0 < t ≤ diam (Z_{y}).

In the present case, we also have that

(6.5)  δ/4 ≤ diam (Z_{y}) ≤ 2δ.

This is Remark 2.15 in [DS3]; grosso modo, the point is that if diam (S^{∗} ∩ B(y, R)) < δ/4 we could contract S^{∗} ∩B(y, R) into a point, and this would contradict the quasiminimality of S.

Cover S ∩ B_{1} with less than C balls B(y, δ/2) centered on S^{∗} ∩ B_{1}, and call Z the corresponding union of sets Z_{y}. Then Z is a uniformly rectifiable Ahlfors-regular set, of diameter comparable to δ and r_{1}, and

(6.6)  S^{∗} ∩ B_{1} ⊂ Z.

[For the reader that would be worried about taking (finite) unions of uniformly rectifiable sets, let us say that this is just a matter of convenience, and we could have worked with the various Z_{y} too.]

We shall not work with uniform rectifiability directly, but with one of its consequences, the “weak geometric lemma”. Set

(6.7)  β_{Z}(y, t) = inf_{P ∈P} { sup {t^{−1} dist (z, P) ; z ∈ Z ∩ B(y, t)}}

for y ∈ Z and t > 0, where P still denotes the set of affine d-planes. This is the same thing as β_{E} in (1.2), but relative to Z. Set

(6.8)  ε = \frac{ε_{0}}{10} ,

where ε_{0} is still the constant in (1.2), and consider the bad set

(6.9)  B(ε) = {(y, t) ; y ∈ Z, 0 < t < δ, and β_{Z}(y, t) > ε}.

The weak geometric lemma says that there is a constant C(ε) > 1, that depends only on ε and the Ahlfors-regularity and uniform rectifiability constants for Z, such that

(6.10)  ∫_{y∈Z∩B(y_{0},R)} ∫_{0<t<R} 1_{B(ε)} \frac{dH^{d}(y)dt}{t} ≤ C(ε) R^{d}

for y_{0} ∈ Z and 0 < R < diam (Z). The simplest reference for this is probably the following. Theorem 2.4 on page 32 of [DS1] says that uniform rectifiability implies the bilateral weak geometric lemma (BWGL). The BWGL is defined on page 32 in [DS1], and the reader may easily check that it is stronger than the weak geometric lemma (or (6.10)). Let us also say that since [DS1] prefers to talk about unbounded Ahlfors-regular sets Z, we may need to apply their Theorem 2.4 to Z′ = Z ∪ P, where P is some d-plane in ℝ^{n} such that dist (P, B_{1}) ≤ r_{1}. It is very easy to check that Z′ is uniformly rectifiable (once you are given the definition), and the weak geometric lemma for Z′ clearly implies (6.10).

So we have (6.10). It will be convenient to set

(6.11)  dμ = \frac{dH^{d}(y)dt}{t}.

Note that because of \frac{dt}{t}, μ is locally infinite (and essentially scale invariant).

Actually, we shall only use (6.10) at one scale. That is, we shall only need to know that

(6.12)  μ(B(ε)) = ∫ ∫_{(y,t)∈B(ε)} \frac{dH^{d}(y)dt}{t} ≤ C(ε) r_{1}^{d} ,

which is an easy consequence of (6.10). [If you are worried about the measure of the part

of B(ε) where diam (Z) < t < δ, note that ∫_{y∈Z} ∫_{10^{−2}δ≤t≤δ} dμ ≤ 10H^{d}(Z) ≤ Cr_{1}^{d}.]

Since μ is infinite, (6.12) really says that B(ε) is small, or equivalently that β_{Z}(y, t) ≤ ε most of the time. The general idea now to use this and the fact that

(6.13)  \begin{aligned}H^{d}(Z ∩ E_{ρ} ∩ B_{0}) & ≥ H^{d}(S^{∗} ∩ E_{ρ} ∩ B_{0}) = H^{d}(F^{∗} ∩ E_{ρ} ∩ B_{0}) \\ & = H^{d}(F ∩ E_{ρ} ∩ B_{0}) ≥ δ_{1}r_{1}^{d}/2, \end{aligned}

by (6.6), (6.2), (4.13), and (4.32), to find some place where E_{ρ} is very flat and get a contradiction. Since apparently we cannot use density results to find directly large balls where E_{ρ} and S^{∗} essentially coincide, we shall have to have resort to a slightly more complicated accounting argument.

Let C_{2} be a large constant, to be chosen soon; C_{2} will depend on many things, including ε_{0}, but this is still all right. Set

(6.14)  G_{0} = {(y, t) ; y ∈ Z ∩ E_{ρ} ∩ B_{0} and C_{2}ρ ≤ t < δ},

and then

(6.15)  G = {(y, t) ∈ G_{0} ; β_{Z}(y, t) ≤ ε}.

To check that G is reasonably large, first note that

(6.16)  μ(G_{0}) = H^{d}(Z ∩ E_{ρ} ∩ B_{0}) ∫_{C_{2}ρ}^{δ} \frac{dt}{t} ≥ \frac{δ_{1}r_{1}^{d}}{2} Log (\frac{δ}{C_{2}ρ}),

by (6.13). Later on, we shall take α_{2} so small (in (4.3)) that this will be as big as we want. But for the moment, let us just record that G = G_{0} \ B(ε), so

(6.17)  μ(G) ≥ μ(G_{0}) − μ(B(ε)) ≥ \frac{δ_{1}r_{1}^{d}}{2} Log (\frac{δ}{C_{2}ρ}) − C(ε) r_{1}^{d} ,

by (6.12). Next we want to give an upper bound for μ(G) that will contradict (6.17).

Our plan is to associate a reasonably large piece of E_{ρ} to each pair (y, t) ∈ G, with reasonable overlap, and then deduce our upper bound from (3.4).

So let (y, t) ∈ G be given. Since β_{Z}(y, t) ≤ ε, we can find a d-plane P such that

(6.18)  dist (z, P) ≤ εt for z ∈ Z ∩ B(y, t).

Besides, y ∈ E_{ρ} by definition of G_{0}. Thus we can find ξ ∈ E such that

(6.19)  |ξ − y| ≤ \sqrt{n} ρ ≤ C_{2}ρ/10 ≤ t/10,

by (3.2), if C_{2} is large enough, and because (y, t) ∈ G_{0}. By (1.2), β_{E}(ξ, t/2) ≥ ε_{0}, and so we can find ζ ∈ E ∩ B(ξ, t/2) such that

(6.20)  dist (ζ, P) ≥ \frac{ε_{0} t}{3} = \frac{10εt}{3}.

Set

(6.21)  D(y, t) = E_{ρ} ∩ B(ζ, εt)

Note that |ζ − y| ≤ |ξ − y| + t/2 ≤ t/10 + t/2 = 6t/10 (by (6.19)), so

(6.22)  D(y, t) ⊂ E_{ρ} ∩ B(y, t).

Also, dist (ζ, Z ∩ B(y, t)) ≥ dist (ζ, P) − εt ≥ \frac{10εt}{3} − ε ≥ 2εt by (6.18) and (6.20), and dist (ζ, Z \ B(y, t)) ≥ 4t/10 because |ζ − y| ≤ 6t/10. Altogether, dist (ζ, Z) ≥ 2εt and

(6.23)  dist (D(y, t), Z) ≥ εt.

These estimates will help with the bounded overlap. But first we want to show that D(y, t) is not too small. We claim that

(6.24)  H^{d}(D(y, t)) = H^{d}(E_{ρ} ∩ B(ζ, εt)) ≥ δ_{1}(εt)^{d}.

The first part comes from (6.21). For the second part, let us try to apply T ND to the ball B(ζ, εt). First we check (3.3) with x_{1} = ζ and r_{1} = εt. Clearly ζ ∈ E, εt ≤ δ ≤ r_{0}, and ρ ≤ C_{2}^{−1}t ≤ (2\sqrt{n})^{−1}αεt because (y, t) ∈ G_{0} and if C_{2} is large enough. If the analogue of (3.4) does not hold, i.e., if H^{d}(E_{ρ} ∩ B(ζ, εt)) > C_{1}(εt)^{d}, we are happy because (6.24) is satisfied (with a much better constant). Otherwise, we get a smaller ball B ⊂ B(ζ, εt) where (3.6) holds for all appropriate families {φ_{t}} of deformations. We take φ_{t}(y) = y everywhere, which satisfies the required conditions trivially, and now (3.6) says that H^{d}(E_{ρ} ∩ B(ζ, εt)) ≥ H^{d}(E_{ρ} ∩ B) ≥ δ_{1}(εt)^{d}. So (6.24) holds in all cases.

We are now ready for the accounting. In the following computations, we assume that all functions are measurable. One could avoid this difficulty by taking exterior measures, or by discretizing the computations, but we prefer to ignore the issue. First write that

(6.25)  μ(G) = ∫ ∫_{(y,t)∈G} \frac{dH^{d}(y) dt}{t} ≤ C ∫ ∫ ∫_{(y,t)∈G ; w∈D(y,t)} \frac{dH^{d}(w) dH^{d}(y) dt}{t^{d+1}}

by (6.11) and (6.24), and where we do not write the dependence on ε either. Next we want to exchange the order of integration. First observe that w ∈ E_{ρ} ∩ B(y, t) ⊂ E_{ρ} ∩ B(x, r) by (6.22), because y ∈ B_{0} by (6.14), and because t ≤ δ < ηr_{1}/100 by (6.14), (5.3), and (4.6). Thus

(6.26)  μ(G) ≤ C ∫_{w∈E_{ρ}∩B(x,r)} I(w) dH^{d}(w)

where the inside integral is

(6.27)  I(w) = ∫ ∫_{(y,t)∈G ; w∈D(y,t)} \frac{dH^{d}(y) dt}{t^{d+1}}.

Now fix w ∈ E_{ρ} ∩ B(x, r) and t ≤ δ, and integrate in y first. Note that y ∈ Z ∩ B(w, t), by (6.14) and (6.22). By local Ahlfors-regularity of Z (see (6.4)), H^{d}(Z ∩ B(w, t)) ≤ Ct^{d}, and when we integrate again in t we get that

(6.28)  I(w) ≤ ∫_{t∈T (w)} H^{d}(Z ∩ B(w, t)) \frac{dt}{t^{d+1}} ≤ C ∫_{t∈T (w)} \frac{dt}{t},

7. A lower bound on Hausdorff dimension

where now we integrate over the set T (w) of radii t for which we can find y such that w ∈ D(y, t). Note that if w ∈ D(y, t), then dist (w, Z) ≥ dist (D(y, t), Z) ≥ εt by (6.23), and on the other hand dist (w, Z) ≤ |w − y| ≤ t. So T (w) ⊂ [dist (w, Z), ε^{−1} dist (w, Z)], I(w) ≤ C, and (6.26) yields

(6.29)  μ(G) ≤ CH^{d}(E_{ρ} ∩ B(x, r)) ≤ CC_{1}r_{1}^{d},

because B(x, r) ⊂ B(x_{1}, r_{1}) and we assumed that (4.2) fails.

When we compare this with (6.17), we get that

(6.30)  \frac{δ_{1}r_{1}^{d}}{2} Log (\frac{δ}{C_{2}ρ}) − C(ε) r_{1}^{d} ≤ CC_{1}r_{1}^{d} ,

which is obviously false if ρ/r_{1} is small enough. We precisely chose our last constant α_{2} so small, depending on all the other constants, that (4.3) implies a contradiction with (6.30).

We finally reached the contradiction that proves Lemma 4.1. □

7. A lower bound on Hausdorff dimension

In this section we deduce Theorem 3.8 from Lemma 4.1. So we continue with the same closed set E as in the statement of Theorem 3.8. First we want to choose a constant C_{1}, so that we can apply the lemma. We give the definition now so that the reader will be convinced that there is no vicious circle, but it will look a little weird.

For each ρ > 0 and each y ∈ ℝ^{n}, call R_{ρ}(y) the union of the skeletons S_{d}(Q), where Q ∈ Q(ρ) meets \overline{B}(y, 10ρ). Clearly H^{d}(R_{ρ}(y)) ≤ Aρ^{d} for some geometric constant A = A(n, d); we choose C_{1} = 10A.

Now we can apply Lemma 4.1. This gives constants r_{0} and α_{2} such that

(7.1)  H^{d}(E_{ρ} ∩ B(x, r)) > C_{1}r^{d}

for all choices of x ∈ E, r ∈ (0, r_{0}), and ρ ∈ (0, α_{2}r]. Choose an integer N such that \frac{1}{2}α_{2} ≤ N < α_{2}; this will make our computations look nicer.

For x ∈ E and r ∈ (0, r_{0}), set ρ = N^{−1}r and pick a maximal collection H(x, r) of points of E ∩ B(x, 2r), subject to the constraint that

(7.2)  |y − z| ≥ 10ρ for y, z ∈ H(x, r), y ≠ z.

We want to show that

(7.3)  ♯H(x, r) ≥ 10N^{d}.

Let us check first that

(7.4)  the sets R_{ρ}(y), y ∈ H(x, r), cover E ∩ B(x, r).

Let z ∈ E ∩ B(x, r) be given. Then z ∈ S_{d}(Q) for some Q ∈ Q(ρ) that meets E. Call y_{1} a point of E ∩ Q. By maximality of H(x, r), we can find y ∈ E such that |y − y_{1}| ≤ 10ρ. Then Q is one of the cubes that meet \overline{B}(y, 10ρ), and S_{d}(Q) ⊂ R_{ρ}(y). This proves (7.4).

Now H^{d}(E ∩ B(x, r)) ≤ ∑_{H(x,r)} H^{d}(R_{ρ}(y)) ≤ Aρ^{d}(♯H(x, r)) = AN^{−d}r^{d}(♯H(x, r)), and so ♯H(x, r) ≥ A^{−1}N^{d}r^{−d}H^{d}(E ∩ B(x, r)) ≥ A^{−1}N^{d}C_{1} = 10N^{d}, by (7.1).

The simplest way to estimate the dimension of E will be to use (7.4) to embed a Cantor set T^{ℕ^{∗}} in E, where T = {1, 2, … 10N^{d}} is a set with 10N^{d} elements. Set

(7.5)  ρ_{k} = N^{−k}r_{0}/2 for k ≥ 0.

We want to define a point x_{j} ∈ E for every choice of j ∈ T^{k} and k ≥ 0.

We start with any point x ∈ E, which we call x_{∅} to start the induction. Then assume that we already constructed the points x_{j}, j ∈ T^{k}, and let us do this for j ∈ T^{k+1}.

Let j ∈ T^{k} be given, and apply (7.3) to the point x_{j} and the radius ρ_{k}. We get a set H(x_{j}, ρ_{k}) of points of E ∩ B(x_{j}, 2ρ_{k}), that lie at mutual distances at least 10ρ_{k+1} from each other. Pick 10N^{d} points in this set, and call them x_{j,1}, x_{j,2}, … x_{j,10N^{d}}. Here we use the convenient notation (j, ℓ) for points of T^{k+1} that start with j.

This completes our definition of x_{j}, j ∈ T^{k}, by induction on k. Note that in the construction above, |x_{j,ℓ} − x_{j}| ≤ 2ρ_{k}. An easy induction gives that

(7.6)  |x_{j′} − x_{j}| ≤ 3ρ_{k} when j ∈ T^{k}, j′ ∈ T^{k′}, k′ ≥ k, and j′ starts with j.

Similarly, recall that the points x_{j,ℓ} lie at distances at least 10ρ_{k+1} from each other. Since the further generations will not modify this by more than 6ρ_{k+2} < ρ_{k+1}, by (7.6), we get that

(7.7)  |x_{j′} − x_{j}| ≥ 9ρ_{k} when the k-th components of j and j′ differ.

Now we put a distance on K = T^{ℕ^{∗}}. If j and j′ are two distinct sequences in K, call k(j, j′) the smallest integer k such that j_{k} ≠ j′_{k}, and then set

(7.8)  dist (j, j′) = ρ_{k(j,j′)} = N^{−k(j,j′)}r_{0}/2.

This defines a distance on K, for which K is closed.

By (7.6), we can define h : K → E by h(j) = lim_{k→+∞} x_{j_{1},j_{2},...j_{k}} , where we write

j = (j_{1}, j_{2}, … j_{k}, …). Moreover, if we apply (7.6) with k = k(j, j′) − 1, we get that |h(j) − h(j′)| ≤ 3ρ_{k(j,j′)−1} = 3N dist (j, j′).

Similarly, (7.7) and a small limiting argument say that |h(j) − h(j′)| ≥ 9ρ_{k(j,j′)} = 9 dist (j, j′). Altogether, h is biLipschitz, and

(7.9)  Hdim(E) ≥ Hdim(K)

Set d^{∗} = Hdim(K). We claim that

(7.10)  d^{∗} = d + \frac{Log 10}{Log N} > 1.

REFERENCES

Indeed it is known that d^{∗} is given by the equation

(7.11)  (10N^{d})^{−k} = N^{−kd^{∗}},

where the left-hand side gives the measure (for the obvious uniform measure on K) of a box of the k-th generation, while the right-hand side is an equivalent of the d^{∗}-th power of the diameter of the box. See [Fa] for a proof of (7.11) and similar computations on self-similar sets.

When we take the Logarithm of both sides of (7.11) and divide by −k, we get Log 10+ d Log N = d^{∗} Log N, which is equivalent to (7.10). Thus Hdim(E) ≥ d^{∗}, by (7.9).

The constant N, and hence also d^{∗}, depends only on the various constants in the construction, as promised after the statement of the theorem. This completes our proof.

References

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[Du] J. Dugundji, \underline{topology}, Allyn and Bacon, Boston, 1966.

[DS1] G. David and S. Semmes, \underline{Analysis of and on uniformly rectifiable sets}, A.M.S. series of Mathematical surveys and monographs, Volume 38, 1993.

[DS2] G. David and S. Semmes, Quasiminimal surfaces of codimension 1 and John domains, Pacific Journal of Mathematics Vol. 183, N 2 (1998), 213-277.

[DS3] G. David and S. Semmes, Uniform rectifiability and quasiminimizing sets of arbitrary codimension, Memoirs of the A.M.S. Number 687, volume 144, 2000.

[Fa] K. Falconer, \underline{Fractal geometry, Mathematical foundation and applications}, John Wiley and Sons, 1990.

[Se1] S. Semmes, A criterion for the boundedness of singular integrals on hypersurfaces, Trans. Amer. Math. Soc. 311 (1989), no. 2, 501–513.

[Se2] S. Semmes, Finding structure in sets with little smoothness, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 875–885, Birkhäuser, Basel, 1995.

[Yu] C. Yue, Dimension and rigidity of quasi-Fuchsian representations, Annals of Math. 143 (1996), 331-355.

Guy David, Mathématiques, Bâtiment 425, Université de Paris-Sud, 91405 Orsay Cedex, France guy.david@math.u-psud.fr