Tangent points of lower content d ‐regular sets and β numbers

Given a lower content d ‐regular set in Rn , we prove that the subset of points in E where a certain Dini‐type condition on the so‐called Jones β numbers holds coincides with the set of tangent points of E , up to a set of Hd ‐measure zero. The main point of our result is that Hd|E is not σ ‐finite; because of this, we use a certain variant of the β coefficient, firstly introduced by Azzam and Schul (Math. Ann. 370 (2018) 1389–1476), which is given in terms of integration with respect to the Hausdorff content.

Rectifiability is a central notion in geometric measure theory. It was first introduced (albeit under a different terminology) by Besicovitch [6] with d = 1 and n = 2; the theory was then extended to general d, n by Federer [14]; we refer the reader to [18] for more details. A classic characterisation of rectifiability is given in terms of approximate tangents. Let x ∈ E, and let V be a d-dimensional affine plane in R n . One says that V is an approximate tangent plane for E at x if lim sup r→0 H d (B(x,r)∩E) r d > 0 and for all 0 < s < 1, we have where X(x, V, s) = {y ∈ R n | d(y − x, V ) < s|y − x|} (and B(x, r) denotes the open ball centred at x and with radius r). Then the following is true. In other words, a set is d-rectifiable if and only if, at least at small scales, it lays close to a d-plane; in Theorem 1.1, 'laying close' is made precise by (1.2) -one, however, can easily think of other conditions that translate 'laying close' into a mathematical quantity, and consequently define a different notion of tangent points and planes. Let us give a few examples.
• For x ∈ E, we say that an affine d-plane V is a C-tangent † plane for E at x, if for all 0 < s < 1, there exists a r 0 > 0 so that for all r < r 0 , (1. 3) A C-tangent point is defined in the obvious way.
• If U, V ⊂ R n , we set In this case, we will call L a tangent plane.
It is immediate to see that if a point is a C-tangent point, then it is an approximate tangent point. With this in mind, let us state the following theorem of Bishop and Jones Theorem 1.2 [7, Theorem 2]. Let E ⊂ C be a Jordan curve. Except for a set of zero H 1 -measure, x ∈ E is a C-tangent point of E if and only if The quantities β E,∞ appearing in (1.7) are the so-called Jones β-numbers -they were introduced by Jones in [17], where he gave a solution to the Analyst's travelling salesman problem (see [20] for a survey), and they are defined as follows: y∈E∩B (x,t) dist(y, L) r , (1.8) where the infimum is taken over all affine d-planes and dist(y, L) = inf q∈L |y − q|. One may think of β E,∞ (x, r) · r as the width of the smallest strip which contains E ∩ B(x, t). Theorem 1.2 † C here is for cone. ‡ H is for Hausdorff. § We do not put any prefix here: this definition will be the most used one in the paper.
where the infimum is taken over all affine d-planes L in R n , and H d ∞ denotes the Hausdorff content.
Azzam and Schul used this version of β number to prove an analyst's travelling salesman theorem for lower content d-regular sets in R n -their result is analogous to the one given by Jones in the plane (see Remark 1.8 for some motivation on the usage of these coefficients). Let us recall what a lower content d-regular set is.
Let us make some remarks.
Remark 1.5. The content of the theorem is really in the equivalence between T1 and T2. The equivalence between the various definition of tangents is known; we will include proofs for completeness. We chose to prove the equivalence T1 ⇐⇒ T2 for convenience: the definition of tangent in (1.11) seems the simplest and easiest to use.
Remark 1.6. The conclusion about the rectifiability of E 0 in Theorem 1.4 can be reached through different methods. For example, see [18,Theorem 15.19] on the equivalence of (T4) and rectifiability. Also, that the Dini condition T1 implies rectifiability is implicit in our proof of Proposition 4.1.
Remark 1.7 (On related works). The relationship between rectifiability and a Dini-type condition on (some type of) Jones β numbers has been an active area of research in the past 30 years. Starting with the already mentioned work of Jones [17], these coefficients have been applied and used to understand geometric aspects of sets and measure in a variety of context; see for example David and Semmes in [10,11] (the reader will find here very interesting connections between harmonic analysis and rectifiability); David and Toro in [12] (see also Ghinassi in [15]), and Edelen, Naber and Valtorta in [13] (these papers relates the β coefficients with parameterisation of Reifenberg flat sets); Schul in [19] for a generalisation of the traveling salesman theorem of Jones on curves to Hilbert spaces; the series of results by Tolsa and Azzam and Tolsa [22,23] and [2] for a characterisation of rectifiability in terms of a Dini square integrability condition on the β coefficients; the series of results by Badger and Schul [3][4][5] for characterisations of 1-rectifiable measure in terms of β coefficients without density assumptions.
Remark 1.8. Let us mention why is the theorem novel. In any situation where one is dealing with sets of dimension larger than 1, the β E,∞ coefficient become rather useless (at least when we want to consider a Dini-type condition on them, see the example in [Fa]). David and Semmes then introduced an averaged version of such numbers, given as (1.14) Note that for β d E,p to even make sense, H d | E needs to be at least σ-finite. However, a generic lower content d-regular set will have dimension strictly larger than d -just as in the case of Theorem 1.2, where Bishop and Jones considered a Jordan curve. For this reason, we decided to use the β coefficient introduced by Azzam and Schul: the integral there is a Choquet integral with respect to the Hausdorff content: it makes sense in any case. Remark 1.9 (On the range of exponents p). The range of exponents p for which Theorem 1.4 is true should not be (so) surprising. The results of David and Semmes for uniformly rectifiable sets and the Traveling Salesman Theorem of Azzam and Schul, for example, hold for the very same range. The 'so' in brackets is there to say that things are not as not-surprising as they seem. The characterisation of d-rectifiable sets in terms of β numbers by Tolsa in one direction (see [22]) and by Azzam and Tolsa in the other (see [2]), holds only for p = 2. The question remained open on whether this result could be extended to the 'expected' range of p: Tolsa's recent examples show that such a characterisation can only hold for p = 2 (see [23]).

Notation
We gather here some notation and some results which will be used later on. We write a b if there exists a constant C such that a Cb. By a ∼ b, we mean a b a. In general, we will use n ∈ N to denote the dimension of the ambient space R n , while we will use d ∈ N, with d n − 1, to denote the dimension of a subset E ⊂ R n , in the sense of lower content regularity.
For two subsets A, B ⊂ R n , we let dist(A, B) := inf a∈A,b∈B |a − b|. For a point x ∈ R n and a subset A ⊂ R n , dist(x, A) := dist({x}, A) = inf a∈A |x − a|. We write B(x, t) := {y ∈ R n | |x − y| < t}, and, for λ > 0, λB(x, t) := B(x, λt). At times, we may write B to denote B(0, 1). When necessary we write B n (x, t) to distinguish a ball in R n from one in R d , which we may denote by B d (x, t). See refer the reader to [18] for more detail on Hausdorff measures and content. For 0 < p < ∞ and A ⊂ R n Borel, we define the p-Choquet integral as See [1, Section 2, Appendix ] for more on Choquet integration.

Coherent families of balls and planes
A main tool which we will use is the theorem by David and Toro on parameterisations of Reifenberg flat sets with holes (see [12]). To state more precisely the result given there, we need to introduce a few more notions and definitions.
Definition 2.1. Set the normalised local Hausdorff distance between two sets E, F to be given by } is a family of planes so that for each k ∈ N, j ∈ J k , P j,k x k j . Moreover, (P 0 , {x k j }, {P j,k }) satisfies the following properties; (D) set B j,k := B(x k j , r k ) and, for λ > 0, The property which we require is that The main technical result of [12] is the following.
is a C -Reifenberg flat set that contains the accumulation set David and Toro also prove an another version of Theorem 2.4, which requires a further assumption. In order to state it, we introduce one more quantity, which will be used to control the rate at which the planes belonging to the CCBP change as we go through locations and scales. We also need to recall from [12,Chapter 4], that the function g of Theorem 2.4, 'appears as the limit of a sequence of functions g k '; such sequence converges uniformly to g. In fact, g is pointwise defined as Moreover, we have the estimate We refer the reader to Chapters 6, equation (6.1) and [12, Lemma 6.1] for more details on this.
The following lemmas are to be found in [1, Section 2]. They will be used later on in the various proofs. . Then

t) be any ball centred on E and let B(y, s) ⊂ B(x, t) be also centred on E with s t. Then
Remark 2.10. At first sight, one could think that the implication T1 ⇒ T2 follows immediately from Lemma 2.9. This is not the case because the infimising planes in (2.6) could change as r ↓ 0: we want to prove instead the existence of a fixed tangent plane so that (1.11) holds true.

Intrinsic cubes and Whitney cubes
In this section, we briefly recall two 'cubes'-tools which will be needed later on. Intrinsic cubes. First David in [9] and then Christ in [8], introduced a construction which allows for a partition of any space of homogeneous type into open subsets which resemble in behaviour the dyadic cubes in R n . Recall that a space of homogeneous type is a set X equipped with a quasi-metric p and a Borel measure μ so that • the balls associated to p are open; • μ(B(x, r)) < ∞ for all x ∈ X, r > 0; • μ satisfies the doubling condition with respect to these balls, that is, there exists a constant A, independent of x and r so that , r)).
In this case, we use Christ's construction: Theorem 2.12 (Christ's cubes). Let (X, p, μ) be a space of homogeneous type. Then there exists a collection of open subsets (Q j,k ), k ∈ Z and j ∈ J k , and constants δ ∈ (0, 1), C 1 < ∞ and a 0 > 0 so that We refer the reader to [8] for a precise definition of x k j . Roughly speaking, this is the centre of the 'cube' Q j,k . Also, the original statement involves a further property which we will not use. Hytönen and Martikainen provided a variant to this construction in [16]. The main difference between the two is that the latter results in an exact partition of the space (X, p, μ), that is However, Hytönen and H. Martikainen's cubes are not open. Taking their interior, however, one recovers Christ's cubes.
We will write To consider the family of cubes of a given scale, we fix k ∈ Z and we write We remark that in what follows we will not have the measure μ. We will only have a distance (the Euclidean one). But this is not a problem, the construction of the cubes Q j,k does not depend on μ.
Whitney cubes. We follow [21, Lemma 2, Chapter 1]. Let F be a closed non-empty subset of R n endowed with the standard Euclidean distance. Then there exists a collection {S k } of closed cubes with disjoint interiors which covers F c (the complement of F ) and whose side lengths are comparable to their distance to F , that is, there exists a constant A so that 3. Equivalence of (T2), (T3) and (T4) In this section, we prove the equivalence (H d -almost everywhere) of the different versions of tangent present in Thereom 1.4. As mentioned above, these equivalences are already known; we add proofs for the sake of completeness.

Lemma 3.1. A point x ∈ E is an approximate tangent point of E if and only it is a tangent point of E.
Proof. Take E to be d-lower content regular (this insures that the hypothesis on the nonvanishing upper density is satisfied). Let us first show that if a point x ∈ E is a tangent point in the sense of (1.6), then there exists an approximate tangent at x. Indeed, suppose not. Then there exists an > 0, a sequence r j ↓ 0 and an 0 < s < 1 so that This implies that for each j ∈ N, we may find a point This is a contradiction, since we assumed that a is a tangent point (in the sense of (1.6).
Let us prove the converse. Let x ∈ E be a point which admits an approximate tangent V . Again, we argue by contradiction; assume that there exists an > 0 and a sequence of radii Then there exists a sequence of points This contradicts the fact that V is an approximate tangent at x.
Proof. Note that if x is an H-tangent point, that is, it satisfies (1.12), than it follows immediately that it is a tangent point, that is, it satisfies (1.11). Let us prove the converse statement; set T := {x ∈ E | x is a tangent point of E}; without loss of generality, we assume 0 < H d (T ) < ∞. On the other hand, set HT := {x ∈ E | x is an H − tangent point of E}. We already know that HT ⊂ T and we want to show that H d (T \ HT ) = 0. Let f : R d → R n be one of the Lipschitz maps covering E 0 and set Then M is a measurable subset (with respect to the d-dimensional Lebesgue measure on R d ).
It suffices to prove that H d -almost every point x ∈ f (M ) is an H-tangent point of E. Let p ∈ M be a density point and such that f is differentiable here. Let us denote the tangent plane of f at p by L p . As p is a density point, then we have that for all > 0, there exists a radius r 0 > 0 such that for any q ∈ B n (p, r), with 0 < r < r 0 , we can find a q ∈ M so that |q − q | < r. From this it follows that, given > 0 and for r > 0 sufficiently small, for any point y ∈ f (B(p, r)), we may find a point y ∈ f (M ) ⊂ E 0 so that |y − y | r. To prove the lemma, we can now proceed as follows. Given > 0, let r > 0 be small enough. Given q ∈ L p , let y q ∈ f (B(p, r)) be so that |y q − q| 2 dist (q, f (B(p, r))). Then we have f (B(p, r) There exists a point y ∈ f (M ) so that |y − y q | r, and so, inf x∈E |y q − x| inf x∈E |x − y| + |y − y q |; note that the first term on the right-hand side of this last inequality vanishes, since y ∈ E. If we now supremize over q ∈ L p ∩ B(x 0 , r), we obtain f (B(p, r))) + r.
dist(q,f (B(p,r))) r → 0 as r → 0. This completes the proof of the lemma.
Let us give some easy remarks.
Remark 4.2. Because of Lemma 2.7, it is enough to prove Theorem 4.1 for p = 1. We may assume that the set has non-zero H d measure, for otherwise there is nothing to prove.
We argue by contradiction. Suppose that there exists a subset Ω ⊂ A with 0 < H d (Ω) and so that no x ∈ Ω is a tangent point of E. If H d (Ω) < ∞, then we already reached a contradiction. Hence, we may take H d (Ω) = ∞. However, by [18,Theorem 8.13], there exists a compact subset K ⊂ Ω with 0 < H d (K) < ∞ and so that no x ∈ K is a tangent point of E. This leads again to a contradiction. Thus, the lemma follows.
For an arbitrary ∈ N with H d (E ) > 0, we now fix to be a compact subset of positive and finite H d -measure. Note that if we prove Proposition 4.1 for E 0 , then the statement in its full generality follows by first saturating E with compact subsets, and then by taking the countable union ∪ E = A. Moreover, by applying a dilation and a translation, we may assume that and that = 1 in (4.1). Thus, we see that, by construction Remark 4.5. Note that E 0 ⊂ E will not necessarily be a d-lower content regular set. However, this will not cause issues, for when estimating angles, we will be always integrating over the whole E.
We now show that one can construct a bi-Lipschitz parameterisation of any such a subset E 0 .

4)
then, denoting by P 0 the plane which minimises β 1,d E (0, 1), there exists a bi-Lipschitz map g : P 0 → R n with constant Lip(g) Ce and such that E 0 ⊂ g(P 0 ).
We will prove Lemma 4.7 in the next few subsections; the proof will consists in checking the hypotheses of Theorem 2.4.

Construction of a CCBP: A to E. Consider a maximal collection {u
We may assume that #(J 0 ) = 1 and that u j,0 = 0. This is because of (4.2). From now on, we will denote L(u k j , r k ), that is, the plane which infimises β 1,d E (u k j , 120r k ), as L j,k .
Fix k, j. We choose a point x k j ∈ E closest to L j,k in E ∩ B(u k j , r k /3). Note that This and Chebyshev inequality give us Inequality (4.7) follows from the containment (4.5). Inequality (4.8) follows from the lower content regularity of E. Equation (4.9) is due to the choice u k j ∈ E 0 . We now define P j,k to be the d-plane parallel to L j,k such that it contains x k j . The family {P j,k } will constitute the collection of planes through which we will be able to verify David and Toro's Theorem.
(A) There is only one element in the family {x 0 j } j∈J0 . Thus, we take P 0 := P (x 0 j , 1) (4.10) in the definition of CCBP.
(B) The family {x k j } satisfies (CCBP1) by construction. This is true up to a constant, but this is unimportant.
(C) {P j,k } k∈N,j∈J k will be the sought for family of d-planes.
(D) Note that Moreover, since x k j were chosen to be at distance strictly less than 1 3 r k from u k j , then B u k j , Thus, This let us conclude that for u ∈ E 0 , we may pick i ∈ J k−1 so that u ∈ B(x k−1 i , 5 3 r k−1 ), and so and therefore that x k j ∈ 2V k−1 . (E) Clearly, (CCBP3) follows from (4.10).

Construction of a CCBP: F to H.
We start with the following, somewhat technical, lemma, to be found in [12,Chapter 12,Lemma 12.7]. It is useful to prove Lemma 4.12, which let us control the angle between planes belonging to different location and scales.
Remark 4.10. We remark that [1] provides a way of controlling angles between planes through β numbers, that is, Lemma 2.16. However, the construction which follows, which is the one given in [12,Chapter 13], adapted to our current beta coefficients β, is needed to check the summability hypothesis (CCBPS). Lemma 4.11. Let x ∈ R n , r > 0, τ ∈ (0, 10 −1 ). Let P 1 , P 2 be two affine d-planes and take d orthogonal unit vectors {e 1 , . . . , e d }. Suppose that, for 0 l d, we are given points a l ∈ P 1 and b l ∈ P 2 , so that Then d x,s (P 1 , P 2 ) Cτ for r s 10 4 r.
Lemma 4.12. Let k ∈ N and j ∈ J k . Take either m = k or m = k − 1 and let i ∈ J m be so that Then, for r k 2 s 5000r k , Proof. Fix k, j ∈ J k . We will now choose d + 1 points which will control the position of L j,k . Let {e 1 , . . . , e d } be an orthonormal basis for the vector space V ∈ G(d, d + 1) parallel to L j,k . Let For 0 l d, it holds that H d ∞ (E ∩ B(p l , C r k )) C d r d k . Thus, we may find a point w l ∈ E ∩ B(p l , C r k ). (4.13) Below, it will be useful to know that To see this, note that Thus, (4.14) holds as long as C < 1/6. By the choice of , this is certainly satisfied.
Again by using the lower content regularity of E and Chebyshev inequality, we may find, for 0 l n, a z l ∈ E ∩ B(w l , r k+2 ) which is the closest to L j,k and thus, as in (4.7)-(4.9), satisfies Note that we used the containment B(w l , r k+2 ) ⊂ B(x k j , r k ) ⊂ B(u k j , 120r k ). Keeping k, j fixed, take either m = k or m = k − 1, and let i ∈ J m be such that Note that Because of (4.16), using lower content regularity (that is, (1.9)), we may choose z l ∈ B(w l , r k+2 ) so that we also have We proceed as follows. Take x = x k j in Lemma 4.11, take r = r k 2 and take We will let P 1 to be P i,k and P 2 to be P i,m . Moreover, we take {e 1 , . . . , e d } to be the orthonormal basis of the d-subspace parallel to P j,k which was used to select the functions z l . Recall that Let q l ∈ L j,k so that dist(z l , L j,k ) = dist(z l , q l ). We have dist(z l , P j,k ) = inf The estimate dist(q l , P j,k ) β 1,d E (u k j , 120r k )r k follows from our initial choice of {x k j } and recalling that P j,k was constructed by translating L j,k by a distance smaller than β 1,d E (u k j , 120r k ). We similarly obtain For 0 l d, set a l to be the point in P j,k for which and similarly, let b l ∈ P i,m so that |z l − b l | = dist(z l , P i,m ). Then Indeed, note that This is due to how we defined functions p l and to (4.13). One the other hand, Since is small, (4.21) is satisfied. Hence, we may apply Lemma 4.11, to obtain that for r k 2 s 5000r k .
(F) We now obtain (CCBP4): that is, for k 0 and for i, j ∈ J k such that dist(x k j , x m i ) 100r k , (4.22) one wants dist x k j ,100r k (P j,k , P i,k ) . Indeed, (4.22) coincides with the premise (4.11) of Lemma 4.12, which can therefore be applied with k = m and choosing s = 100r k , to obtain and thus dist x k j ,100r k (P j,k , P i,k ) , since u k j ∈ E 0 . (G) We verify (CCBP5) with k − 1: first, note that (4.11) is satisfied since we only need dist(x k j , x i,k−1 ) < 2r k−1 = 2r m . Then we apply (4.12) with the choice s = 20r k−1 so to have (H) To see (CCBP6), we apply Lemma 4.12 with k = m = 0 and we recall that diam(E 0 ) < 1, thus (4.11) is immediately satisfied. Moreover, recall from (4.10) and the line below, that P 0 is in fact P j,0 for some j ∈ J 0 . Then (CCBP6) follows. Pickz ∈ E 0 so that We note that 11r k .  4). Inequality (4.25) is also due to the choice ofz and by the fact that u k j ∈ E 0 . For (4.27), recall that y = f k (z) ∈ 10B j,k , and that dist(u k j , x k j ) 1 3 r k .
, then x is a tangent point of E (in the sense of (T2)).
Proof. Without loss of generality, let us assume that P 0 = R d . Set M := g −1 (E 0 ); note that M is a measurable set with respect to the d-dimensional Lebesgue measure on R d and clearly 0 < L d (M ) < ∞. Let p 0 ∈ M be a density point and moreover assume that g is differentiable there -that the set of points in M with these characteristics has full measure. It follows from the Lebesgue differentiation theorem that for all > 0, there exists an r 0 > 0 such that if p ∈ B d (p 0 , r), 0 < r < r 0 , then there exists a point p ∈ M such that |p − p | < r.
Set x 0 = g(p 0 ). Because g is bi-Lipschitz, there exists a constant c 1, depending only on Lip(g), such that B d (p 0 , cr) ⊂ g −1 (B n (x 0 , r)); furthermore, since we chose p 0 to be a density point, for any y ∈ g (B d (p 0 , cr)), there exists a y ∈ f (M ) so that |y − y | C r. In particular, we can find d points y 1 , . . . , y d ∈ f (M ) such that they are linearly independent (with good constant) and such that dist(y j , L p ) C r for all j = 1, . . . , d.
Let us now argue by contradiction. Suppose that there exists a point z ∈ E with dist(z, L) 1000 r. Then z is linearly independent with respect to the family y 1 , . . . , y d and therefore there cannot exist a d-plane V so that dist(y j , V ) C r for all j and dist(z, V ) C r. But this contradicts the fact that β d E,∞ (x 0 , cr) d -which must hold from Lemma 2.9. The ensuing contradiction proves the lemma.
Proof. It is enough to show that for any x ∈ E, Let r k t 10r k . Then recall from Lemma 2.8, that This let us conclude that The corollary then follows from Proposition 4.1.
Given any bounded set E, we can translate and dilate it so that it is contained in the unit ball centred at 0. Hence, Corollary 4.16 immediately gives one implication of Theorem 1.4.
Remark 4.17. One could think of an alternative way of proving this direction of Theorem 1.4; it was suggested to the author by Azzam. It goes as follows. Take a subset E 1 ⊂ E where the Jones function is finite. First, reduce the problem to a subset E 0 ⊂ E 1 of positive and finite d-dimensional Hausdorff measure -as was done in Lemma 4.3. Second, Frostmann's Lemma (for example, see [18,Theorem 8.8]) guarantees the existence of a measure μ such that if μ is finite, it is supported on a compact subset of E 0 and it is upper d-regular, that is, μ(B(x, r)) r d for any x ∈ R n and r > 0. Third, this implies that μ is bounded above by the d-dimensional Hausdorff content. Hence, one can bound above the standard β numbers with the content β numbers. Fourth, this let us apply [2, Theorem 1.1] and obtain rectifiability for E 0 . Hence, one can conclude as in Subsections 3.5 and 3.6.
Such a proof would be shorter and clearer It has, however, a drawback: as [2, Theorem 1.1] holds only for p = 2, one could obtain Theorem 4.1 for 2 p < ∞. This is the only reason why we kept the current (longer and more technical) proof.

(T2) implies (T1)
Consider the decomposition of E into the intrinsic cubes given in Theorem 2.12. For a cube Q ∈ D, set The following lemma will be useful during the proof; it can be found in Mattila's book [18,Chapter 15].
The following is an elementary fact, which we prove for completeness. Proof. Recall the definition of C-tangent in 1.3; if the lemma did not hold, for each d-dimensional affine plane V , we could find a θ ∈ (0, π/2) such that for all r > 0, there existed a y ∈ X(x, V ⊥ , θ, r) ∩ E; such a y would have dist(y, V ) sin(θ)|y − x|. One can easily see that this implies that lim sup r↓0 sup y∈B(x,r)∩E dist(y, V )/r sin(θ).
Denote by T (E) the set of tangent points of E. We can assume without loss of generality that H d (T (E)) > 0, for otherwise there is nothing to prove. If x ∈ T (E), denote its tangent by L x . Fix θ ∈ (0, π/2) and let {L k } k∈N ⊂ be a dense countable subset of the Grassmannian, {r } ∈N be a dense, countable subset of (0,1). Set , r ), and , then there exists an r > 0 and an affine d-plane so that X(x, L ⊥ , θ, r) ∩ E = ∅; since {L k } is dense, for each δ > 0, we may find an k ∈ N so that dist H (L k ∩ B, (L x − x) ∩ B) < δ, for, say, B = B(0, 1). It is clear then, that we may find an r r so that X(x, L ⊥ k , θ, r ) ∩ E = ∅.
Remark 5.4. Without loss of generality, we can assume that K k, is compact and that diam(K k, ) r /2. By Lemma 5.2, each K k, can be covered by the image of a bi-Lipschitz map f : Π L k (K k, ) → R n with Lip(f ) = 1 cos(θ) .
Lemma 5.5. Fix k, and consider K k, as above. Let p be given as in Proposition 5.1. Then for H d -almost every point x ∈ K k, , Denote by Q 0 the minimal cube in D such that K ⊂ 3Q 0 ; we may assume that diam(Q 0 ) Sublemma 5.6. We have Proof. We have that The second equality is an application of Fubini-Tonelli's theorem. For the inequality, note that we are integrating over K. Such set is not necessarily Ahlfors regular, for the density may be zero in some balls. However, the upper d-regularity is still maintained, since it is a subset of a d-dimensional Lipschitz graph.
Proof. Clearly, β p,d E (Q) 2 1. Also recall that any Q ∈ L is contained in B(0, 1). Moreover, for each generation, there can be at most one cube which contains Q 0 . Hence, the sum (5.9) is really a finite sum of bounded terms, and therefore must be finite.
Let us now deal with the sum over the family S. Without loss of generality, take L k (recall that K = K k, ) to be R d . We extend f (as given in Remark 5.4) to the whole R d ; if we call g such an extension, it is a standard fact that Lip(g) ∼ n,d Lip(f ). Set We apply Lemma 2.11 to obtain Note that, because G is the Lipschitz image of a plane, then Before estimating B in (5.10), we will construct a 'Whitney decomposition' for Q 0 \ K. We proceed as follow. Because Π R d (K) is compact, there exists a Whitney decomposition of Π R d (K) c in R d ; let us denote such a decomposition by W d . For S ∈ W d , set and Proof. Let y, z ∈ T S ⊂ Q 0 \ K. For brevity, let us write Π := Π R d and K := Π(K). Pick p = p(y) ∈ K such that |p − Π(y)| 2 dist(Π(y), K), (5.15) and set x(y) := f (p) ∈ K. (5.16) We choose q = q(z) ∈ K and x(z) = f (q) ∈ K is the same manner. Now, let L y (respectively, L z ) be the d-plane parallel to R d which contains x(y) (respectively, x(z)). Denote by Π y (respectively, Π z ) the orthogonal projection onto L y (respectively, L z ). Then set y := Π y (y) (5.17) z := Π z (z). (5.18) Then, |y − z| |y − y| + | y − z| + | z − z|. We see that |y − y| = |(y − x(y)) − ( y − x(y))| = dist(y − x(y), R d ).
On the other hand, if we let y ∈ T S , x(y) as in (5.15)  Furthermore, note that We state one last lemma before proving the summability of the error term. Proof. For y ∈ T S , let x(y) and p = p(y) as above. Then dist(y, G) |y − g(p)| = |y − x(y)|. If we now argue as in (5.26), the lemma follows.
Let us now go back to the proof of Lemma 5.5. Recall that we had to give an estimate for the error term B in (5.10).
This concludes the proof of Lemma 5.5 and therefore of Proposition 5.1.