Quantifying Curvelike Structures of Measures by Using L 2 Jones Quantities

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1 Quantifying Curvelike Structures of Measures by Using L 2 Jones Quantities GILAD LERMAN Courant Institute Abstract We study the curvelike structure of special measures on R n in a multiscale fashion More precisely, we consider the existence and construction of a sufficiently short curve with a sufficiently large measure Our main tool is an L 2 variant of Jones β numbers, which measure the scaled deviations of the given measure from a best approximating line at different scales and locations The Jones function is formed by adding the squares of the L 2 Jones numbers at different scales and the same location Using a special L 2 Jones function, we construct a sufficiently short curve with a sufficiently large measure The length and measure estimates of the underlying curve are expressed in terms of the size of this Jones function c 2003 Wiley Periodicals, Inc 1 Introduction We explore the curvelike structure of special measures on R n by using multiscale geometric analysis and following a work of Jones [14] The translation of our objective into a mathematical language is not immediate Before doing it, we would like to describe our motivation The main problem we have in mind is that of curve learning (and, more generally, manifold learning ; see, eg, [1, 6, 13, 28, 30]) Given a data set in R n,it is natural to ask whether this set or some of its subsets are well approximated by sufficiently short curves If the answer is positive, then we would like to study the underlying curves or analyze their smoothness properties In this paper we transcribe the above problem to the study of the curvelike structure of special measures The manifoldlike structures of measures will be investigated in a joint work with Jones [16] Numerical results and applications to data sets will be presented in [18] (see also [19]) The most general (and abstract) notion of curvelike structure of measures is that of weak one-dimensional rectifiability, that is, the existence of a countable union of rectifiable curves of full measure (see Section 2 for further explanation) Note that this definition does not take into account any quantitative estimates like the lengths of the curves The latter estimates are the scope of quantitative rectifiability Communications on Pure and Applied Mathematics, Vol LVI, (2003) c 2003 Wiley Periodicals, Inc

2 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1295 The theory of quantitative rectifiability was created by Peter Jones [14] in order to solve a continuous version of the traveling salesman problem This version examines the existence and construction of a curve (a Lipschitz image of a compact interval) with nearly minimal length containing a given compact set in R n Jones theory was further extended by different authors Kate Okikiolu [22] generalized [14, first part of theorem 1] to R n (instead of R 2 ) Guy David and Stephen Semmes [9, 10] developed a theory for embedding a d-dimensional Ahlfors-regular set, lying in R n,inad-dimensional regular manifold with a small d-volume (d an integer, 1 d < n; for the definition of Ahlfors regularity, see, eg, [10, definition 113] or Section 2 where d = 1) They formulated the notion of uniform rectifiability and showed its relation with different quantitative estimates (see, eg, [10, section 14]; their notes [11] are written for the nonspecialist) Xiang Fang [12], Hervé Pajot [23, 24, 25], and J C Léger [17] expanded the theory further Applications of quantitative rectifiability to various problems in geometric measure theory and potential theory can be found in [2, 3, 4, 5, 7, 8, 15] One of the attractive features of this theory is the use of multiscale analysis Jones [14] introduced the multiscale β numbers, which record normalized L approximation errors of a set by lines at different scales and different locations David and Semmes [9] used L p approximations instead of the L ones for similar multiscale analysis of d-dimensional Ahlfors-regular sets (or possibly Ahlforsregular measures) In this paper we extend the above theory to a wide class J n of one-dimensional rectifiable measures Given a measure µ in J n, we fit a sufficiently short curve with a sufficiently large measure µ This problem is different from that of fitting a sufficiently short curve to the whole given set in the theory mentioned above We use L 2 variants of Jones β numbers These β 2 numbers generalize the ones of David and Semmes [10] for any locally finite Borel measure Following Bishop and Jones [4], we form the square function J 2 by adding the squares of β 2 numbers from different scales and the same location In order to avoid some technical difficulties in proving the relevant theorems, we modify our initial β 2 numbers The corresponding modified square function is denoted by Ĵ 2 For a given measure µ, the two functions J 2 and Ĵ 2 are comparable in L 1 (µ) By applying a technique from [16], one can use this comparability to replace the function Ĵ 2 by J 2 in the theory presented here A main idea of this paper is the relation between the size of Ĵ 2 of a given measure µ in J n and one-dimensional quantitative estimates of µ In Theorems 48 and 410 we show that if Ĵ 2 is well controlled, then there exists a sufficiently short curve with a sufficiently large measure Moreover, the optimal length and measure of such a curve can be estimated by using the size of Ĵ 2 In Section 5 we construct a measure for which these estimates are indeed optimal Due to the general setting, some of the estimates and constants in this paper are not practical for various subclasses of measures For example, in Section 5

3 1296 G LERMAN we show that the estimates of Theorems 48 and 410 are far from optimal when applied to one-dimensional Ahlfors-regular measures Nonetheless, following [14, 22, 24] we can still show in this special case how to use the size of Ĵ 2 to obtain the optimal estimates It would be appealing for us to find out the sharp dependence of the different constants in Theorems 48 and 410 on the dimension n (we ask the same question when replacing Ĵ 2 by J 2 in these theorems) We are also interested in the following two related tasks: First, find large subclasses of rectifiable measures with constants depending weakly on n (see, eg, Talagrand [29] and Pestov [26] for typical obstacles with large n) Second, replace the rigid dyadic grids by flexible grids adapted to the given measure so that the constants are reduced substantially and so that such grids can be easily used in numerical algorithms for data sets (see [18, 19] for the kind of numerical algorithms we have in mind) The paper is organized as follows In Section 2 we list some general definitions and notation More specialized definitions appear in the relevant sections In Section 3 we describe the extended dyadic grids and verify their main attractive property In Section 4 we introduce L 2 versions of Jones quantities for analyzing locally finite Borel measures on R n and define the class of measures J n We then formulate the two theorems of this paper The estimates of the theorems versus the optimal estimates are exemplified in Section 5 In Section 6 we verify a special case of Theorem 48 Based on this result, we conclude both Theorems 48 and 410 in Section 7 2 Some Basic Notation and Definitions We shall work in the Euclidean n-space R n We assume that the space is endowed with a locally finite Borel measure µ Here are the basic notation and definitions used throughout this paper Let A be a subset of R n We denote by Ā the closure of A, bya c the complement of A, and by A the boundary of A If u is any vector in R n, then A + u = {v + u v A} The restriction of the measure µ to the set A is designated by µ A We use H 1 to denote the one-dimensional Hausdorff measure and dist H to denote the Hausdorff distance (see, eg, Mattila [20]) We denote by B(x, t) a ball with center at x and radius t and by Q(x, l) a cube with sides parallel to the axes, center at x, and side length 2 l If Q is a cube in R n, then l(q) denotes its length and C Q denotes the cube with the same center as Q, side length C l(q), and sides parallel to the sides of Q Assume next that µ(q) >0 The center of mass of the cube Q is denoted by z Q and defined by the formula z Q := Q z dµ(z) µ(q)

4 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1297 A best L 2 line in the cube Q is a line L Q that attains the minimum of the integral Q dist2 (z, L Q )dµ(z) among all lines in R n Note that a best L 2 line for a given cube is not necessarily unique Denote by H Q the set of all best L 2 lines for the cube Q A function f :[a, b] R n is called a Lipschitz function if there is a constant C > 0 such that f (x) f (y) C x y for all x, y [a, b] Denote by f Lip[a,b] the infimum of all such possible constants C A set Ɣ in R n is a rectifiable curve if there exist a segment [a, b] and a Lipschitz function f :[a, b] R n such that Ɣ = f ([a, b]) Equivalently, a rectifiable curve is a compact, connected set with finite one-dimensional Hausdorff measure (for a proof of the equivalence, see, eg, [10, theorem 18]) The length of a rectifiable curve Ɣ, denoted by l(ɣ), is defined to be H 1 (Ɣ) We refer to the measure H 1 Ɣ as the arc length measure on Ɣ We use the notation supp, dist, and diam as short for support, distance, and diameter, respectively We designate absolute constants larger than 1 by capital letters (mainly C) and very small absolute constants by ε and δ We may use the same letter with the same subscript to denote different constants at different places in the text We denote by C(n) constants that depend on the dimension n When using a constant often, we might omit its dependence on n in order to simplify notation We say that A is approximately less than B,orA B,ifA C B, where C is an absolute constant that may vary from line to line Similarly, we say that A and B are comparable, ora B, ifa B and B A Let f and g be two functions defined on a domain D We say that f is approximately less than g everywhere, or f g, if f (x) g(x) for all x D Similarly, f and g are comparable everywhere, or f g, if f (x) g(x) for all x D We use the notation n, n, and n whenever the constants of comparability depend (or might depend) on the dimension n We say that a measure µ in R n is weakly one-dimensional rectifiable if and only if there exists a countable union of rectifiable curves in R n of full measure If, in addition, µ is absolutely continuous with respect to H 1, then we omit the word weak (see Preiss [27] for a different definition and a useful criterion) A measure µ is one-dimensional Ahlfors-regular if and only if there exists a constant C = C(µ) such that for any cube (or ball) Q = Q(x, r) in R n, where x supp(µ) and r diam(supp(µ)), the following inequality is satisfied: C 1 l(q) µ( Q) C l(q) AsetA is one-dimensional rectifiable if and only if the measure H 1 A is one-dimensional rectifiable Similarly, it is one-dimensional Ahlfors-regular if and only if the measure H 1 Ā is one-dimensional Ahlfors-regular

5 1298 G LERMAN 3 Definitions and Properties of Grids Throughout the paper we will use dyadic cubes and grids extensively We say that Q R n is a dyadic cube with side length 2 j if [ k1 Q = 2, k ) [ kn j 2 j 2, k ) n + 1, j 2 j where k 1,,k n and j are arbitrary integers Denote by D j the dyadic grid at the scale 2 j, that is, the collection of all dyadic cubes whose side length l(q) equals 2 j Let Q be a half-closed, half-open cube in R n, and let T : R n R n be the affine transformation that maps [0, 1) n onto Q We define the normalized dyadic grid with respect to Q by D(Q) = T ( j=0 D j) In order to avoid some edge effects of dyadic grids, we use an extended grid It is the union of some shifted dyadic grids This extended grid was introduced to the theory of quantitative rectifiability by Kate Okikiolu [22] DEFINITION 31 (Extended normalized grids D(Q 0 ) and D j (Q 0 ) wrt Q 0 )IfQ 0 is a cube in R n, then D(Q 0 ) = ( Q 0 + l(q ) 0) e 3 and D j (Q 0 ) = e {0,1} n D { Q : Q D(Q 0 ) and l(q) = l(q } 0) 2 j The new collection of cubes, D(Q 0 ), has the property that any point in R n is the center of a cube in D(Q 0 ) with a given length This property can be formulated more precisely as follows: PROPOSITION 32 Let Q 0 be a cube in R n Ifx R n and j 0, then there exists a cube Q in D j (Q 0 ) such that x 2 3 Q PROOF: We first prove the claim for n = 1, so that both Q and Q 0 are intervals We distinguish between two cases: Case 1 j is an even number, j 0 In this case 2 j 1 (mod 2) and thus 3 ( (31) D j (Q 0 ) = D j (Q 0 ) D j (Q 0 ) + 1 ) 3 l(q 0), j = 0, 2, 4, Consequently, by using appropriate shifting and scaling, it is sufficient to consider the special case where Q 0 =[0, 1), j = 0, and x [0, 1) Assume this reduced

6 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1299 case and note that the following cube Q satisfies the desired proposition: [ 2 3, 1 3 ) if 0 x < 1 6 Q = [0, 1) if 1 6 x < 5 6 [ 1 3, 4 3 ) if 5 6 x < 1 Case 2 j is an odd number, j 1 In this case 2 j 2 (mod 2) and thus 3 (32) D j (Q 0 ) = D j (Q 0 ) (D j (Q 0 ) l(q 0)), j = 1, 3, Consequently, by using appropriate shifting and scaling, it is sufficient to consider the special case where Q 0 =[0, 1), j = 1, and x [0, 1 ) Assume this reduced 2 case and note that the following cube Q satisfies the desired proposition: [ 1 6, 1 3 ) if 0 x < 1 12 Q = [0, 1 2 ) if 1 12 x < 5 12 [ 1 3, 5 6 ) if 5 12 x < 1 2 The above two cases conclude the proof for n = 1 Assume next that n > 1 Fix x R n and j 0 Denote Q 0 = I 0 1 I 0 n and x = (x 1,,x n ) For each i, i = 1,,n, let I i be an interval in D j (I 0 i ) such that x i 2 3 I i Note that Q := I 1 I n is a cube in D j (Q 0 ) such that x 2 3 Q We next extend the grid D(Q 0 ) to include cubes with side length larger than l(q 0 ) The idea is due to Peter Jones DEFINITION 33 (Cubes Q L (Q) and Q R (Q)) IfI is the interval I =[a, b), then I L (I ) = [ a 2 3 (b a), a + 4 (b a)) 3 and I R (I ) = [ a 1 3 (b a), a (b a)) If Q is a half-closed, half-open cube in R n of the form Q = I 1 I n, then Q L = I L (I 1 ) I L (I n ) and Q R = I R (I 1 ) I R (I n ) DEFINITION 34 (Cubes Q + j (Q), j 0) If Q is a cube in R n, then the cubes Q + j (Q), j 0, are formed recursively as follows: Q if j = 0 Q + j (Q) = Q L (Q +( j 1) (Q)) if j = 1, 3, 5, Q R (Q +( j 1) (Q)) if j = 2, 4, 6,

7 1300 G LERMAN The new grids, D + j Q 0, j 0, are defined as follows: (33) D + j Q 0 := D(Q + j (Q 0 )), j 0 Observe that D + j +( j+1) Q 0 D Q 0, j 0 Consequently, define the extended dyadic grid D + Q 0 that contains cubes from all scales by the formula D + Q 0 = j 0 D + j Q 0 In this paper we fix a cube Q 0 and frequently use the cubes in the grid D +3 Q 0 For simplicity of notation, we designate this grid by D 4 L 2 Theory of Quantitative Rectifiability In this section we present an L 2 theory of one-dimensional quantitative rectifiability for a certain class J n of locally finite Borel measures on R n This theory originates from the L theory of quantitative rectifiability [4, 14] We fix a cube Q 0 in R n throughout the rest of the paper (in Section 6 we denote the underlying cube by Q 1 ) Let µ be a locally finite Borel measure on R n Following [4, 14] and [9, 10], we study the one-dimensional properties of µ around Q 0 in a multiscale fashion We start by defining the β 2 numbers that record normalized L 2 approximation errors of µ by lines at different scales and locations Scales and locations are given by cubes in the grid D D +3 Q 0 The Jones function J 2 is formed by adding the squares of the β 2 numbers at different scales and at the same location We employ a variant of J 2 : Ĵ 2 The two functions J 2 and Ĵ 2 are comparable in L 1 (µ) We define a class J n of measures whose Jones functions Ĵ 2 are well controlled around Q 0 (see equation 45) We show that if µ is a measure in J n, then there exists a curve (in a neighborhood of the cube Q 0 ) with a sufficiently short length and a sufficiently large measure µ We estimate the short length and large measure in terms of the size of the function Ĵ 2 These ideas are formulated precisely in Theorems 48 and 410 A concrete construction of the underlying curve is described in their proof (see Sections 6 and 7) The L 1 (µ) comparability of J 2 and Ĵ 2 together with an observation from [16] imply that Theorems 48 and 410 are also valid when replacing Ĵ 2 by J 2 41 L 2 Jones Quantities and Related Definitions We first define a one-dimensional L 2 version of Jones β numbers (see also [19] for a d-dimensional version) For Ahlfors-regular measures, these β 2 numbers are comparable to the ones of David and Semmes [10]

8 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1301 DEFINITION 41 (Jones β 2 numbers) If µ is a locally finite Borel measure on R n and Q is a cube in R n, then ( ( ) 1 dist(z, L) 2 ) 1/2 min dµ(z) if µ(q) >0 β 2 (Q) β µ 2 (Q) = lines L µ(q) l(q) Q 0 otherwise Recall that Q 0 is a fixed cube in R n and that D denotes the grid D +3 Q 0 (see equation (33)) Following [4], we define the L 2 Jones function with respect to Q 0 as follows: DEFINITION 42 (L 2 Jones function wrt the cube Q 0 )Ifµ is a locally finite Borel measure on R n and Q 0 is a fixed cube in R n, then for any x R n (41) J 2 (x) J Q 0,µ 2 (x) = Q D β 2 2 (Q) χ Q(x), where χ Q (x) denotes the indicator function of the cube Q We next modify the L 2 Jones numbers and function defined above Recall that a best L 2 line in a cube Q is a line L Q that attains the minimum of the integral Q dist2 (z, L Q )dµ(z) among all lines A best L 2 line is not necessarily unique We define the L 2 Jones number of a cube with respect to another cube as follows: DEFINITION 43 (L 2 Jones number β 2 (Q 1, Q 2 ))Ifµ is a locally finite Borel measure on R n, and Q 1 and Q 2 are two cubes in R n such that Q 1 Q 2 and H Q2 is the set of all best L 2 lines in Q 2, then (42) β 2 (Q 1, Q 2 ) β µ 2 (Q 1, Q 2 ) = ( ( ) dist(z, L Q2 ) 2 ) dµ(z) 1/2 sup if µ(q 1 )>0 L Q2 H Q2 l(q 1 ) µ(q 1 ) Q 1 0 otherwise j 1 The following definitions use two integers j0 and j 1 = j 1 (n) where 2 j 0 We fix these constants in the beginning of Section 6 DEFINITION 44 (The set P(Q)) IfQ 0 is a cube in R n and Q is a cube in D + Q 0, then P(Q) P Q 0(Q) is the set of all cubes Q in D + Q 0 that contain Q and satisfy the length estimate 2 j 1 l(q ) l(q) 2 j 0 The modified Jones numbers ˆβ 2 and function Ĵ 2 are defined as follows:

9 1302 G LERMAN DEFINITION 45 (Modified Jones numbers ˆβ 2 )Ifµ is a locally finite Borel measure on R n, Q 0 is a cube in R n, and Q is a cube in D, then ˆβ 2 (Q) ˆβ Q 0,µ 2 (Q) = sup β 2 (Q, ˆQ) ˆQ P(Q) DEFINITION 46 (Modified Jones function Ĵ 2 wrt the cube Q 0 )Ifµ is a locally finite Borel measure on R n, then (43) Ĵ 2 (x) Ĵ Q 0,µ 2 (x) = ˆβ 2 2 (Q) χ Q(x) Q D REMARK 47 We did not worry about efficiency when defining the set P(Q) It is possible to replace this set by a subset containing fewer cubes, so that the techniques of Sections 6 and 7 also apply to the corresponding modified Jones function 42 Theory We explain here how to use the function Ĵ 2 of a measure µ and a cube Q 0 to determine the solution of the following problem: Find a sufficiently short curve with a sufficiently large measure µ in a neighborhood containing the cube Q 0 We also discuss related optimization problems Our emphasis is on the ability to use the function Ĵ 2 to solve the above problem We do not care here about the optimal estimates of particular cases or about the best constants The main result of this paper is stated in Theorem 410 A weaker version of this result is formulated as follows: THEOREM 48 There exist positive constants C 1 = C 1 (n), C 2 = C 2 (n), C 3,C 4 = C 4 (n), and C 5 such that the following property is satisfied: If µ is a locally finite Borel measure, Q 0 a fixed cube in R n, Ĵ 2 Ĵ Q 0,µ 2, and M is a positive number such that Ĵ 2 (x) M for all x supp(µ) C 5 Q 0, then there exists a curve Ɣ, Ɣ C 5 Q 0, with length at most C 1 e C 2 M l(q 0 ) and measure at least C 1 3 e C 4 M µ(q 0 ) These two exponential bounds are both achieved by certain measures (up to logarithmic comparability) REMARK 49 We are usually interested in the case where the support of µ is contained in Q 0 The proof of Theorem 48 contains two parts The first part validates the theorem when M = δ, a sufficiently small constant (compare with Léger [17, proposition 12]) It is described in Section 6 The second part uses stopping-time arguments and the first part in order to prove the theorem for any value of M Itis described in Section 7

10 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1303 The theorem provides estimates for some optimization problems For example, fix a measure µ on R n, a cube Q 0 in R n, and a positive integer L Among all curves contained in C 5 Q 0 and with length L (or less than or equal to L), find the ones with maximum measure In view of the above theorem, if Ĵ 2 M and L C 1 e C2 M, then µ(ɣ) (44) r max (µ, L) := max l(ɣ)=l µ(q 0 ) l(q 0) (C 1 C 3 ) 1 e (C 2+C 4 ) M L Ɣ C 5 Q 0 A special case of this problem is the bank robber problem for atomic measures (see, eg, [21]) The dual problem is obtained by fixing a constant m, 0< m µ(q 0 ), and looking for the curve of shortest length among all curves contained in C 5 Q 0 with measure m Discrete analogues of this problem are the k-mst and the quota-driven salesman problems (see, eg, [21]) A further optimization can be applied to the above problems The corresponding version for the first problem goes as follows: Find the largest value of L among all values maximizing (or nearly maximizing) r max (µ, L) Denote this value by L opt Also denote by Ɣ opt any curve such that Ɣ opt C 5 Q 0 and l(ɣ opt ) = L opt We remark that the ratio r max (µ, L) is a decreasing function of L Therefore the last optimization problem is interesting only when r max (µ, L) is constant around L = 0 (or slowly varying around L = 0 when using the nearly optimal version) In this case the problem has the interpretation of finding the longest curve with maximum uniform linear density The exponential estimates implied by Theorem 48 for the above optimization problems are usually not satisfactory for particular examples However, there exist measures for which these bounds are achieved (but not necessarily with the constants C 1, C 2, C 3, and C 4 ) In Section 5 we provide three examples where the lower exponential bound of r max in equation (44) is obtained In one of these examples the unique curve Ɣ opt achieves the exponential length and measure estimates of Theorem 48 The next theorem slightly generalizes Theorem 48 We verify it in Section 76 by slightly modifying the proof of the above theorem THEOREM 410 There exist positive constants C 0 = C 0 (n), C 1 = C 1 (n), C 3, and C 5 such that the following property is satisfied: If µ is a locally finite Borel measure, Q 0 a fixed cube in R n, Ĵ 2 Ĵ Q 0,µ 2, and A is a positive number such that e C 0 Ĵ 2 (x) dµ(x) A µ(q 0 ), C 5 Q 0 then there exists a curve Ɣ, Ɣ C 5 Q 0, with length at most C 1 A l(q 0) and measure at least C 1 3 A 1 µ(q 0 )

11 1304 G LERMAN REMARK 411 Theorem 48 is a special case of the above theorem Indeed, if Ĵ 2 (x) M for all x supp(µ) C 5 Q 0, then Ĵ 2 (x) M for all x C 5 Q 0Now set A := e C 0 M so that the condition of the latter theorem is satisfied Theorem 48 is then concluded with the following constants: C 2 C 4 := C 0, C 1 = C 1, C 3 = C 3, and C 5 = C 5 REMARK 412 In the very special case of one-dimensional Ahlfors-regular measures, the function e C 0 Ĵ 2 in the above integral can be replaced by 1 + C 0 Ĵ 2,or equivalently 1 + Ĵ 2 (see Theorem 51 for a stronger statement) REMARK 413 In general, it is impossible to replace the function e C 0 Ĵ 2 in the above integral by a polynomial function of Ĵ 2 We verify this proposition in example 4 of Section 5 We next define the most general class of measures (with respect to the fixed cube Q 0 ) for which Theorem 410 applies Denote I C (µ) I Q 0 C (µ) := e C Ĵ2(x) dµ(x), C 5 Q 0 where Ĵ 2 Ĵ Q 0,µ 2 Fix C0, the minimal (or nearly minimal) constant among all constants C 0 n for which Theorem 410 is satisfied Define the class of measures J (with respect to the cube Q 0 ) as follows: (45) J n J n,q 0 = { µ : µ is a locally finite Borel measure on R n and I Q 0 C0 (µ) < } REMARK 414 It is possible to elaborate on the proof of the above theorem to obtain that all measures in J n are weakly one-dimensional rectifiable inside the cube Q 0 (see [16] for a more general result) We end this section by showing that the L 1 norms of J 2 and Ĵ 2 are comparable This fact can be used to replace Ĵ 2 by J 2 in the above two theorems (the way of doing it follows from [16]) LEMMA 415 If µ is a locally finite Borel measure on R n,q 0 is a cube in R n, and J 2 and Ĵ 2 are defined with respect to Q 0, then J 2 (x)dµ(x) n Ĵ 2 (x)dµ(x) PROOF: We first develop two simple expressions for the integrals J 2 (x)dµ(x) and Ĵ 2 (x)dµ(x)

12 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1305 Let Q be a cube in D Recall that H Q denotes the set of all best L 2 lines for Q Note that if L Q H Q, then ( ) 1 dist(z, L Q ) 2 dµ(z) if µ(q) >0 β2 2 (Q) = µ(q) l(q) Q 0 otherwise Combine the above equation with equation (41) to obtain that if {L Q } Q D is any collection of best L 2 lines in {H Q } Q D, then (46) J 2 (x)dµ(x) = Q D ( 1 dist(y, L Q ) µ(q) l(q) ) 2 χ Q (y)dµ(y)χ Q (x)dµ(x) The application of Fubini s theorem to equation (46) results in the following simple expression for J 2 (x)dµ(x): (47) J 2 (x)dµ(x) = ( ) dist(y, L Q ) 2 dµ(y) l(q) Q D Q A similar estimate for Ĵ 2 (x)dµ(x) is derived as follows: If Q is a cube in D, fix a cube ˆQ(Q) in P(Q) and a best L 2 line ˆP Q for ˆQ(Q) so that the following equation is satisfied: ( ) 1 dist(z, ˆP Q ) 2 ˆβ 2 2 (Q) = dµ(z) if µ(q) >0 µ(q) l(q) Q 0 otherwise By combining the above equation with equation (43) and Fubini s theorem, we obtain the following expression for Ĵ 2 (x)dµ(x): (48) Ĵ 2 (x)dµ(x) = ( ) dist(y, ˆP Q ) 2 dµ(y) l(q) Q D Q Note that β 2 (Q) ˆβ 2 (Q) for any cube Q in D Thus in order to prove the lemma, it is sufficient to control Ĵ 2 (x)dµ(x) by J 2 (x)dµ(x) The relevant inequality follows from two observations First, ( ) dist(z, ˆP Q ) 2 ( ) dµ(z) 4 j 1 dist(z, ˆP Q ) 2 dµ(z) l(q) l(q) Q ˆQ(Q)

13 1306 G LERMAN Second, if Q is a cube in D, then #{Q : ˆQ(Q ) = ˆQ(Q)} C(n) for some large constant C(n) By combining the above two equations with equations (47) and (48), we obtain that Ĵ 2 (x)dµ(x) 4 j 1 C(n) J 2 (x)dµ(x) and thus prove the lemma 5 Examples of Sets and Their L 2 Jones Quantities In this section we introduce five examples where we clarify some of the definitions of the previous section and also Theorems 48 and 410 In each one of the examples there is a special measure or a class of measures on R n and a fixed cube Q 0 in R n We estimate the corresponding Jones function Ĵ 2 = Ĵ Q 0,µ 2 and also evaluate the sharp estimates for the optimization problems discussed in the previous section We then compare these estimates with the ones implied by Theorem 48 (if applicable) In the first three examples the lower bound in equation (44) is optimal up to logarithmic comparability Moreover, in the third example both exponential estimates of Theorem 48 are sharp for the curve Ɣ opt The fourth example implies that the conditions controlling the size of Ĵ 2 in Theorems 48 and 410 cannot be replaced with L p bounds of Ĵ 2 Finally, we discuss the special class of one-dimensional Ahlfors-regular measures In this example, one has a linear bound in M for the length of a curve with full measure 51 Example 1 This example is practically the same as the optimal example of [4] Fix ε>0 and assume for simplicity that n = 2 Construct the curves {Ɣ i } i 0 as follows The curve Ɣ 0 is a fixed line segment in R 2 Assume that the curve Ɣ i 1, i 1, has been constructed and that it is piecewise linear Form Ɣ i by replacing each line segment I composing Ɣ i 1 with the two opposite sides of an isosceles triangle with base I and height ε l(i ) (by opposite we mean a negative scalar product between the current height vector and the previous-stage height vector that was used in forming the line segment I ) Fix a sufficiently large integer N, N 1/ε 2, and denote by µ N the arc length probability measure of Ɣ N That is, dµ N = dh1 ƔN l(ɣ N ) Let Q 0 be a cube containing the curve Ɣ N such that l(q 0 ) l(ɣ 0 ) We make the following observations: First, l(ɣ N ) l(ɣ 0 ) = (1 + ε2 ) N 2 and consequently log l(ɣ N ) l(ɣ 0 ) N ε2 2

14 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1307 Second, there exists a constant C such that if Ɣ is any curve in R 2, then µ N (Ɣ ) (51) µ N (Q 0 ) l(ɣ 0) N ε 2 e C l(ɣ N ) Third, (52) M := max Ĵ 2 N ε 2 x Q 0 Indeed, if Q is a cube in D, then ˆβ 2 (Q) ε Also, if Q Ɣ N =, then ˆβ 2 (Q) = 0 Equations (51) and (52) imply that the lower bound in equation (44) is optimal (up to logarithmic comparability) if L l(ɣ N ) and M := M Note that there exist curves that obtain both exponential bounds of Theorem 48 up to logarithmic comparability (take, eg, Ɣ to be a subcurve of Ɣ N with length (1 + ε 2 ) N/4 ) However, the unique curve Ɣ opt Ɣ N only obtains the exponential bound of the length estimate but not the one of the measure estimate (µ N (Ɣ opt ) = 1) We remark that it is possible to construct an analogous n-dimensional example where Ĵ 2 is proportional to the dimension n The corresponding sequence of curves {Ɣ i } i 0 is formed by replacing recursively each edge of Ɣ i 1, i 1, with n smaller intervals of different directions in R n 52 Example 2 Fix ε>0and a large integer N, N 1 ε log 1 Assume for simplicity that ε n = 2 Define { 1 I i =[0, 1] and dµ N = 8 i }, 0 i N, I N+1 =[0, 1] {0}, N ε (1 ε) i dh 1 Ii + (1 ε) N+1 dh 1 IN+1 i=0 Let Q 0 be a cube in R 2 containing the support of µ N and with side length 2 Note that µ N (R 2 ) = 1, µ N (I N+1 ) = (1 ε) N+1, µ N (I i ) ε, i = 0,,N The assumption N ε 1 log ε 1 implies that µ N (I i ) µ N (I N+1 ), i = 0,,N Therefore the ratio µ N (Ɣ)/l(Ɣ) is optimized for curves Ɣ that are line segments contained in [0, 1] {0} Moreover, (53) r max (µ N, L) = 2 µ N (I N+1 ) = 2 (1 ε) N+1 2 e N ε C, 0 < L 1, and Ɣ opt [0, 1] {0} We next compare the above upper bound for r max (µ N, L), 0< L 1, with the lower bound given in equation (44) Note that Ĵ 2 (x) i ε + C for any x I i, i = 0,,N + 1

15 1308 G LERMAN Therefore, (54) M := max Ĵ 2 (x) N ε x Q 0 Equations (53) and (54) imply that the lower bound in equation (44) is optimal (up to logarithmic comparability) when using M M Also note that the measure of Ɣ opt decays exponentially in M like the estimate of Theorem 48; however, l(ɣ opt ) = 1 53 Example 3 We combine the previous two examples to obtain a measure with a unique curve Ɣ opt such that Ɣ opt achieves both the length and measure estimates of Theorem 48 Fix ε>0 and 1 ε N 1 ε log 1 ε Let Ɣ N = N+1 where γ i is the i th iteration of the von Koch type curve described in example 1 with parameter ε and with the initial interval γ 0 := [0, 1] { 1 8 i } Define dµ N = N i=0 i=0 γ i, ε (1 ε) i dh1 γi l(γ i ) + (1 ε)n+1 dh1 γn+1 l(γ N+1 ) Let Q 0 be a cube in R 2 containing the support of µ N with side length comparable to 1 Note that Ɣ opt = γ N+1, µ N (Ɣ opt ) = (1 ε) N+1 e N ε C, l(ɣopt ) = (1 + ε) N 2 N ε e C Also note that M := max Ĵ 2 N ε x Q 0 Therefore by taking M = M, the two exponential bounds of Theorem 48 are obtained by the curve Ɣ opt but with different constants 54 Example 4 This example can be thought of as a limit of example 2 as N It shows that the integral condition in Theorem 410 cannot be replaced with an L p bound of J 2, where 0 < p < Fix 0 <ε 1and set { } 1 I i := [0, 1], i = 0, 1, Define the probability measure µ as follows: dµ = ε (1 ε) i dh 1 Ii i=0 8 i

16 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1309 Fix Q 0 a cube in R 2 containing the support of µ with side length 2 Note that Ĵ 2 (x) i ε + C for any x I i Consequently, Ĵ 2 L (µ) =, e C 0 Ĵ 2 (x) dµ(x) =, Ĵ 2 BMO(µ) <, and Ĵ 2 L p (µ) = O(p + 1), 0 < p < Also note that r max (µ, L) <2 ε for any L 0 On the other hand, taking ε to zero does not affect the finite values of Ĵ 2 L p (µ), 0 < p <, and Ĵ 2 BMO(µ) We thus conclude that it is impossible to replace the uniform bound of Ĵ 2 in Theorem 48 or the exponential integral bound in Theorem 410 by an L p or BMO norm of Ĵ 2 55 Example 5 Fix a cube Q 0 in R n Assume that µ is a one-dimensional Ahlfors-regular probability measure on R n with compact support Furthermore, assume that (55) supp(µ) Q 0, diam(supp(µ)) n l(q 0 ) = 1, and M := Ĵ 2 L (µ) < where Ĵ 2 Ĵ Q 0,µ 2 Note that in this special case β 2 (Q) n ˆβ 2 (Q) for all cubes Q in D ( D D +3 Q 0 ) and consequently Ĵ 2 n J 2 Moreover, (56) β2 2 (Q) l(q) n Q D Q D β2 2 (Q) µ(q) = J 2 (x)dµ(x) n Ĵ 2 (x)dµ(x) M Combine equations (55) and (56) with the length estimate in [23] (this estimate can be adapted for Ahlfors-regular measures instead of sets) to obtain consequently, inf Ɣ rectifiable, µ(ɣ)=1 l(ɣ) n M + 1 ; r max (µ, L) n M, 0 < L inf Ɣ rectifiable, µ(ɣ)=1 l(ɣ) In this case, the estimate in equation (44) is far from optimal if M 1 In this example, the optimal characterization of curvelike structure by Jones quantities follows from [14, theorem 1], [22] and [24, theorem 13] By applying equation (56), we formulate this characterization in terms of the Jones function Ĵ 2 (x) as follows:

17 1310 G LERMAN THEOREM 51 [14, 22, 24] If Q 0 is a cube in R n, µ a one-dimensional Ahlforsregular probability measure on R n with compact support satisfying equation (55), and Ĵ 2 Ĵ Q 0,µ 2, then there exists a rectifiable curve with full measure if and only if Ĵ 2 dµ(x) < Moreover, l(ɣ) n inf Ɣ rectifiable, µ(ɣ)=1 6Theδ Theorem (1 + Ĵ 2 )dµ(x) In this section we formulate and prove Theorem 63 It is a special case (see Remark 62) of Theorem 48, where M = δ, a small constant NOTATION AND DEFINITIONS 61 Throughout this section we fix the cube Q 1 instead of Q 0 This notation is consistent with that of Section 7, where Q 1 is used for the δ-construction We also redefine here the grid D as follows: D := D +0 Q 1 D(Q 1 ), whereas in other sections D D +3 Q 0 Note that the Jones quantities ( ˆβ 2 and Ĵ 2 ) are redefined here with respect to the cube Q 1 and the grid D +0 Q 1 (instead of Q 0 and D +3 Q 0 ) REMARK 62 The redefinition of the grid D (see above notation and definitions) makes the estimates of the following theorem sharper than those of the special case of Theorem 48, where M = δ THEOREM 63 There exist positive constants C 1 = C 1 (n), C 2 = C 2 (n), C 4 = C 4 (n), ε 1 = ε 1 (n), and δ 0 = δ 0 (n) such that the following proposition is satisfied: If µ is a locally finite Borel measure on R n,q 1 is a cube in R n, δ δ 0, and (61) Ĵ 2 (x) Ĵ Q 1,µ 2 (x) δ for any x supp(µ) (1 + ε 1 ) Q 1, then there exists a curve Ɣ such that l(ɣ) C 1 e C 2 δ l(q 1 ) and µ(ɣ (1 + ε 1 ) Q 1 ) e C 4 δ µ(q 1 ) We build the curve Ɣ suggested by the above theorem in a multiscale fashion At each stage l of the basic construction we form a piecewise linear curve Ɣ l,a stopping-time region S l, and a strip E l around Ɣ l Sl c The length of edges in Ɣ l and the thickness of E l are of order l(q 1 ) C l L The general construction is obtained by restarting the basic construction repeatedly at stopping-time cubes contained in S l It results in curves Ɣ N, N 1, and strips around it Ẽ N S N, N 1, with thickness of order l(q 1 ) C N L The estimates of the length and measure of Ɣ are based on simple geometric arguments together with martingale-type and stopping-time techniques The geometric estimates follow mainly from three elementary theorems: Pythagoras theorem, Jensen s inequality, and Chebyshev s inequality Jensen s and Chebyshev s inequalities are used in verifying the following local properties:

18 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1311 LEMMA 64 If µ is a locally finite Borel measure on R n, Q and ˆQ are cubes in R n such that Q supp(µ) ˆQ supp(µ), LˆQ is a best L 2 line in ˆQ, and z Q denotes the center of mass of the cube Q, then dist(z Q, L ˆQ ) β 2 (Q, ˆQ) l(q) PROOF: This is a simple application of Jensen s inequality as follows: dist 2 (z Q, L ˆQ ) = dist2 ( Q Q z dµ(z) µ(q), L ˆQ ) dist 2 (z, L ˆQ )dµ(z) µ(q) β2 2 (Q, ˆQ) l(q) 2 LEMMA 65 If µ is a locally finite Borel measure on R n, Q and ˆQ are cubes in R n such that Q supp(µ) ˆQ supp(µ), and L ˆQ is a best L 2 line for ˆQ, then µ{x Q : dist(x, L ˆQ ) ε l(q)} 1 ε 2 β2 2 (Q, ˆQ) µ(q) PROOF: This is a straightforward application of Chebyshev s inequality Indeed, µ{x Q : dist(x, L ˆQ ) ε l(q)} 1 ε dist 2 (x, L ˆQ ) dµ(x) 2 l(q) 2 µ(q) µ(q) Q 1 ε 2 β2 2 (Q, ˆQ) µ(q) The martingale-type idea of the proof is used in the basic construction and is described heuristically as follows Following Bishop and Jones [4], we form a sequence of functions {F l } l 0 such that 0 = F 0 F 1 F l δ We show that there exists a sufficiently large constant C 2 = C 2 (n) such that the sequence e C 2 F l, l 0, is a supermartingale in the following sense: e C2 Fl(x) ds(x) e C2 Fl 1(x) ds(x), l 1 Ɣ l 1 Ɣ l This estimate follows mainly from Lemma 64 and Pythagoras theorem (see Section 63) By applying it repeatedly, we obtain the following uniform bound on the lengths of the approximating curves: l(ɣ l ) e C 2 δ l(ɣ 0 )

19 1312 G LERMAN Similarly, we show that there exists a sufficiently large constant C 4 = C 4 (n) such that the sequence e C 4 F l, l 0, is a submartingale in the following sense: e C 4 F l (x) dµ(x) e C 4 F l 1 (x) dµ(x), l 1 E l S l E l 1 S l 1 This estimate follows mainly from Lemma 65 (see Section 64) By applying it repeatedly, we obtain the following uniform lower bound on the measures of the approximating curves: µ(e l S l) e C 4 δ µ(q 1 ) If there are no stopping-time cubes (S l =, l 0), then a limit argument concludes the length and measure estimates of the curve Ɣ itself from the above estimates Otherwise, we normalize the above estimates of the basic construction at each stopping-time cube and control the sum of the lengths of the stopping-time cubes to conclude the length and measure estimates for Ɣ We remark that condition (61) in Theorem 63 implies that (62) ˆβ 2 (Q) < δ for any Q D The above weak condition is used in our proof, but it is not sufficient to imply the theorem Indeed, let K be a generalized von Koch arc in the plane with angles of size δ, and let s be the Hausdorff dimension of K (see, eg, example 2 in Section 5 for constructing a sequence of curves converging to K ; in that example ε = δ) Construct an s-dimensional Frostman s measure µ with support K (see, eg, [20, proof of theorem 88]) This measure satisfies equation (62) However, any subset of the plane with positive such measure has an infinite length Different constants appear in the proof, sometimes with the same notation They might depend on the dimension of the ambient space n The following large constants are used extensively: A 0, A 1 = A 1 (n), A 2 = A 2 (n), C L = C L (n), j 0, and j 1 = j 1 (n) We have initialized them as follows: A 0 = 5, A 1 = A 1 (n) is the smallest integer greater than 6480 n e such that log 2 A 1 is an integer, A 2 = 128 A 1, C L = A 1, j 0 = 2, j 1 = 16 log 2 A 1, and ε 1 = ε 1 (n) = 2/(3 C L 2) We also initialize the constant δ 0 = δ 0 (n) so that δ 0 = 1/(6336 e C 3 L )Wefix throughout this section a constant δ such that δ δ 0 Our choice of constants is not optimal We do not worry here about the most desirable numbers We recall that the definitions of the set P(Q) and the Jones quantities ˆβ 2 and Ĵ 2 depend on the fixed values of the constants j 0 and j 1 In Section 61 we list properties of the l th level basic construction and present the zeroth-level construction In Section 62 we describe the basic inductive construction at stage l, l 1 We verify the relevant length and measure estimates in Sections 63 and 64, respectively In Section 65 we formulate the general construction by restarting the basic construction at stopping-time cubes Finally, in

20 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1313 Section 66 we validate the length and measure estimates of the general construction and consequently conclude Theorem 63 Some of the highly technical but elementary computations of this section appear in the appendix 61 Properties of the Basic Construction In this section we describe the zeroth-level basic construction and list some properties of the l th -level construction In Section 62 we build the sets of level l, l 1, while assuming that the sets of level l 1, l 1, possess the properties given here The construction implies that the same properties are then satisfied at level l The following sets appear at stage l of the basic construction: Ɣ l, M l, E l, L l, M l, E l, S l, and S l The curve Ɣ l approximates the ultimate curve Ɣ outside S l 1 at scale of order diam(k ) C l L The set M l, l 1, is a collection of cubes surrounding the curve Ɣ l 1 outside S l 1 The union of all cubes in M l is the set E l The set L l is composed of centers of masses of cubes in M l We use the points in L l as vertices of the curve Ɣ l The set S l is a collection of disjoint stoppingtime cubes The union of these cubes is denoted by S l and is referred to as the l th -level stopping-time region The sets M l and E l are obtained from M l and E l by excluding the stopping-time cubes in S l NOTATION 66 Denote l j = l(q 1), j 0 C j L The letter l is used both for denoting length and for indexing levels This convention results in the following notation: l l 1 and l l DEFINITION 67 ( Phantom point x with respect to x and y) Ifx and y are two different points in R n, then a point x in R n is a phantom point with respect to x and y if and only if the following equations are satisfied: x Q(x, 4 A 0 l l 1 ) and [x, y] [ x, y] DEFINITION 68 (Trivial and nontrivial components of Ɣ l 1 S c l 1 ) A connected component γ of Ɣ l 1 S c l 1 is trivial if and only if γ L l 1 = Similarly, γ is nontrivial if and only if γ L l 1 = The zeroth-stage construction goes as follows: Fix a cube ˆQ 1 in P(Q 1 ) such that l( ˆQ 1 ) = 2 j 0 l(q 1 ) and dist( Q 1, ˆQ 1 ) 1 C L l(q 1 ) The existence of such a cube follows from Proposition 32 Fix a best L 2 line for ˆQ 1 and denote it by L ˆQ 1 Define (63) Ɣ 0 = L ˆQ 1 (9 A 0 ) Q 1, M 0 =, M 0 ={Q 1}, E 0 = E 0 = Q 1, (64) L 0 = A 0 Q 1 L ˆQ 1 {z 1, z 2 }, and S 0 = S 0 = We remark that some of the above definitions are artificial They are chosen so that we do not need to distinguish between the first-level construction and the ones

21 1314 G LERMAN at higher levels In particular, there is no special meaning to the fact that M 0 is empty The properties of the sets of level l 1, l 1, are listed as follows All of them are satisfied when l = 1 (1) If x L l 1, then (65) Q(x, A 0 l l 1 ) L l 1 \{x} = (2) If l > 1, Q M l 1, and z Q is its center of mass, then there exists a point x L l 1 Sl 1 c such that z Q Q(x, A 0 l l 1) (3) If l > 1 and x L l 1 Sl 1 c, then there exists a cube Q in M l 1 such that x is its center of mass (4) The cubes in S l 1 are disjoint (5) The connected components of the set Ɣ l 1 Sl 1 c are piecewise linear curves with obtuse angles between neighboring edges (6) If γ is a nontrivial connected component of Ɣ l 1 Sl 1 c, then there are at least two points in γ L l 1 (7) If γ is a nontrivial, connected component of Ɣ l 1 Sl 1 c and z 1,,z m, m 2, are the points in γ L l 1 indexed so that the union m j=2 (z j 1, z j ) is disjoint and contained in Ɣ l 1, then (66) z j 1, z j+1 Q(z j, 4 A 1 l l 1 ) for all 2 j m 1 (8) If γ is a nontrivial connected component of Ɣ l 1 Sl 1 c, the points z 1,, z m, m 2, are defined as in property 7, z 0 is the phantom point with respect to z 1 and z 2, and z m+1 is the phantom point with respect to z m and z m 1 Then m+1 γ [z j 1, z j ] j=1 (9) If γ is a nontrivial connected component of Ɣ l 1 Sl 1 c, the points z 0,,z m, z m+1, m 2, are defined as in property 8, and Q is a cube in D with side length l l such that ( m+1 ) Q γ \ [z j 1, z j ] =, j=1 then Q E l 1 = (10) If γ is a trivial connected component of Ɣ l 1 Sl 1 c and Q is a cube in D with side length l l such that Q γ =, then Q E l 1 = (11) If γ is a nontrivial connected component of Ɣ l 1 Sl 1 c and the points z 0,,z m, z m+1, m 2, are defined as in property 8, then for any point x in

22 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1315 m+1 j=1 [z j 1, z j ] and a different nontrivial connected component γ of Ɣ l 1 Sl 1 c, Q(x, 4 A 1 l l 1 ) γ = 62 The Basic Construction for Stage l In this section we construct the sets M l, E l, L l, Ɣ l, S l, S l, M l, and E l, l 1 We assume that M l 1, E l 1, L l 1, Ɣ l 1, S l 1, S l 1, M l 1, and E l 1 have been formed The idea is to build the l th -level sets locally around each point in L l 1 and then patch up the local parts to obtain the whole sets We define the sets M l and E l as follows: { } Q : Q D, l(q) = l (67) M l = l,µ(q) >0, Q E l 1 =, and there exists x Ɣ l 1 such that x 2 3 Q and (68) E l = Q Q M l Before defining the set L l, we present some notation and formulate some properties of the curve Ɣ l 1 (Lemma 610) These properties are needed in order to properly define the set L l NOTATION AND DEFINITIONS 69 We refer to the nontrivial connected components of the set Ɣ l 1 Sl 1 c as its segments Throughout the rest of this section, we fix a segment γ of Ɣ l 1 Sl 1 c We also fix the corresponding points z 0,,z m, z m+1, m 2, defined in properties 7 and 8 of Section 61 If l > 1 and 2 j m, set a cube Q j in D with side length A 2 l l 1 satisfying the equation ( zj 1 + z j (69) Q j Q, A ) l l 1 Fix a best L 2 line for Q j and denote it by L Q j Ifl = 1, then m = 2 and ˆQ 2 is the fixed cube ˆQ 1 in P(Q 1 ) (see Section 61) The line L ˆQ 2 is the best L 2 line L ˆQ 1 in ˆQ 1 such that Ɣ 0 L ˆQ 1 Let H j, j = 1,,m, be the hyperplane containing z j and bisecting the angle between the line segments [z j 1, z j ] and [z j, z j+1 ]LetH 0 be the hyperplane containing z 0 and parallel to H 1, and let H m+1 be the hyperplane containing z m+1 and parallel to H m Denote by B j 1, j,1 j m + 1, the closed region bounded between the hyperplanes H j 1 and H j+1 Also denote by I j 1, j and Ĩ j 1, j,1 j m + 1, the regions ( ) zj 1 + z j I j 1, j = B j 1, j Q, 4 A 1 l l 1 \ H j 1 2

23 1316 G LERMAN and Ĩ j 1, j = Define the set T Q j as follows: min( j+3,m+1) i=max( j 3,1) I i 1,i T Q j = { Q : Q M l 1 M l, Q Ĩ j 1, j =, and Q j P(Q) } Fix a cube Q j max in T Q j that satisfies the equality ( ) ( ) β 2 Q j max, Q j = max β 2 Q, ˆQ j Q T Q j Observe that ( ) ( ) (610) β 2 Q j max, ˆQ j ˆβ 2 Q j max We next formulate some properties of the fixed segment γ with vertices z 0,,z m+1 (see Notation and Definitions 69) LEMMA 610 The following properties are satisfied: (i) If 2 j m and max(1, j 2) p min(m, j + 2), then dist(z p, L ˆQ j ) ( ) (611) β 2 Q j l max, ˆQ j l 1 Also, dist(z 0, L ˆQ (612) 2 ) ( ) 9 β 2 Q 2 l max, ˆQ 2 l 1 and dist(z m+1, L ˆQ (613) m ) ( ) 9 β 2 Q m l max, ˆQ m l 1 (ii) If 0 j m + 1 and max(1, j 1) p min(m + 1, j + 1), then (614) cos(ang([z p 1, z p ], H j )) 3 A 0 β 2 ( Q j max, ˆQ j ) PROOF: (i) If l = 1, then this property is trivial due to the fact that z 1, z 2 Ɣ 0 L ˆQ 2 (see equation (64)) If, on the other hand, l > 1, then equation (611) follows mainly from Lemma 64 Indeed, fix integers p and j such that 1 j m and max(1, j 2) p min(m, j + 2) Recall that z p L l 1 Sl 1 c Apply repeatedly equation (66) and obtain that ( ) zj 1 + z j z p Q, 10 A 1 l l 1 2 Let Q be a cube in M l 1 such that z p is its center of mass (see property 3 in Section 61) The above equation implies that Q ˆQ j

24 QUANTIFYING CURVELIKE STRUCTURES OF MEASURES 1317 Apply Lemma 64 to the cubes Q and ˆQ j and conclude equation (611) We next verify equation (612); equation (613) is derived similarly Note that (615) dist(z 0, L ˆQ 2 ) dist(z 1, L ˆQ 2 ) + dist(z 0, z 1 ) sin θ, where θ is the angle between the line segment [z 1, z 2 ] (or equivalently [z 0, z 1 ]) and the line L ˆQ 2 Equation (611) implies the following estimate: (616) sin θ 2 β 2(Q 1 max, ˆQ 1 ) l l 1 dist(z 1, z 2 ) Recall that (617) z 0 Q(z 1, 4 A 0 l l 1 ) and that (618) z 2 Q(z 1, A 0 l l 1 ) By applying equations (611) and (616) through (618) to equation (615), we conclude equation (612) (ii) Equation (611) and the separation properties of the points {z i } i=1 m (see equations (65) and (66)) imply that (619) sin(ang([z p 1, z p ], L ˆQ j )) 2 A 0 β 2 ( Q j max, ˆQ j ), where 1 j m and max(1, j 1) p min(m, j + 1); consequently, sin(ang([z p 1, z p ], [z p, z p+1 ])) 4 A 0 β 2 ( Q j max, ˆQ j ), 2 p m 1 We prove (614) from the above equation and the fact that cos(2 δ/a 0 ) 2 3 We next construct the sets L l (γ ) for each segment γ of Ɣ l 1 Sl 1 c (see Notation and Definitions 69) The set L l is formed as the union of the sets L l (γ ) over all segments γ of Ɣ l 1 Sl 1 c Let A l (γ ) be the following set of centers of masses: { ( m+1 ) } A l (γ ) = z Q : Q M l, Q I j 1, j = Let L l (γ ) be a subset of A l (γ ) that is maximally separated by l distances A 0 l l That is, (620) Q(x, A 0 l l ) L l (γ ) \{x} = for all x L l (γ ) and (621) Q(x, A 0 l l ) A l (γ ) x L l (γ ) j=1

25 1318 G LERMAN Note that if x L l (γ ), then there exists a unique j, 1 j m + 1, such that x I j 1, j This observation follows from Lemmata 64 and 610 The curve Ɣ l is built around each segment γ of Ɣ l 1 Sl 1 c by using the set L l (γ ) as vertices Before presenting the construction, we introduce some technical definitions NOTATION AND DEFINITIONS 611 Denote by L j 1, j the line containing the segment [z j 1, z j ] and by P j 1, j the projection onto L j 1, j The linear ordering on L l (γ ) is defined as follows Let x 1, x 2 L l (γ ) and assume that x 1 I j1 1, j 1 and x 2 I j2 1, j 2 If j 1 < j 2, then x 1 x 2 If j 1 = j 2, then we order x 1 and x 2 according to their projections on L j 1, j ; that is, let t 1 and t 2 be defined by the equation P j 1 1, j 1 x i = z j1 1 + t i (z j1 z j1 1), i = 1, 2 We say that x 1 x 2 if and only if t 1 t 2 We order the points v l 1,,vl k in L l(γ ), where k = k(γ ), so that v l 1 vl k We also define the points v l 0 and vl k+1 as follows: vl 0 = z 0 and v l k+1 = z m+1 The points in L l (γ ) are classified into three different types We say that v l i, i = 1,,k, is a type 1 point if (622) v l i 1,vl i+1 / Q( v l i, 4 A 1 l l ) We say that vi l, i = 1,,m, is a type 2 point if either one of the following conditions is satisfied: (623) vi+1 l Q( vi l, 4 A ) 1 l l and vi 1 l / Q( vi l, 4 A ) 1 l l or (624) vi 1 l Q( vi l, 4 A ) 1 l l and v l i+1 / Q( v l i, 4 A 1 l l ) We say that vi l, i = 1,,m, is a type 3 point if vi 1 l,vl i+1 Q( vi l, 4 A ) 1 l l We next extend the sequence {vi l}k+1 i=0 to a larger sequence {vl i }N+1 i=0 of vertices of Ɣ l around the corresponding segment γ of Ɣ l 1 Sl 1 c Ifvl i,1 i k, is a type 2 point and if it satisfies equation (623), then denote by ṽi l the phantom point with respect to vi l and vl i+1 and modify the whole sequence as follows: vi l := ṽi l, vl i+p := vl i+p 1, 1 p k + 1 i If vi l,1 i k, is a type 2 point that satisfies equation (624), then denote by ṽl i the phantom point with respect to vi l and vi 1 l and modify the whole sequence as follows: vi+1 l := ṽl i, vl i+p := vl i+p 1, 1 < p k + 1 i If v = v l i, i = 0,,N, define v+ = v l i+1 Similarly, if v = vl i, i = 1,,N + 1, define v = v l i 1