Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces

The aim of this note is to generalize to the class of non collapsed RCD(K,N) metric measure spaces the volume bound for the effective singular strata obtained by Cheeger and Naber for non collapsed Ricci limits in \cite{CheegerNaber13a}. The proof, which is based on a quantitative differentiation argument, closely follows the original one. As a simple outcome we provide a volume estimate for the enlargement of Gigli-DePhilippis' boundary (\cite[Remark 3.8]{DePhilippisGigli18}) of ncRCD(K,N) spaces.


Introduction
In the last years the theory of metric measure spaces (X, d, m) satisfying the Riemannian curvature dimension condition has undergone several remarkable developments. After the introduction, in the independent works [33,34] and [30], of the curvature dimension condition CD(K, N) encoding in a synthetic way the notion of Ricci curvature bounded from below and dimension bounded above, the de nition of RCD(K, N) metric measure space was proposed and extensively studied in [4,21,22] (see also [7] for the equivalence between the RCD * (K, N) and the RCD(K, N) condition) in order to single out spaces with Hilbert-like behaviour at in nitesimal scale. The in nite dimensional counterpart of this notion had been previously investigated in [3]. In particular, due to the compatibility of the RCD condition with the smooth case of Riemannian manifolds with Ricci curvature bounded form below and to its stability with respect to pointed measured Gromov-Hausdor convergence, limits of smooth Riemannian manifolds with Ricci curvature uniformly bounded from below and dimension uniformly bounded from above are RCD(K, N) spaces. The study of Ricci limits was initiated by Cheeger and Colding in the nineties in the series of papers [8][9][10][11] and has seen remarkable developments in more recent years (see for instance [18]). Since the above mentioned pioneering works, it was known that the regularity theory for Ricci limits improves adding to the lower curvature bound a uniform lower bound for the volume of unit balls along the converging sequence of Riemannian manifolds: this is the case of the so called non collapsed Ricci limits. In particular, as a consequence of the volume convergence theorem proved in [17], it is known that the limit measure of the volume measures is the Hausdor measure on the limit metric space (while this might not be the case for a general Ricci limit space).
Inspired by the theory of non collapsed Ricci limits, De Philippis and Gigli proposed in [20] a notion of non collapsed RCD(K, N) metric measure space (X, d, m) (ncRCD(K, N) for short) asking that m = H N , the Ndimensional Hausdor measure over (X, d). Let us remark that this class of spaces had already been studied by Kitabeppu in [28]. Let us point out that recently examples of metric measure spaces which are ncRCD but not non collapsed Ricci limits have been built: hence a gap widens between the two theories. Nevertheless in [20] the authors were able to prove that many of the structural results valid for non collapsed Ricci limits hold for ncRCD spaces. In particular, building upon [19], it is possible to prove that any tangent cone to a ncRCD space is a metric cone. Letting then R ⊂ X be the set of those points where the tangent cone is the N-dimensional Euclidean space, following [9] it is possible to introduce a strati cation S ⊂ · · · ⊂ S N− = S = X \ R of the singular set S, where, for any k = , . . . , N − , S k is the set of those points where no tangent cone splits a factor R k+ . Adapting the arguments of [9], in [20] the Hausdor dimension estimate dim H S k ≤ k was obtained.
In [13] a quantitative and e ective counterpart of the above mentioned strati cation of the singular set was introduced letting, for any k = , . . . , N − and for any r, η > , S k η,r be the set of those points x ∈ X where the scale invariant Gromov-Hausdor distance between the ball Bs(x) and any ball of the same radius centered at the tip of a metric cone splitting a factor R k+ is bigger than η for any r < s < . While in the classical strati cation points are separated according to the number of symmetries of tangent cones, in the quantitative one they are classi ed according to the number of symmetries of balls of xed scales therein centered. In particular, the e ective singular strata might be non empty even in the case of smooth Riemannian manifolds while in that case there is no singular point.
Starting from [13] a number of properties for the e ective singular strata on non collapsed Ricci limit spaces have been obtained. In particular, in the very recent [16], the authors were able to prove k-recti ability of the classical singular stratum S k building on the top of some new volume estimates for the e ective strata. The aim of this note is twofold. On the one hand our main result Theorem 2.4 generalizes to the class of ncRCD the volume estimate for the e ective singular strata obtained by Cheeger and Naber in [13] (which is easily seen to be stronger than the above mentioned Hausdor dimension estimate dim H S k ≤ k), on the other hand we give detailed proofs (in the metric context) of some of the results that therein were just stated. Let us point out that Theorem 2.4 has already an application in the proof of [32,Theorem 5.8].
Let us remark that the proof of the volume estimate, which closely follows the one for Ricci limits, provides an instance of the so called quantitative di erentiation technique that, although being quite recent in its formulation, has already a broad range of applications in the regularity theory in various di erent geometric and analytic contexts. In general, quantitative di erentiation allows to bound the number of locations and scales at which a given geometric con guration is far away from any element of a class of special con gurations. In the case of our interest special con gurations are the conical ones. We refer to [12] for a general survey about quantitative di erentiation and detailed list of references to the recent applications of this tools in the various contexts.
This note is organised as follows: in section 1 we list a few basic de nitions and results useful when dealing with ncRCD metric measure spaces. Most of the results are stated without proof and references are indicated. We provide instead proofs for the "almost volume cone implies almost metric cone" Theorem 1.12 and the "almost cone splitting" Theorem 1.17, since we were not able to nd any reference in the literature. In section 2 we give a complete proof of the volume bound for the e ective singular strata following the same strategy introduced by Cheeger and Naber in the setting of non collapsed Ricci limit spaces.
for any f ∈ Lip(X) we shall denote its slope by We will use the standard notation L p (X, m), for the L p spaces and L n , H n for the n-dimensional Lebesgue measure on R n and the n-dimensional Hausdor measure on a metric space, respectively. We shall denote by ωn the Lebesgue measure of the unit ball in R n . The Cheeger energy Ch : L (X, m) → [ , +∞] associated to a m.m.s. (X, d, m) is the convex and lower semicontinuous functional de ned through and its niteness domain will be denoted by W , (X, d, m). Looking at the optimal approximating sequence in (1.1), it is possible to identify a canonical object |∇f |, called minimal relaxed slope, providing the integral representation Any metric measure space such that Ch is a quadratic form is said to be in nitesimally Hilbertian and from now on we shall always make this assumption, unless otherwise stated. Let us recall from [3,23] that, under this assumption, the function and, in that case, we put ∆f = h. It is easy to check that the de nition is well-posed and that the Laplacian is linear (because Ch is a quadratic form). .

RCD(K, N) metric measure spaces
The notion of RCD(K, N) m.m.s. was proposed and extensively studied in [4,21,23] (see also [7] for the equivalence between the RCD and the RCD * condition), as a nite dimensional counterpart to RCD(K, ∞) m.m.s. which were introduced and rstly studied in [3] (see also [2], dealing with the case of σnite reference measures). We point out that these spaces can be introduced and studied both from an Eulerian point of view, based on the so-called Γ-calculus, and from a Lagrangian point of view, based on optimal transportation techniques, which is the one we shall adopt in this brief introduction.
Let us start recalling the so-called curvature dimension condition CD(K, N). Its introduction dates back to the seminal and independent works [30] and [33,34], while in this presentation we closely follow [6].
De nition 1.1 (Curvature dimension bounds). Let K ∈ R and ≤ N < +∞. We say that a m.m.s. (X, d, m) is a CD(K, N) space if, for any µ , µ ∈ P(X) absolutely continuous w.r.t. m with bounded support, there exists an optimal geodesic plan Π ∈ P(Geo(X)) such that for any t ∈ [ , ] and for any N ≥ N we have where (e t ) Π = ρ t m, µ = ρ m, µ = ρ m and the distortion coe cients τ t K,N (·) are de ned as follows. First we de ne the coe cients [ , ] × [ , +∞) (t, θ) → σ (t) K,N (θ) by The main object of our study in this paper will be RCD(K, N) spaces, that we introduce below.
De nition 1.2. We say that a metric measure space (X, d, m) satis es the Riemannian curvature-dimension condition (it is an RCD(K, N) m.m.s. for short) for some K ∈ R and ≤ N < +∞ if it is a CD(K, N) m.m.s. and the Banach space W , (X, d, m) is Hilbert.
Note that, if (X, d, m) is an RCD(K, N) m.m.s., then so is (supp m, d, m), hence in the following we will always tacitly assume supp m = X.
We assume the reader to be familiar with the notion of pointed measured Gromov Hausdor convergence (pmGH-convergence for short), referring to [35,Chapter 27] for an overview on the subject. Remark 1.3. A fundamental property of RCD(K, N) spaces, that will be used several times in this paper, is the stability w.r.t. pmGH convergence, meaning that a pmGH limit of a sequence of (pointed) RCD(K, N) spaces is still an RCD(K, N) m.m.s..
We recall that any RCD(K, N) m.m.s. (X, d, m) satis es the Bishop-Gromov inequality: for any < r < R and for any x ∈ X, where v K,N (r) (1.4) In particular (X, d, m) is locally uniformly doubling, that is to say, for any R > there exists C K,N,R > such that m(B r (x)) ≤ C K,N,R m(Br(x)) for any x ∈ X and for any < r < R. (1.5) We refer to [35,Theorem 30.11] for the proof of the Bishop-Gromov inequality in the setting of metric measure spaces satisfying the curvature dimension condition.

. Non collapsed RCD(K, N) spaces
Let us recall the de nition of non collapsed RCD(K, N) m.m.s., as introduced in [20] (see also [28], where Kitabeppu rstly investigated this class). Now we are ready to state the volume convergence theorem [20,Theorem 1.2] in this setting and other de nitions which will be useful for our aims. De nition 1.6 (Metric cone). Given a metric space (Z, d Z ) we de ne the metric cone C(Z) over Z to be the completion of R + × Z endowed with metric Thanks to the Bishop-Gromov inequality (1.3), the following de nition can be given, following [20].
De nition 1.7 (Bishop-Gromov density). Given K ∈ R, N ∈ [ , +∞) and an RCD(K, N) space (X, d, m), for any x ∈ X we let the Bishop-Gromov density at x be de ned by

. Almost volume cone implies almost metric cone
It is possible to prove a rigidity result about Bishop-Gromov inequality in RCD( , N) spaces which, roughly speaking, tells us that if we have equality of Bishop-Gromov ratios at two di erent radii then, at a certain scale, the space is isometric to a metric cone. This result is proven in [19, Theorem 1.1, Theorem 4.1] in the case of RCD( , N) and RCD(K, N) spaces respectively but we will state (part of) it here only in the case K = .
Theorem 1.10 (Volume cone implies metric cone). Let N ∈ ( , +∞) and (X, d, m) be an RCD( , N) space. Suppose there exist x ∈ X and R > r > such that Then, if the sphere S R (x) contains at least 3 points, we conclude that N ≥ and that there exists (Z, dz , mz) an is isometric to the closed ballB R (z) in the metric cone built over Z, where z is the tip of the cone. This isometry sends x to z. If the sphere S R (x) consists of two points, thenB R (x) is isometric to − R , R with an isometry which sends x to 0 while if the sphere S R (x) contains one pointB R (x) is isometric to , R with an isometry which takes x to 0. Remark 1.11. If X is a Riemannian manifold with metric g, Ric ≥ K and dim = N, the existence of x ∈ X and R > r > such that implies that the ball B R (x) with the Riemannian metric is isometric (in the Riemannian sense) to the ball B R ( ) in the model with metric g K,N .
as base space. It is important to note that in general this Riemannian isometry implies that the two balls are only locally isometric and the Riemannian isometry could not extend to a metric isometry, which is the reason why in the statement of the previous theorem we have R instead of R. To see that in general the Riemannian isometry given by the rigidity in the Riemannian case does not extend to a global isometry, consider a cylinder in R with sections of diameter 1. Then take a point x on it and a ball of radius R = centered at x. Even if (1.10) holds with any r < , K = and N = and the cylinder is a at surface in R it is simple to see that the ballB (x) is not isometric to the euclidean ballB ( ) in R .
The previous rigidity theorem gives the possibility to deduce, arguing by compactness, an almost rigidity theorem. In fact Cheeger and Colding proved in [8] a result of this avour: if in a Riemannian manifold with a bound from below on Ricci curvature the Bishop-Gromov ratios at two radii R and r are almost equal, then the closed ball of radius R in the manifold is close, in the sense of Gromov-Hausdor distance, to the closed ball of radius R around the tip of a suitably chosen metric cone. We can now rephrase and prove this result in the non smooth context, arguing by compactness and using Theorem 1.10.

Theorem 1.12 (Almost volume cone implies almost metric cone -nc version). Given ε
such that there exist δ > R > r > , with r R = η, and x ∈ X satisfying , (1.12) then there exists (Z, dz , mz) an RCD(N − , N − ) space with diam Z ≤ π such that, beingB R (z) the closed ball of radius R around the tip z of the metric cone built over Z, then and with for eachB R i (z) closed ball of radius R i around the tip z of any metric cone built over Z, an RCD(N − , N − ) space with diam Z ≤ π. If we suitably rescale the metric on these spaces . Read in these spaces (1.14) becomes for eachB (z) closed ball of radius around the tip z of any metric cone built over an RCD Here we tacitly exploited the fact that a metric cone is isometric to any rescaling of itself with center in the tip. We also know that, there exist C > and c > depending only on K and N such that because of the non collapsed condition (2.3), the bound on the density (1.9) and the fact that, for is bounded uniformly from above and below. By compactness we have that, up to subsequences, where (X∞, d∞, m∞, x∞) is a ncRCD( , N) space as a consequence of the volume convergence (see Theorem 1.5) and (1.18). Passing to the limit (1.16), taking into account Since dim H (X∞) = N > we can exclude the degeneracy cases in Theorem 1.10, thus we obtain the existence of (Z, dz , mz) an RCD(N − , N − ) space with diam Z ≤ π such thatB (x∞) is isometric to the closed ball B (z) in the metric cone built over Z, where z is the tip of the cone. Then which contradicts (1.17). Remark 1.13. With the same proof, when we work in the class of ncRCD( , N) spaces, we obtain the same statement as before with the constraint R < instead of R < δ.

. Almost cone splitting
De nition 1.14. Given metric spaces (X, De nition 1.15. Let us x N ≥ a number which has the meaning of the upper bound of the dimension of our m.m.s. Given a metric space (X, d), we de ne the t-conicality of the ball Br(x) as De nition 1.16. Following [13] we de ne the ε − (t, r) conical set in B (x ) as 23) where N is de ned in De nition 1.15.

Theorem 1.17 (Cone splitting, quantitative version).
For all K ∈ R, N ∈ [ , +∞), < γ < , δ < γ − , and for all τ, ψ > there exist < ε = ε(N, K, γ, δ, τ, ψ) < ψ and < θ = θ(N, K, γ, δ, τ, ψ) such that the following holds. Let (X, d, m) be an RCD(K, N) m.m.s., x ∈ X and r ≤ θ be such that there exists an εr-GH equivalence where Tτr(·) is the tubular neighbourhood of radius τr, then for some cone R l+ Theorem 1.17 is a quantitative version of the following statement: if a metric cone with vertex z is a metric cone also with respect to z ≠ z, then it contains a line. It can be rigorously stated in the setting of RCD spaces as follows.
Proof. The sought conclusion can be achieved through two intermediate steps.
Step 1. Aim of this rst step is to prove that (X, d, m) contains a line passing through I(ẑ) (and therefore with non trivial component on the C(Z) factor). In order to do so we wish to prove that the ray connecting ( ,z) to I(ẑ) actually extends to a line. Indeed, taking into account the fact that locally around I(ẑ) it is a geodesic we obtain that the cross sectionẐ contains two points z , z such that dẐ(z , z ) = π. Hence, considering now the ray emanating from I(ẑ) and passing through ( ,z), which corresponds to the point z without loss of generality, we obtain that alsoZ contains pointsz ,z such that dZ(z ,z ) = π, otherwise the ray above would not be minimizing around ( ,z). Hence, as we claimed, there is a line in X passing through I(ẑ) and ( ,z).
Step 2. The sought conclusion about the additional splitting follows from what we proved in Step 1 applying Lemma 1.20 below. To conclude it su ces to observe that the split factor is still a metric cone since the whole space is. Indeed if a product R l × Z is a metric cone, then it can be viewed as a metric cone on his sphere of radius and Z is the cone over the intersection of this sphere with the section { } × Z. Remark 1.19. Let us remark that the same conclusion of Proposition 1.18 above holds true under the following weaker assumption: with the same notation adopted above, there exist z ∉ R l × {z} and an isometry I : Br(ẑ) → Br(z) for some r > d(z, ( ,z)) such that I(ẑ) = z. This stronger statement can be checked with no modi cation w.r.t. the proof we presented above. Proof. We just brie y outline the strategy of the proof.
Let us begin by observing that in the case l = the statement corresponds to the splitting theorem, proved in this generality in [22]. If l ≥ we wish to prove that the existence of a line with non constant Y-component implies the existence of a splitting function on the Y-factor and therefore the conclusion. In order to do so, rst we build the Busemann function u associated to the given line. From [22] we know that ∆u = and |∇u| = and then from the Bochner formula, Hess u = (see [24] and [25]). Let us denote furthermore by x , . . . , x l the coordinate functions of the Euclidean factor R l . We claim that there exist real numbers a , . . . , a l such that f . = u − a x · · · − a l x l is a non constant function with constant minimal upper gradient, independent of the Euclidean variable and with vanishing Hessian. Indeed we can de ne a i . = ∇u · ∇x i . These numbers are constant because of the fact that Hess u = and Hess x i = . Then taking f as before, from ∆u = it follows ∆f = and from Hess u = it follows Hess f = ; while from the fact that |∇u| = , it follows, because of the tensorization, that |∇f | is constant. Such a function induces a splitting function on (Y , d Y , m Y ) and the sought conclusion can be obtained applying the results in the appendix of [26], which inspires the fortchoming Lemma 1.21. One has to verify that f is not constant: if not u will be an a ne function on R l and then the Busemann function of a line entirely contained in the rst factor R l . This is not possible since the line associated to the Busemann function u had a non-trivial projection over the second factor, while in this case u would be the Busemann function associated to a line in the factor R l .
Analogously, if there exist functions u , . . . , u l : X → R such that for all i = , . . . , l it holds ∆u i = and |∇u i | = in (X, d, m) as before, and ∇u i · ∇u j = for all ≤ i < j ≤ l, then (X, d, m) is isomorphic to Proof. From the improved Bochner formula [24, Corollary 3.3.9] it follows that Hess u = . Then one can consider the regular Lagrangian ow X t (see [5] for the de nition of regular Lagrangian ow) associated to ∇u. Since ∆u = and Hess u = we can use [1, Theorem 1.9, (iv)] to deduce that for every x, y ∈ X, Then, since for every x ∈ X d dt u(X t (x)) = ∇u · ∇u(X t (x)) = , it follows that for x ∈ X, u(X t (x)) − u(x) = t. Using this information, jointly with the fact that u has a -Lip representative, being |∇u| = , and the fact that d(X t (x), x) ≤ t because of |∇u| ≤ , it follows that for every x ∈ X and t > , d(X t (x), x) = t.
where ( , z * ) is a vertex of the cone R l × C(Z). Furthermore we can nd x ∈ B δ (x) such that B − δγ −N (x )¹ is isometric to the ball centred in the tip of a metric cone and A compactness argument, which is due to Gromov, together with the rescaling and stability properties of the RCD(K, N) condition (see Remark 1.3), yields that Tan(X, d, m, x) is non empty for every x ∈ X and its elements are all RCD( , N) pointed m.m.s..
In the special case in which (X, d, m) is non collapsed any tangent cone has a conical structure, we refer to [20] for the proof of this result. As a consequence of the structural property proved in [31] it is simple to see that if (X, d, m) is a ncRCD (K, N) m.m.s. then N is integer and the regular set 1 Note that − δγ −N > δ which will be important to end the proof by applying a localized version of Proposition 1.18 around x and ( , z * ) (see also Remark 1.19).
satis es m(X \ R) = . The singular set of X is the complement of R. In [9] Cheeger and Colding, inspired by the strati cation results of geometric measure theory, introduced a way to stratify the singular set of a non collapsed Ricci limit according to the maximal dimension of the Euclidean factor split o by a tangent space. This de nition can be given also in the context of ncRCD(K, N) spaces and reads as follows: De nition 1.24. Let (X, d, m) be a ncRCD (K, N) m.m.s.. Given x ∈ X and ≤ k ≤ N we say that x ∈ S k if no tangent space of (X, d, m) at x splits o isometrically a factor R k+ .
Note that we have the inclusions Proof. We refer to [9,Theorem 4.7] for the proof of this result for non collapsed Ricci limits and to [20,Theorem 1.8] for its generalization to ncRCD spaces. Let us just recall here that the proof is based on a dimension reduction argument and on the use of the splitting theorem [22], together with Theorem 1.23. Remark 1.27. It is possible to nd examples of non collapsed Ricci limit spaces of dimension such that S is dense (see for instance [16,Subsection 3.4]). Hence, in general, H k is not locally nite when restricted to S k .

Volume bound for the quantitative strata . Statement and basic consequences
A quantitative counterpart of the strati cation in De nition 1.24 was introduced in [13] in the setting of non collapsed Ricci limit spaces. The de nition extends to the case of ncRCD spaces with no modi cation.

De nition 2.1.
For any η > and any < r < , de ne the k th -e ective stratum S k η,r by where Bs ( , z * ) denotes the ball in R k+ × C(Z) centered at ( , z * ) with radius s.
Since it plays a role in the sequel of the note, we point out here that, given metric spaces (X, d X ) and (Y , d Y ), the notions "d GH (X, Y) ≤ ε" and "there exists an ε-GH equivalence between X and Y" are only equivalent up to a multiplicative constant which, however plays no role for the sake of our discussion. We refer to [35,Chapter 27] for more details about this point.
Let us observe now that S k η,r ⊂ S k η ,r , if k ≤ k , η ≤ η and r ≤ r (2.1) Indeed, if y ∈ S k then y ∈ r S k η,r for some η > and it is trivial to see that r S k η,r ⊂ S k . The classical strati cation is built separating points according to the in nitesimal symmetries of the space. The quantitative strati cation instead is based on how many symmetries there are on balls of a de nite size at any point. Remark 2.2. In (2.2) we can consider just the union over < η < ε for some ε > xed, or even over a countable sequence η i → . Remark 2.3. Let us remark that on a smooth Riemannian manifold the strata S k are all empty, instead the e ective strata S k η,r are non trivial.
Let us state the main result of this note, which extends to the synthetic framework the result proved for non collapsed Ricci limit spaces in [13]. As we already pointed out in the introduction, this statement has already been useful, very recently, in the proof of [32,Theorem 5.8], dealing with stability properties for the boundary of non collapsed RCD(K, N) spaces.
then, for all x ∈ X and < r < / , it holds Let us make a few remarks about (2.4). First we wish to prove that it implies the standard Hausdor dimension estimate dim H (S k ) ≤ k. To do so let us observe that the ηr-enlargement of S k η,r is a subset of S k η,r , that is to say Tηr(S k η,r ) To check (2.5) it is enough to use the triangle inequality: take x ∈ Tηr(S k η,r ), by de nition there exists x ∈ S k η,r such that d(x, x ) < ηr, hence we have d GH (Bs(x), Bs(( , z * ))) ≥ d GH (Bs(x ), Bs(( , z * ))) − d GH (Bs(x ), Bs(x)) ≥ ηs − ηs = ηs for any R k+ × C(Z) with z * tip of C(Z) and every r ≤ s ≤ , where in the last inequality we used x ∈ S k η,r and d GH (Bs(x ), Bs(x)) ≤ d(x, x ) < ηr ≤ ηs. With (2.5) at our disposal we can strengthen (2.4) obtaining a volume estimate of the ηr-enlargement of the quantitative strata In particular, (2.6) implies that On the other hand the Bishop-Gromov inequality and (2.3) guarantee where c > depends only on K and N. Thus where c > depends only on α and we used (2.10) in the last passage. Letting δ → we obtain the sought conclusion. Let us also mention that, even though (2.4) is stronger than dim H (S k ) ≤ k it does not imply one of the problems being the term r η appearing at the right hand side of (2.4). An improvement in this direction is one of the fundamental results in [16].

. . Estimate for the r-enlargement of the boundary
In [20]  of X, the space is as close as we like to the conical structure. To this aim we need a lemma which, together with the almost rigidity result about metric cones proved in Theorem 1.12, will give us the sought result. In this lemma we use a technique reminding the general machinery of quantitative di erentiation (see [12]). We recall here the de nition of conicality given in De nition 1.15.
De nition 2.8. Given a metric space (X, d), we de ne the t-conicality of the ball Br(x) as Moreover, given any R > r ≥ R > r , it holds and also by Remark 1.9. Monotonicity of the logarithm tells that so that it is su cient to choose j ≥ log δ log γ + i .

. . Construction of the covering and conclusion
From now on we x x ∈ X and our aim is to construct a good covering of S k η,r ∩ B (x ) in order to give a bound on H N S k η,r ∩ B (x ) . We recall here the de nition of conical sets given in De nition 1.16.
De nition 2.14. Following [13] we de ne the ε − (t, r) conical set in B (x ) as (2.31) where N is de ned in De nition 1.15.
The following lemma, whose proof is postponed to the next subsection, is a key ingredient for the proof of Theorem 2.4.

then the minimal number of balls of radius
Proof of Theorem 2.4. We can reduce ourselves to prove the sought estimate with r = − γ j for every j ∈ N, for a xed < γ(K, N, η) < / which will be chosen later. Indeed, suppose that there exist < γ(K, N, η) < / and c(K, N, v, η) such that, for every j ∈ N, Then, given < r < / , we can nd j such that − γ j+ < r ≤ − γ j . Since s → S k η,s is increasing, we easily obtain . Let us prove (2.32). We will call j-uple an element of { , } j . From now on we will denote any j-uple with entries in { , } with T j and the i-th entry of this j-uple with T j i . Also |T j | will indicate the number of 's in this j-uple. Let us x j ∈ N. To each x ∈ B (x ) we can associate T j (x) a j-uple with entries in { , } as follows: For any j-uple T j with entries in { , } we let An immediate consequence of Corollary 2.13 is that if E T j is not empty for some j-uple T j , then Recalling that a j-uple de ned starting from a point according to (2.33) has a 1 in the i-th entry if and only if N γ −N B γ i (x) ≥ ε, the estimates of Corollary 2.13 applied with k = γ −N , gives the sought result. The bound obtained in (2.35) allows to estimate the number of non empty sets E T j by j j . Indeed, the number of possible choices of j positions in a string with j ≥ j entries is and the estimate holds also in the case j < j since in that case the j-uples are at most j which is less than the right hand side in the previous equation since j < j .
Let us de ne now inductively on j the covering of S k η, − γ j ∩ B (x ) in such a way that where B T j is a union of balls of radius − γ j . For j = we let B ( ) be the union of the minimum amount of balls of radius − γ with centers in S k η, − γ ∩ E ( ) needed to cover S k η, − γ ∩ E ( ) , if this intersection is not empty. Then we let B ( ) be the union of the minimum amount of balls of radius − γ with centers in S k η, − γ ∩ E ( ) which we need to cover S k η, − γ ∩ E ( ) , if this intersection is not empty. Now for any j > and for any T j for which E T j is not empty, we want to de ne B T j . Let us consider the (j − )-uple T j− which we obtain by dropping the last entry in T j . For each ball B − γ j− (x) in B T j− , we take the minimum amount of balls of radius − γ j with centers in S k η, − γ j ∩ E T j ∩ B − γ j− (x) needed to cover S k η, − γ j ∩ E T j ∩ B − γ j− (x), if this intersection is not empty.
The next step in order to achieve the volume estimate (2.4) aims to bound the cardinality of the families B T j . We claim that for any such family, setting Q . = n + j , the number of balls needed can be controlled by for some constants c (N, K) ≥ c (N) > . To this aim we just observe that (2.37) follows from the way in which we constructed the covering, after the appropriate choice of ε forced by Lemma 2.15, by means of an induction argument. Indeed the factor with exponent Q in (2.37) arises from the at most j +n scales on which the assumptions of Lemma 2.15 are not satis ed and therefore we are forced to cover with c γ −N balls (this possibility is guaranteed by (1.3)). The factor with exponent j − Q instead arises from the remaining scales on which Lemma 2.15 applies and we can cover with less than c γ −k balls. Recapitulating what we obtained so far, we proved that there exist constants c (K, N) ≥ c (N) > and a natural number j such that, for any natural j, the set S k η, − γ j ∩ B (x ) is contained in the union of at most j j non empty families of balls. Furthermore, each of the families above contains at most (c γ −N ) Q (c γ −k ) j−Q balls of radius − γ j .
Let us see how (2.4) can be obtained starting from these results. First we let γ = γ(η) , where c is given by Lemma 2.15. Then we observe that c j = γ j − η , j j ≤ c(N, K, v, η)(γ j ) − η and up to choose η small enough < γ < / . The considerations above, together with the volume comparison yielding H N (B − γ j (x)) ≤ c (N, K)( − γ j ) N , give the estimate In view of what we observed at the beginning of the proof, the estimate above gives the desired result when η is small enough, this in turn implies the general case thanks to (2.1).

. . Proof of the covering lemma via cone splitting
Aim of this subsection is to prove Lemma 2.15. The key tool in proving it will be the e ective almost cone splitting theorem proved in Theorem 1.17 that we restate here for the reader convenience.