A new class of $({\cal H}^k,1)$-rectifiable subsets of metric spaces

The main motivation of this paper arises from the study of Carnot-Carath\'eodory spaces, where the class of 1-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree $k$, which are H\"older but not Lipschitz continuous when $k>1$. Replacing Lipschitz curves by this kind of curves we define $({\cal H}^k,1)$-rectifiable sets and show a density result generalizing the corresponding one in Euclidean geometry. This theorem is a consequence of computations of Hausdorff measures along curves, for which we give an integral formula. In particular, we show that both spherical and usual Hausdorff measures along curves coincide with a class of dimensioned lengths and are related to an interpolation complexity, for which estimates have already been obtained in Carnot-Carath\'eodory spaces.


Introduction
The main motivation of this paper arises from the study of Carnot-Carathéodory spaces. Recall that such a metric space (M, d) is defined by a sub-Riemannian manifold (M, D, g), where M is a smooth manifold, D a subbundle of T M and g a Riemannian metric on D. The absolutely continuous paths which are almost everywhere tangent to D are called horizontal and their length is obtained as in Riemannian geometry integrating the norm of their tangent vectors. The distance d is defined as the infimum of length of horizontal paths between two given points. By construction, only horizontal paths may have finite length and may be Lipschitz with respect to the distance. In contrast to the Euclidean case, both properties are independent on the regularity: all smooth non-horizontal paths have infinite length and are not Lipschitz. This gives rise to two kind of questions.
The first query concerns the measure of non-horizontal paths: what kind of notion is the best suited? One of our motivation is that, from an intrinsic point of view, computing measures of paths should allow to determine metric invariants of curves and thus recover metrically the structure of the manifold [14]. Since non-horizontal paths have a metric dimension greater than one (see [17]), Hausdorff measures are the most natural candidates. However they pose two problems: first they can hardly be computed (except for specific cases [1]), second they do not appear as integrals along the path, which is what we expect for a measure generalizing the notion of length.
The second question comes from geometric measure theory. A typical problem in this field is whether it is possible to characterize the geometric structure of a set using only measures. This gave rise to the notion of rectifiable sets, which is based on Lipschitz functions, and to density results in Euclidean (see [6,9,24] and [21] for a complete presentation) and general metric spaces [18]. In the context of Carnot-Carathéodory spaces rectifiable sets have been studied in Heisenberg groups (see [11,22]) and a different notion of rectifiability was proposed in [19]. However, in these spaces the class of Lipschitz paths is quite poor and does not include non-horizontal smooth curves which consequently are not rectifiable in the usual sense. To take into account the latter curves we need to define rectifiability through a larger class of paths, intrinsically characterized by the distance.
In this paper we address these issues in any metric space, not only in Carnot-Carathéodory ones, by defining a class of curves in the spirit of ( [2,18]). Namely, we introduce curves on a metric space (M, d) that are continuously metric differentiable of degree k (m-C 1 k for short) as continuous is well-defined and continuous (see Definition 1). In an Euclidean space, this definition is useless since the class of m-C 1 k curves with non-zero measure is empty when k > 1 (see Proposition 2). However, in the sub-Riemannian context, for integer values of k this class of curves contains some smooth non-horizontal paths (see Proposition 1).
For m-C 1 k curves we can compute different kind of measures. First, we examine the Hausdorff measures: the usual ones H k and the spherical ones S k . Second, we study the k-dimensional length of a curve γ : [a, b] → M introduced in [8] and defined by Third, we consider a measure based on approximations by finite sets called interpolation complexity (see [13,16]). The first result of the paper (Theorem 1) states that for an injective m- . It also provides a relation between H k (C ) and the interpolation complexity. On the one hand, Theorem 1 gives an integral formula for the Hausdorff measure. On the other hand, it essentially implies that the considered measures are equivalent. Another interesting property of injective m-C 1 k curves with non-zero k-dimensional measure is that the k-dimensional density of H k ⌊ C exists and is constant along the curve (see Proposition 3).
We define (H k , 1)-rectifiable sets as sets that are covered, up to H k -null sets, by countable unions of m-C 1 k curves (see Definition 3). This notion is modeled on the definition of (H k , k)-rectifiable sets in R n , which are sets that are covered, up to H k -null sets, by countable unions of image of C 1 maps from R k to R n . Thanks to the properties of m-C 1 k curves, we show a density result for sets that are rectifiable according to our definition. Namely, the second main theorem of the paper (Theorem 2) states that if a set S is H k -measurable and satisfies H k (S) < +∞, then being (H k , 1)-rectifiable implies that the upper and lower densities of H k ⌊ S are bounded by positive constants. Theorem 2 is inspired by the result proved in Federer [10,Th. 3.2.19], which states that for a H k measurable subset E of the Euclidean n-space (H k , k)-rectifiability implies that the measure H k ⌊ E has k-dimensional density equal to 1 at H k -almost every point of E. The converse of this fact was proved for k = 1 and for a general measure µ in [23]. Much later, Preiss showed not only that the converse holds true for any k, but also a stronger result: there exists a constant c > 1 (depending only on n and k) such that if then E is (µ, k)-rectifiable. Our Theorem 2 implies that an estimate of the type above is satisfied by (H k , 1)-rectifiable sets. An open question is whether an analogous of Preiss' result still holds in non-Euclidean metric spaces with our definition of (H k , 1)-rectifiability. Another open problem is to show a Marstrand's type result (see [20,Th. 1]) for (H k , 1)-rectifiable subsets, at least in Carnot-Carathéodory spaces. In Section 2.2 we construct m-C 1 k curves in sub-Riemannian manifolds having nonzero k-dimensional measure for integer values of k ≥ 1. When the curve is absolutely continuous, it is easy to see that being m-C 1 k with non-vanishing k-dimensional measure implies that k is an integer (see Corollary 1). The question is whether such result holds true without assuming absolute continuity.
The structure of the paper is the following. In Section 2 we give the definition of m-C 1 k curves in metric spaces and construct them in Carnot-Carathéodory spaces. We then study measures along curves. In Section 3.1 we recall different notions of measures. In Section 3.2 we show an auxiliary result for m-C 1 k injective curves with nonzero k-dimensional measure. In Section 3.3 we analyse m-C 1 k curves in (the Euclidean space or a) Riemannian manifold. The main theorem concerning injective m-C 1 k curves is proved in Section 3.4. Some possible generalizations to non m-C 1 k curves are discussed in Section 3.5. Finally, in Section 4 we define (H k , 1)-rectifiable sets and prove the density result.
2 m-C 1 k curves Throughout the paper (M, d) denotes a metric space.

Definitions
Let γ : [a, b] → M be a continuous curve, where a, b ∈ R, and let k ≥ 1 be a real number.
exists and is finite. In this case, we call this limit the metric derivative of degree k of γ at t and we define moreover the k-dimensional infinitesimal measure of γ at t as When γ is not m-differentiable of degree k at t we set meas k t (γ) = +∞. For the case k = 1, the notion of metric derivative is classical, see [3,Def. 4.1.2]. The kdimensional infinitesimal measures of curves were introduced in the context of sub-Riemannian geometry in [8].
Note that if γ is m-differentiable of degree k at t then, for any k ′ , Therefore, for any k ′ > k, meas k ′ t (γ) = 0. If moreover meas k t (γ) > 0, then for any k ′ < k meas k ′ t (γ) = +∞. Definition 2. Given k ≥ 1, we say that γ is differentiable of class m-C 1 k on [a, b] (m-C 1 k for short) if for every t ∈ [a, b] the curve is m-differentiable of degree k at t and the map t → meas k t (γ) is continuous.
Clearly, γ is m-C 1 k if and only if the limit in (1) exists and depends continuously on t. We shall see in the next section that when a smooth structure on M exists, m-C 1 k curves need not be differentiable in the usual sense. The following lemma states that they are Hölder continuous of exponent 1/k as functions from an interval to the metric space (M, d).
where ǫ t (s) tends to zero as s tends to zero uniformly with respect to t.
This is a direct consequence of the continuity of t → meas k t (γ) on the compact interval [a, b].

Construction of m-C 1 k curves
In this section we consider a class of metric spaces which are also smooth manifolds and construct smooth m-C 1 k curves on them with non-vanishing metric derivative of degree k for some integer values of k. The analysis of this class of spaces is the main motivation of this paper.
Let (M, d) be a metric space defined by a sub-Riemannian manifold (M, D, g), i.e., M is a smooth manifold, D a subbundle of T M , g a Riemannian metric on D, and d is the associated sub-Riemannian distance. We assume that Chow's condition is satisfied: let D s denote the R-linear span of brackets of degree < s of vector fields tangent to D 1 = D; then, at every p ∈ M , there exists an integer r = r(p) such that D r(p) (p) = T p M , that is, The proof of this proposition is based on the notions of nilpotent approximation and privileged coordinates (see [5]) and some results in [8]. We do not give the complete argument, but only the underlying ideas. All the facts that here are simply claimed are already established and complete proofs can be found in the cited literature.
Sketch of the proof. Since γ([a, b]) is equiregular, the integers w i defined by do not depend on t. We define for s ≥ 0 the dilation δ s : R n → R n by δ s z = (s w 1 z 1 , . . . , s wn z n ).
Moreover, locally there exist n vector fields Y 1 , . . . , Y n whose values at each γ(t) form a basis of T γ(t) M adapted to the filtration (3) at γ(t), in the sense that, for every integer i ≥ 1, to zero as s tends to zero uniformly with respect to t.
The coordinates φ t are privileged at γ(t) and the distance d t is the sub-Riemannian distance associated with a nilpotent approximation at γ(t). Set φ t (γ(t)) = (γ 1 (t), . . . , γ n (t)). By the construction in the proof of [8, Le. 12], the limit exists at every t and is equal to Using the properties of d t , we have As a consequence, meas k t (γ) exists and is equal to d t (x(t), 0) k . Since the components of x(t) are continuous and the distance d t depends continuously on t, γ is m- Let us explain the construction in Proposition 1 through an example.
Example 1. Consider the Heisenberg group, that is, the sub-Riemannian manifold (R 3 , D, g) where D is the linear span of the vector fields and g = dx 2 + dy 2 . Denote by d the Carnot-Carathéodory distance associated with the Heisenberg group. Recall that d is homogeneous with respect to the dilation δ λ (x, y, z) = (λx, λy, λ 2 z), λ ≥ 0. and it is invariant with respect to the group law Then γ is of class m-C 1 2 and meas 2 t (γ) is a positive constant. This is a consequence of Proposition 1 as γ is smooth and, for all t ∈ R,γ(t) ∈ D 2 (γ(t)). Let us compute explicitly meas 2 t (γ). Notice first that d(γ(t + s), γ(t)) = d((0, 0, s), (0, 0, 0)), since d is invariant with respect to the group law. Hence, using the homogeneity of d and the fact the value of the distance resulting from an isoperimetric problem. Note that such a computation can be generalized to any contact sub-Riemannian manifold, see [8,Th. 22].
Remark 1. Note that the equiregularity assumption is essential to obtain the continuity of d t and φ t with respect to t. In particular the proof of [8,Le. 12] is not valid without this hypothesis 1 . This assumption has also an intrinsic meaning. Indeed it is shown in [8] that the k-dimensional measure meas k t (γ) can actually be defined through the distance on the metric tangent space to (M, d) at γ(t). Since the metric tangent space does not vary continuously with respect to t around C -singular points, where C = γ ([a, b]), in general non equiregular curves may not be m-C 1 k . Note that for every integer k ∈ {1, . . . , r(p)}, where p is regular, there exist C 1 equiregular curves with tangent vector belonging to D k \ D k−1 . As a consequence, for such integers k the class of m-C 1 k curves with non-vanishing metric derivative of degree k is not empty. For instance, this is the case in the Heisenberg group for k = 2, and in the Engel group (see below) for k = 2, 3. On the contrary, the next proposition states that in the Riemannian case, i.e., when D = T M , the class of m-C 1 k curves with non-vanishing derivative is empty except for k = 1 (the proof of Proposition 2 is postponed to Section 3.3). . Then γ is horizontal, i.e., it is absolutely continuous andγ(t) ∈ D 1 (γ(t)) almost everywhere on [a, b]. To see this, remark that by construction, γ is Lipschitz with respect to the sub-Riemannian distance. The metric g defined on D can be extended (at least in a tubular neighbourhood of γ([a, b])) to a Riemannian metricg on T M . In this way we obtain a Riemannian distance on M which is not greater than the sub-Riemannian distance. Hence γ is Lipschitz with respect to the chosen Riemannian distance which in turn implies that γ is absolutely continuous. Therefore, by [7,Pr. 5] γ is horizontal, i.e.,γ(t) ∈ D 1 (γ(t)) almost everywhere on [a, b].
Using Proposition 1, this fact can be partially generalized to the case k > 1 under the following form.
Corollary 1. Let k ≥ 1 and let γ : [a, b] → M be equiregular and of class m-C 1 k , with meas k t (γ) ≡ 0. If γ is absolutely continuous, then k is the smallest integer m such thatγ(t) ∈ D m (γ(t)) almost everywhere.
In particular Corollary 1 states that if meas k t (γ) ≡ 0 then k is an integer, provided that γ is absolutely continuous. An open question is whether the latter condition is necessary. If this were not the case then we would obtain a Marstrand's type Theorem [20, Th. 1] for m-C 1 k curves: indeed we shall see in Proposition 3 that along injective m-C 1 k curves with non-vanishing k-dimensional measure the density of H k exists and is constant.
Notice that if γ is m-C 1 k and k ′ ≥ k, then γ is m-C 1 k ′ . Define k γ ≥ 1 as the infimum of k ≥ 1 such that γ is m-C 1 k . Then k γ need not be an integer as it is shown in the next example. Moreover, γ is not necessarily m-C 1 kγ . Example 3. Consider the sub-Riemannian structure of Example 2 and the curve γ(t) = (0, 0, W (t), 0). Then k γ = 2/ξ may be any real number greater than 2 (see (4)), but γ is not m-C 1 kγ .

Measures along curves
This section is devoted to compute Hausdorff (and spherical Hausdorff) measures of continuous curves and to establish a relation with the k-dimensional length and with the complexity.

Different notions of measures
Denote by diam S the diameter of a set S ⊂ M . Let k ≥ 0 be a real number. For every set A ⊂ M , we define the k-dimensional Hausdorff measure H k of A as H k (A) = lim ǫ→0 In the Euclidean space R n , k-dimensional Hausdorff measures are often defined as 2 −k α(k)H k and 2 −k α(k)S k , where α(k) is defined from the usual gamma function as α(k) = Γ( 1 2 ) k /Γ( k 2 + 1).
This normalization factor is necessary for the n-dimensional Hausdorff measure and the Lebesgue measure coincide on R n . For a given set A ⊂ M , H k (A) is a decreasing function of k, infinite when k is smaller than a certain value, and zero when k is greater than this value. We call Hausdorff dimension of A the real number dim H A = sup{k : H k (A) = ∞} = inf{k : H k (A) = 0}.
Note that H k ≤ S k ≤ 2 k H k , so the Hausdorff dimension can be defined equally from Hausdorff or spherical Hausdorff measures. When the set A is a curve, another kind of dimensioned measures can be obtained from the integration of k-dimensional infinitesimal measures. Let γ : [a, b] → M be a continuous curve and C = γ ([a, b]). For k ≥ 1, we define the k-dimensional length of C as where meas k t (γ) is as in Definition 1 (these lengths were introduced in [8] in the sub-Riemannian context). Thanks to the properties of meas k t (γ), Length k (γ) is a decreasing function of k, infinite when k is smaller than a certain value, and zero when k is greater than this value. We call this value the length dimension of C .
Another way to measure the set C is to study its approximations by finite sets (see [16] and [14, p. 278]). Here we only consider approximations by ǫ-chains of C , i.e., sets of points q 1 = γ(a), . . . , The interpolation complexity σ int (C , ǫ) is the minimal number of points in an ǫ-chain of C . This complexity has been computed in several cases in [12].

m-C 1
k curves with non-vanishing k-dimensional measure In this section we prove the following proposition about m-C 1 k curves with non-vanishing k-dimensional measure. This result is the first step to prove Theorem 1.
and for every q ∈ C  (a, b). Also, they hold for unbounded intervals. Therefore, thanks to the regularity of L 1 and H k measures, equations (6), (8) are still verified when we replace C by γ(A), for any measurable set A ⊂ [a, b].
If we drop the injectivity assumption we obtain the following weaker result. Proof of Corollary 2. Since meas k t (γ) = 0 for every t ∈ [a, b], γ is locally injective. Hence [a, b] is the disjoint union of a finite family of intervals I i such that γ| I i is injective. For each i there exists a measurable subset A i ⊂ I i such that C = ∪γ(A i ) and the sets γ(A i ) are pairwise disjoint. Using Remark 3, formula (6) applies to each γ(A i ). Since H k (C ) = i H k (γ(A i )) and S k (C ) = i S k (γ(A i )), we obtain (9).
The proof of Proposition 3 is based on the following result for bi-Hölder continuous curves.
for every t, t + s ∈ [0, T ] with |s| < η. Then and, for every t ∈ [0, T ] and r > 0 small enough, Proof. Let ǫ > 0 be smaller than δ + η 1/k . We denote by N the smallest integer such that T ≤ N ( ǫ δ + ) k and define t 0 , . . . , t N by It also results from inequality (15) that γ(t 0 ), . . . , γ(t N ) is an ǫ-chain of C which implies Taking the limit as ǫ → 0 in the preceding inequalities, we find We now prove converse inequalities for H k and σ int . Fix ǫ > 0 and consider a countable family S 1 , S 2 , . . . of closed subsets of M such that C ⊂ i S i and diam S i ≤ ǫ. For every i ∈ N, we set I i = γ −1 (S i ∩ C ). As γ is injective, if ǫ is small enough then it results from (10) that for any t, t ′ in In the same way, an ǫ-chain γ(t 0 ) = γ(0), . . . , γ(t N ) = γ(T ) of C satisfies N ǫ k ≥ T δ k − since the injectivity of γ assures that It follows that ǫ k σ int (C , ǫ) ≥ δ k − T . Taking the limit as ǫ → 0 in this inequality and in (16), we find H k (C ) ≥ δ k − T and lim inf ǫ→0 + ǫ k σ int (C , ǫ) ≥ δ k − T , which completes the proof of (11) and (12). The first inequality in (13) always holds. Before proving the second one, let us recall a standard result in geometric measure theory (see for instance [10, 2.10.18, (1)]). Let X be a metric space and µ be a regular measure on X such that the closed balls in X are µ-measurable. If lim sup y∈B(x,r) r→0 + µ(B(y, r)) (diam B(y, r)) k ≥ λ, for every point x ∈ X, then µ(X) ≥ λS k (X).
We will apply this result to the metric space (C , d| C ) and to the measure µ = H k ⌊ C .
If t ∈ [0, T ], then for r > 0 small enough, there holds The diameter of this set then satisfies Moreover, applying (11) to the curve γ restricted to [t − r k δ k Thus we have, for every point γ(t ′ ) ∈ C , lim sup Finally, formula (14) results from (11) applied to the restrictions of γ in (17).

Proof of Proposition 3. By definition,
Note that the k-dimensional length does not depend on the parameterization [8,Le. 16]. Thus, up to a reparameterization by the k-length, we assume that γ is defined on the interval [0, T ], with T = Length k (C ), and that meas k t (γ) ≡ 1. Fix δ > 0. Then, by Lemma 1, there exists η > 0 so that the hypothesis of Lemma 2 is satisfied with δ − = 1 − δ and δ + = 1 + δ. We let δ tends to zero in inequalities (11)- (14) and the proposition follows. Remark 4. Another way to measure C using approximations by finite sets is to consider ǫ-nets, i.e., sets of points q 1 , . . . , q n ∈ M such that the union of closed balls B(q i , ǫ) covers C , and the metric entropy e(C , ǫ) which is the minimal number of points in an ǫ-net of C . Under the assumptions of Proposition 3, the following estimates can be deduced for a m-C 1 k curve:

The Riemannian case
Let us come back to the case where (M, d) is a Carnot-Carathéodory space associated with a sub-Riemannian manifold (M, D, g). A consequence of Proposition 3 is that if the structure is Riemannian, i.e., D = T M , then the class of m-C 1 k curves having non-zero metric derivative of degree k is empty if k > 1, as stated in Proposition 2.
Proof of Proposition 2. Let s ∈ [a, b] such that meas k s (γ) = 0. Thus, restricted to a small enough neighbourhood I = [s − δ, s + δ] of s, the curve γ is injective and meas k t (γ) = 0 on I. Up to reparameterizing γ| I , we may assume moreover meas k t (γ) ≡ 1 on I. Also, it is sufficient to consider the case M = R n and d is the Euclidean distance on R n . Denote by C the set γ(I). By Proposition 3, we have 0 < H k (C ) < +∞ and for every t ∈ I lim r→0 H k (C ∩ B(γ(t), r)) 2r k = 1.
Moreover, by Lemma 1, there exist 0 < ρ < 1 such that The proposition then results from the lemma below.
Lemma 3. Let k ≥ 1 and let γ : [a, b] → R n be a bi-Hölder curve of exponent 1/k, i.e., Proof. Under the assumptions of the lemma, Marstrand's Theorem [20,Th. 1] assures that k ∈ N and k ∈ {1, 2, . . . , n}. Applying Preiss' result [24], there exists a countable family of k-dimensional Hence there exists a density point γ(t 0 ) ∈ C ∩ N , that is, a point such that is the open ball in N centered at γ(t 0 ) of radius r. Let us identify N with {(x 1 , . . . , x n ) ∈ R n : x k+1 = · · · = x n = 0} by choosing local coordinates around γ(t 0 ). Using the inequalities (18) and the density point t 0 it is not hard to prove that there exists a bi-Lipschitz homeomorphism from B N (γ(t 0 ), δ − ) endowed with the Euclidean distance to (−1, 1) endowed with the distance | · | 1/k (see for instance the argument in the proof of [4,Pr. 4.12]). Since the topological dimension of B N (γ(t 0 ), δ − ) is k, then k must be equal to 1.

Comparison of measures for m-C 1 k curves
Next theorem generalizes the first part of Proposition 3 to the case when the metric derivative may vanish. Namely, it compares the H k measure and the S k measure of sets that are images of m-C 1 k curves. Also, it provides a relation among such measures, the k-length and the behaviour of the complexity of the curve. If moreover H k (C ) > 0 or Length k (C ) > 0, then for every k ′ ≥ 1 Remark 5. When M is a sub-Riemannian manifold, in many cases Gauthier and coauthors (see [12] and references therein) computed the interpolation complexity of curves as integral of some geometric invariants. Jointly with Theorem 1, these results provide a way of computing Hausdorff measures of curves as well as a geometric interpretation of Hausdorff and infinitesimal measures. If moreover γ is injective then H k (γ(A)) = Length k (γ(A)).
Remark 6. Let us consider the case where γ is an injective m-C 1 k curve. Recalling the definition of Length k , we have that is, we have an integral formula for the k-dimensional Hausdorff measure. Moreover, Theorem 1 implies that the Hausdorff dimension k H of C coincides with the length dimension of C . If in addition H k H (C ) (or Length k H (C )) is finite, then Corollary 3 implies that H k H ⌊ C is absolutely continuous with respect to the push-forward measure 2 γ * L 1 and that its Radon-Nikodym derivative is meas k t (γ).
γ is not m-C 1 k at t). A standard argument of measure theory allows to show the following fact. Assume that [a, b] \ I k is an open subset of [a, b] of full L 1 measure and that H k (γ(I k )) = 0. Then the conclusions of Corollary 3 still hold. Moreover, if γ is injective, then the equalities between H k (C ), S k (C ), and Length k (C ) as in Theorem 1 hold true. The result on the complexity is not valid anymore, since lim ǫ→0 ǫ k σ int (C , ǫ) is not a measure. A curve C satisfying the properties above actually appears as a particular case of (H k , 1)-rectifiable set, which will be studied in the next section.
It is however worth to mention a consequence of the result claimed above (and of Proposition 1) in the context of Carnot-Carathéodory spaces. Let (M, D, g) be a sub-Riemannian manifold, γ : [a, b] → M be an absolutely continuous injective curve, and C = γ ([a, b]). Let m C ≥ 1 be the smallest integer such thatγ(t) ∈ D m C (γ(t)) almost everywhere. We denote by I C the set of points t ∈ [a, b] such that either γ is not C 1 at t or γ(t) is C -singular. When the sub-Riemannian manifold is equiregular, it is already known [14, p. 104] that the Hausdorff dimension of a one-dimensional submanifold C is the smallest integer k such that T q C ⊂ D k (q) for every q ∈ C. Corollary 4 generalizes this fact.
Any injective m-C 1 k curve being bi-Hölder (see Lemma 1), it is also natural to generalize our results to such curves. Consider then a bi-Hölder curve of exponent 1/k, i.e. a curve γ : [a, b] → M , i.e., there exist positive constants δ − and δ + such that, for every t, t + s ∈ [a, b], δ − |s| 1/k ≤ d γ(t), γ(t + s) ≤ δ + |s| 1/k . For such a curve C = γ([a, b]), the k-dimensional length does not always exists but Lemma 2 gives estimates of H k (C ) and S k (C ) in function of T = b − a, and the following result for upper and lower density: Let us remark that there is no hope to obtain a density result such as (8) for bi-Hölder curves. Indeed Assouad proved in [4] that, for any k < n, there exist bi-Hölder curves of exponent 1/k from (−1, 1) to R n (both endowed with a Euclidean metric). When k > 1 the density of these curves cannot be constant, for otherwise Lemma 3 would yield a contradiction. This strongly hints that there is not a Rademacher's-type result in this context, that is, being bi-Hölder of exponent 1/k does not imply being m-C 1 k almost everywhere. In what follows, in particular in the definition of (H k , 1)-rectifiability, we will work with m-C 1 k curves and not with bi-Hölder curves. The drawback is that our definitions will not be invariant under bi-Lipschitz equivalence of metric spaces since the m-C 1 k property is not invariant under such equivalence, contrarily to the bi-Hölder property. However we think that rectifiable sets should be defined as sets which admit almost everywhere a metric derivative. As noticed above, the use of bi-Hölder curves would not guarantee such a property. our notion of (H k , 1)-rectifiable sets. Namely, is condition (25) for H k -almost every q ∈ S ⊂ M L 1 (Ĩ i ∩ γ −1 i (B(q, r))) r k ≥ L 1 (Ĩ i ∩ B(t, r k /(1 + δ) k ))) r k .