Embedding snowflakes of Carnot groups into bounded dimensional Euclidean spaces with optimal distortion
aa r X i v : . [ m a t h . M G ] A p r EMBEDDING SNOWFLAKES OF CARNOT GROUPS INTO BOUNDEDDIMENSIONAL EUCLIDEAN SPACES WITH OPTIMAL DISTORTION
SANG WOO RYOO
Abstract.
We show that for any Carnot group G there exists a natural number D G suchthat for any 0 < ε < / G, d − εG ) admits a bi-Lipschitz embeddinginto R D G with distortion O G ( ε − / ). This is done by building on the approach of T. Tao(2018), who established the above assertion when G is the Heisenberg group using a newvariant of the Nash–Moser iteration scheme combined with a new extension theorem fororthonormal vector fields. Beyond the need to overcome several technical issues that arisein the more general setting of Carnot groups, a key point where our proof departs fromthat of Tao is in the proof of the orthonormal vector field extension theorem, where weincorporate the Lov´asz local lemma and the concentration of measure phenomenon on thesphere in place of Tao’s use of a quantitative homotopy argument. Introduction
A map f : ( X, d X ) → ( Y, d Y ) between two metric spaces ( X, d X ) and ( Y, d Y ) is said tohave distortion at most D if there exists a constant C > Cd X ( x , x ) ≤ d Y ( f ( x ) , f ( x )) ≤ CDd X ( x , x ) , ∀ x , x ∈ X. For 0 < ε <
1, the (1 − ε )-snowflake of a metric space ( X, d X ) is defined to be the metricspace ( X, d − εX )(it is clear that d − εX also defines a metric on X ).The metric spaces that we will focus on this paper are Carnot groups. Following [10], aCarnot group is a 5-tuple ( G, δ λ , ∆ , k · k , d G ), where: • The Lie group G is a stratified group, i.e., G is a simply connected Lie group whoseLie algebra g admits the direct sum decomposition g = V ⊕ V ⊕ · · · ⊕ V s , where V s +1 = 0 and V i +1 = [ V , V i ] for i = 1 , · · · , s . • For each λ ∈ R + , the linear map δ λ : g → g is defined by δ λ | V i = λ i id V i , i = 1 , · · · , s. • The bundle ∆ over G is the extension of V to a left-invariant subbundle:∆ p := ( dL p ) e V , p ∈ G. Date : April 17, 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Carnot group, snowflake embedding, Nash-Moser iteration, Lov´asz local lemma,concentration of measure.Acknowledgment: This work will be part of a doctoral dissertation under the supervision of Assaf Naorat Princeton University. I thank him for suggesting the problem of establishing the results of [15] in thesetting of general Carnot groups, and for helpful discussions. • The norm k · k is initially defined on V , and is then extended to ∆ as a left-invariantnorm: k ( dL p ) e ( v ) k := k v k , p ∈ G, v ∈ V . • The metric d G on G is the Carnot-Carath´eodory distance associated to ∆ and k · k ,i.e., d G ( p, q ) := inf (cid:26)Z k ˙ γ ( t ) k dt : γ ∈ C ∞ pw ([0 , G ) , γ (0) = p, γ (0) = q, ˙ γ ∈ ∆ (cid:27) , p, q ∈ G, where C ∞ pw ([0 , G ) consists of the piecewise smooth functions from [0 ,
1] to G .One of the simplest examples of a noncommutative Carnot group is the Heisenberg group H . It has been shown in [15] that, for 0 < ε < /
2, one can embed the snowflake ( H , d − ε H )into a bounded dimensional Euclidean space with optimal distortion O ( ε − / )(more pre-cisely, Theorem 1.1 below for the case G = H was proven in [15]). The goal of this paperis to show that the methods of [15] extend to the more general setting of Carnot groups. Theorem 1.1.
For each Carnot group G , there exists a natural number D G such thatfor every < ε < / there exists an embedding of ( G, d − εG ) into R D G with distortion O G ( ε − / ) . We have not attempted to optimise D G , but from the analysis in this paper it will be clearthat D G = Ω(23 n h ), where n h is the Hausdorff dimension of G , i.e., n h = P sr =1 r dim V r .We will not work in the large epsilon regime ≤ ε < < ε < A , where A is a verylarge number. For A ≤ ε < , a construction of Assouad [2] gives such an embedding; weare thus only interested in the small epsilon regime.One standard consequence of the above theorem, which was also observed in [15], is thefollowing corollary. Corollary 1.1.
Let G be a Carnot group, and suppose Γ ⊂ G is a discrete subgroup of G , where for any two distinct points γ , γ ∈ Γ one has d G ( γ , γ ) ≥ . Let D G be as inTheorem 1.1, and for R ≥ define the discrete ball B Γ (0 , R ) := { γ ∈ Γ : d G (0 , γ ) < R } .Then there exists an embedding of the discrete ball B Γ (0 , R ) with the induced metric d G into R D G of distortion O G (log / R ) . This follows from Theorem 1.1 because on B Γ (0 , R ) with R ≥
2, the metric d G is com-parable to d − / log RG .Some history behind Theorem 1.1: Carnot groups are special cases of doubling metricspaces, where a metric space ( X, d X ) is said to be K -doubling for some natural number K if for any x ∈ X and R > y , · · · , y K ∈ X such that B R ( x ) ⊂ K [ j =1 B R/ ( y j ) , where B R ( x ) := { z ∈ X : d X ( x, z ) < R } is the ball of radius R centered at x ; we say that( X, d X ) is doubling if ( X, d X ) is K -doubling for some natural number K . In the seminalpaper [2], Assouad showed that for 0 < ε < , the (1 − ε )-snowflake of a K -doubling metricspace admits an embedding into ℓ O K ( ε − O (1) )2 with distortion O K ( ε − / ). Here, the distortion O ( ε − / ) is sharp for the Heisenberg group H ; see [12, Section 4] for a proof of this fact. NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 3
After a line of research [1, 3, 4, 5, 6] working on optimising the embedding, it has beenshown in [12] that one can construct snowflake embeddings of any doubling metric spaceinto bounded dimensional Euclidean spaces at the cost of slightly worsening the distortion:more precisely, one can embed the (1 − ε )-snowflake of a K -doubling space into ℓ O (log K )2 with distortion O ( log Kε ), or more generally, for any 0 < δ ≤
1, into ℓ O ( log Kδ )2 with distortion O (( log Kε ) δ ). One may then further inquire whether we can take the target dimensionto be uniformly bounded in 0 < ε < while simultaneously attaining the best possibledistortion O ( ε − / ). We state this separately as a question. Question 1.1 (Assouad’s theorem with optimal distortion and bounded dimension) . Forany natural number K ≥ , does there exists a natural number D ( K ) such that if ( X, d X ) isa K -doubling metric space and < ε < , then there exists an embedding of the snowflake ( X, d − εX ) into R D ( K ) with distortion O K ( ε − / ) ? For completeness we state a sharper version of the above question, which is motivatedby the fact that we must have the lower bound D ( K ) & log K . Question 1.2 (Assouad’s theorem with optimal distortion and optimal dimension) . Forany natural number K ≥ , does there exists a natural number D ( K ) = O (log K ) such thatif ( X, d X ) is a K -doubling metric space and < ε < , then there exists an embedding ofthe snowflake ( X, d − εX ) into R D ( K ) with distortion O K ( ε − / ) ? These questions are relevant to finding counterexamples a question raised by Lang andPlaut in [9], which, under our notation, can be stated as follows.
Question 1.3 (Lang–Plaut problem, [9]) . For any natural number K ≥ , does there existsa natural number D ( K ) such that if X is a subspace of ℓ that is K -doubling under themetric it inherits from ℓ , then there exists an embedding of ( X, k · k ) into R D ( K ) with O K (1) distortion? As observed in [12], if an embedding as in Question 1.1 fails to exist for the Heisenberggroup H , then the Lang–Plaut problem would be answered in the negative, since it isknown that H admits (1 − ε )-snowflake embeddings into ℓ with distortion O ( ε − / ) withthe additional property that the doubling constant of the image is uniformly bounded.Nevertheless, it has been shown in [15] that an embedding as in Question 1.1 for H exists,i.e., Theorem 1.1 for G = H is true. Thus the Heisenberg group (or more precisely, thedoubling ℓ images of the snowflakes thereof) fails to serve as a counterexample to the Lang–Plaut problem. To highlight this connection, we pose in addition the following question. Question 1.4.
For any natural number K ≥ , does there exists a natural number K ′ ( K ) such that if ( X, d X ) is a K -doubling metric space and < ε < , then there exists anembedding of the snowflake ( X, d − εX ) into ℓ with distortion O K ( ε − / ) with the additionalproperty that image of X is K ′ ( K ) -doubling? Of course, the connection is that if Question 1.4 has a positive answer while Question 1.1has a negative answer, then the Lang–Plaut problem (Question 1.3) must have a negativeanswer.The main purpose of this paper is to expand upon our partial knowledge of Question1.1, by answering it in the positive in the setting of Carnot groups. We begin by carryingthe methods of [15] into the setting of Carnot groups, and whenever a tool of [15] becomes
SANG WOO RYOO inadequate in the setting of Carnot groups, we replace it with a tool more suitable inthe general language of doubling metric spaces. More specifically, the key tools of [15]are a new variant of the Nash–Moser iteration scheme, and a certain extension theoremfor orthonormal vector fields using a quantitative homotopy argument. When generalizingto the case of Carnot groups, one potential source of trouble is the fact that arbitraryCarnot groups might have arbitrarily large step size s , which could complicate its geometricproperties and make the tools of [15] fail. It will be shown in Section 3 that the Nash–Moseriteration scheme directly generalizes in the setting of Carnot groups. However, in Section4, it will be clear that there are some obstructions to the quantitative homotopy argument.Nevertheless, we will prove the orthonormal vector field extension theorem even for generaldoubling metric spaces, using the concentration of measure phenomenon for the sphere andthe Lov´asz Local Lemma.Ultimately, we would like to answer Question 1.1 for general doubling metric spaces; thispaper is an intermediate step in such an investigation, showing that it is at least true forCarnot groups. We plan to address the case of general doubling metric spaces in futurework, by possibly adapting some of the proof methods of this paper. In particular, theorthonormal vector field extension theorem for doubling metric spaces (Theorem 4.2) seemsa promising starting point for future work, and we would have to find either a metricanalogue or a replacement for the Nash–Moser iteration scheme.We now briefly overview some of the results and strategies of Assouad [2], Naor andNeiman [12], and Tao [15] for constructing snowflake embeddings, and describe how theseideas connect to the proof strategy of this paper.The starting point of constructing snowflake embeddings is the classical fact that if X isa metric space, A >
1, 0 < ε < , and { φ m : X → ℓ } m ∈ Z is a collection of maps such that | φ m | ≤ A m and φ m is 1-Lipschitz, then the Weierstrass sum(1.1) Φ = X m ∈ Z A − mε φ m is (1 − ε )-H¨older, with the H¨older norm bounded by O ( ε − ). If one assumes in additionthat the φ m takes values in mutually orthogonal subspaces of ℓ , then we can bound the(1 − ε )-H¨older norm by O ( ε − / ). In the case where X is a doubling metric space, Assouad[2] constructed functions φ m with the above properties and with the additional propertythat if d ( p, q ) ≍ A m then | φ m ( p ) − φ m ( q ) | & A m . Then Φ satisfies the H¨older lower bound | Φ( p ) − Φ( q ) | & d ( p, q ) − ε for any p, q ∈ X , and thus is an embedding of the (1 − ε )-snowflakeof X into ℓ with distortion O ( ε − / ).When improving upon Assouad’s theorem to reduce the target dimension, one usuallykeeps the idea of using the Weierstrass sum (1.1) to guarantee the upper bound but needsto be more clever to enforce the lower bound. Note that for d ( p, q ) ≍ A m , the sum P n
Notation.
Following widespread convention, A . B means A ≤ CB for a universalconstant C , and A ≍ B means ( A . B ) ∧ ( B . A ). If the constant C depends on otherparameters, this is denoted using subscripts, e.g., A . C ,N B means A ≤ C ( C , N ) B where C ( C , N ) depends only on C and N , and A ≍ C B means A . C B and B . C A .We will select absolute constants in the following order: • A sufficiently large natural number C > G . • A sufficiently large dyadic number N depending on G and C . • A sufficiently large dyadic number A = 2 M depending on G and C and N .2.2. Basic linear algebra.
Denote by | · | the Euclidean metric and by h· , ·i the Euclideaninner product on Euclidean spaces R D .If T : R D → R D is a linear map, we also denote by | T | the Frobenius norm of T . Also,for 1 ≤ n ≤ D , the exterior power V n R D can be identified with R ( Dn ), and so we can alsodefine a Euclidean norm | · | and a Euclidean inner product h· , ·i on V n R D . With this normon V n R D , the Cauchy-Binet formula tells us that for v , · · · , v n ∈ R D , | v ∧ · · · ∧ v n | = det ( v i · v j ) ≤ i,j ≤ n = det( T T ∗ ) , where T : R D → R n is the linear map T ( u ) := ( u · v , · · · , u · v n ) . More generally, the polarized Cauchy-Binet formula tells us that for u , · · · , u n , v , · · · , v n ∈ R D , (cid:28) u ∧ · · · ∧ u n , v ∧ · · · ∧ v n (cid:29) = det ( u i · v j ) ≤ i,j ≤ n . It is not difficult to see that we have a Cauchy-Schwarz-like inequality: for every v , · · · , v n ∈ R D and 1 ≤ i < n ≤ D , we have | v ∧ · · · ∧ v n | ≤ | v ∧ · · · ∧ v i || v i +1 ∧ · · · ∧ v n | . We will simply refer to this as the Cauchy-Schwarz inequality in the rest of this paper.
SANG WOO RYOO
Some metric space geometry.
Let (
X, d ) be a metric space. For any f : X → R D we denote the norms k f k C := sup x ∈ X | f ( x ) | , k f k Lip := sup x,y ∈ X, x = y | f ( x ) − f ( y ) | d ( x, y ) . These norms satisfy certain algebraic properties. If f, g : X → R D then(2.1) k f · g k Lip ≤ k f k C k g k Lip + k f k Lip k g k C . Also, if f : X → R and f ≥ c > k /f k Lip ≤ c − k f k Lip , (2.2) k p f k Lip ≤ √ c k f k Lip . (2.3)One can use these properties to see that for f : X → R D with | f | ≥ c > (cid:13)(cid:13)(cid:13)(cid:13) f | f | (cid:13)(cid:13)(cid:13)(cid:13) Lip ≤ (cid:13)(cid:13)(cid:13)(cid:13) | f | (cid:13)(cid:13)(cid:13)(cid:13) C k f k Lip + (cid:13)(cid:13)(cid:13)(cid:13) | f | (cid:13)(cid:13)(cid:13)(cid:13) Lip k f k C ≤ ( c − + c − k f k C ) k f k Lip . For δ >
0, a subset N δ ⊂ X is a δ -net if for any distinct x, y ∈ N δ one has d ( x, y ) ≥ δ .By Zorn’s lemma, δ -nets which are maximal with respect to inclusion exist, and if N δ is amaximal δ -net then we have the covering X = [ x ∈N δ B δ ( x ) . An immediate consequence of the doubling property is that if X is a K -doubling metricspace and m ≥
0, then for any δ -net N δ we have(2.5) |N δ ∩ B m δ ( x ) | ≤ K m +1 ∀ x ∈ X. Function spaces on Carnot groups.
We will assume that the norm k · k on V isan inner product on V (in other words, we may assume G is a sub-Finsler Carnot group).We may do so by using John’s ellipsoid theorem, which allows us to replace k · k by aninner product norm while introducing distortion at most √ k , which is acceptable since thisis independent of the amount ε of snowflaking.We fix a left-invariant orthonormal basis X , · · · , X k of V with respect to k · k . If φ : G → R D is a differentiable function, we let ∇ φ : G → R kD denote the horizontalgradient ∇ φ := ( X φ, · · · , X k φ ) . By iteration, we have ∇ m φ : G → R k m D for any m ≥
1, if φ is m times differentiable. Werecall the C norm k φ k C = sup p ∈ G | φ ( p ) | , and define the higher C m norms k φ k C m := X ≤ j ≤ m k∇ j φ k C . For a fixed spatial scale
R > C mR norm to be the rescaled norm k φ k C mR := X ≤ j ≤ m R j k∇ j φ k C . NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 9
Given a H¨older exponent 0 < α < k φ k ˙ C ,α := sup p,q ∈ G,p = q | φ ( p ) − φ ( q ) | d ( p, q ) α and the higher H¨older norms k φ k C m,α := k φ k C m + k∇ m φ k ˙ C ,α and more generally the rescaled H¨older norm k φ k C m,αR := k φ k C mR + R m + α k∇ m φ k ˙ C ,α . One may easily verify k φ k C m,αR . k φ k C m +1 R . By an iterated application of the product rule, one can verify the algebra properties k φψ k C mR . m k φ k C mR k ψ k C mR , k φψ k C m,αR . m k φ k C m,αR k ψ k C m,αR . These inequalities continue to hold when φ and ψ are vector-valued and we take the wedgeproduct or the dot product, where the constants do not depend on the dimension of thecodomain of φ and ψ : k φ · ψ k C mR . m k φ k C mR k ψ k C mR , k φ · ψ k C m,αR . m k φ k C m,αR k ψ k C m,αR , and k φ ∧ ψ k C mR . m k φ k C mR k ψ k C mR , k φ ∧ ψ k C m,αR . m k φ k C m,αR k ψ k C m,αR . More generally, one can observe that these algebra properties continue to hold when wereplace the above norms by norms of the form k φ k C m { Rj } mi =0 := X ≤ j ≤ m R j k∇ j φ k C , where { R j } mi =0 is a ‘sequence of spatial scales’, i.e., a sequence of positive real numbers, thatis log-concave: R i R j ≥ R i + j . Examples of such norms include(2.6) k φ k + R k∇ φ k C m , or k φ k C + k∇ φ k C m /R , R ≥ . Some Carnot group geometry.
Recall the decomposition g = V ⊕ V ⊕ · · · ⊕ V s .We will denote dim G = n , dim V i = k i , and k = k . We also denote the Hausdorffdimension n h := P si =1 ik i . We will assume s ≥
2, since if s = 1, then G is just a finite-dimensional Euclidean space, and near-optimal snowflake embeddings of Euclidean spaceswere constructed in [2]. This will entail n h ≥
4, as we must have k ≥ k ≥ ≤ r ≤ s , we fix a basis X r, , · · · , X r,k i of V i and extend them to left-invariantvectors over G . For r = 1, we simply denote X ,i = X i .As G is nilpotent and simply connected, the exponential map exp : g → G is a diffeo-morphism. Recall that we have defined the scaling maps δ λ : g → g for λ > δ λ | V i = λ i id V i , i = 1 , · · · , s. One may then define the dilation δ λ : G → G so that it commutes with exp: δ λ ◦ exp = exp ◦ δ λ . One can compute that δ λ is the unique Lie group automorphism δ λ : G → G such that( δ λ ) ∗ = δ λ . Moreover, δ λ interacts with the left-invariant vector fields as follows: X r,i ( φ ◦ δ λ ) = λ r ( X r,i φ ) ◦ δ λ , r = 1 , · · · , s, i = 1 , · · · , k r . The special case r = 1 tells us that δ λ is a scaling in the Carnot-Carath´eodory metric: d G ( δ λ ( p ) , δ λ ( p ′ )) = λd G ( p, p ′ ) , p, p ′ ∈ G. By iteration we can also deduce ∇ m ( φ ◦ δ λ ) = λ m ( ∇ m φ ) ◦ δ λ . One can parametrize G by R n , by first identifying G with g via the exponential mapexp, and then identifying g with R n via the basis { X r,i } ≤ r ≤ s, ≤ i ≤ k r . We will denote thecorresponding canonical basis as { f r,i } ≤ r ≤ s, ≤ i ≤ k r .We may define a weighted degree for polynomials in x r,i by assigning degree r to x r,i . Itis clear that δ λ acted upon a homogeneous polynomial of degree m is just multiplication by λ m , so the differential operator X r,i acts on polynomials by reducing the weighted degreeby r in each term. One can also see, using the scaling δ λ , that d G exp( s X r =1 k r X i =1 x r,i X r,i ) , e G ! ≍ G s X r =1 k r X i =1 | x r,i | /r . One can express the group law in this coordinate system using the Baker-Campbell-Hausdorff formula gh = s X m =1 ( − m − m X r + s > ... r m + s m > [ g r h s g r h s · · · g r m h s m ]( P mj =1 ( r j + s j )) · Q mi =1 r i ! s i ! , where the sum is finite since G is of step s , and we have used the following notation:[ g r h s · · · g r m h s m ] = [ g, [ g, · · · [ g | {z } r , [ h, [ h, · · · [ h | {z } s , · · · [ g, [ g, · · · [ g | {z } r m , [ h, [ h, · · · h | {z } s m ]] · · · ]] . Thus we can see that s X r =1 k r X i =1 x r,i X r,i ! s X r =1 k r X i =1 x r,i X r,i ! = s X r =1 k r X i =1 x r,i X r,i ! where x r,i = x r,i + x r,i + (homogeneous polynomial of { x r ′ ,i ′ } r ′ For r > B r := { h ∈ G : d G ( h, e G ) < r } , and for g ∈ G and r > B r ( g ) := { h ∈ G : d G ( h, g ) < r } = gB r , where the last equality follows from left-invariance of the metric.A simple volumetric argument gives the following bound for any δ -net N δ :(2.7) |N δ ∩ B R ( p ) | ≤ ( 2 Rδ + 1) n h , p ∈ G, R > . Littlewood-Paley theory on Carnot groups. A basic tool used in [15] was aLittlewood-Paley theory for functions defined on the Heisenberg group. One can easilymodify the argument in [15] to show the following. For a positive number N and a C function φ : G → R D , one can construct the Littlewood-Paley projection P ≤ N φ : G → R D ,which is a C ∞ function, and the variants P 1] that equals 1 on[ − / , / N > 0, the Littlewood-Paley projection P ≤ N using theabove functional calculus by the formula P ≤ N := ϕ ( L/N )The proof of the properties listed in Theorem 2.1 is essentially the same as presented inTheorem 6.1 of [15]; although it was done for the case G = H , one can follow the proofwith minimal adjustments.3. Nash–Moser Perturbation for a bilinear form For two given C functions φ, ψ : G → R D , we define the bilinear form B ( φ, ψ ) : G → R k as B ( φ, ψ ) := ( X φ · X ψ, · · · , X k φ · X k ψ ) . Later, when constructing good embeddings of the Carnot group G , we will encounter thefollowing situation. Given ψ : G → R D with certain regularity properties, so that ψ ‘represents’ the geometry of G at scale A and above, we will need to find a ‘nontrivial’solution φ : G → R D to(3.1) B ( φ, ψ ) = 0 , i.e., X i φ · X i ψ = 0 for all i = 1 , · · · , k . This way, the Pythagorean theorem will tell us that |∇ ( φ + ψ ) | = |∇ φ | + |∇ ψ | , which, coupled with an Assouad-type summation technique,will give us optimal control on the growth of |∇ ψ | and hence provide us with the optimaldistortion rate O ( ε − / )(note that Assouad [2] achieved this orthogonality and hence theoptimal distortion by allowing the φ and ψ to take values in different direct sum componentsof the target space, but thereby losing control on the dimension of the target space). Here,when we say that φ is ‘nontrivial’, we mean that ψ + φ also has the regularity properties of ψ but at scale 1 instead of A . Attempts to solve this system (3.1) directly using the Leibnizrule and linear algebra gives less control on the smoothness on φ than that on ψ , which isunsuitable for iteration. The solution proposed by [15] was to first find a nontrivial andapproximate solution ˜ φ to (3.1), or more precisely a solution to the low-frequency equation(3.2) B ( ˜ φ, P ≤ N ψ ) = 0 . This way, we have control on all levels of smoothness of P ≤ N ψ (by Theorem 2.1), and hencealso on ˜ φ . This ˜ φ will be constructed in later sections. Once we have this approximatesolution ˜ φ , [15] then proposed to use a variant of the Nash–Moser iteration scheme to findsmall perturbations of ˜ φ , which are small enough to preserve the non-triviality of ˜ φ , andwhich allows us to solve the original equation (3.1).Our goal in this section is to show that the Nash–Moser iteration scheme of [15] carrieson to the general setting of Carnot groups without obstruction. The rest of this sectionis a repetition of Section 7 of [15]; we have reproduced the entire argument here in orderto book-keep certain calculations that arise from higher-dimensional matrix operations, aswell as to state and verify various estimates in the case of Carnot groups. NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 13 Because the Nash–Moser iteration process produces error terms, we will need to considera slightly more general setting. Given ψ : G → R D and F = ( F , · · · , F k ) : G → R k , weconsider the problem of finding a solution φ : G → R D to(3.3) B ( φ, ψ ) = F. One easy way to solve (3.3) is to solve the system ( φ · X i ψ = 0 ,φ · X i X i ψ = − F i , i = 1 , · · · , k, for then X i φ · X i ψ = X i ( φ · X i ψ ) − φ · X i X i ψ = F i . This system is solvable if { X i ψ, X i X i ψ } ≤ i ≤ k are pointwise independent.More precisely, for each p ∈ G define the linear map T ψ ( p ) : R D → R k by T ψ ( p ) v := ( v · X ψ ( p ) , · · · , v · X k ψ ( p ) , v · X X ψ ( p ) , · · · , v · X k X k ψ ( p )) , v ∈ R D . If { X i ψ, X i X i ψ } ≤ i ≤ k are pointwise independent, i.e., if T ψ ( p ) has full rank, or equiva-lently(by the Cauchy-Binet formula) if | X ψ ( p ) ∧ · · · ∧ X k ψ ( p ) ∧ X X ψ ( p ) ∧ · · · ∧ X k X k ψ ( p ) | > , then we can define the pseudoinverse T ψ ( p ) − : R k → R D of T ψ ( p ) by the formula T ψ ( p ) − := T ψ ( p ) ∗ ( T ψ ( p ) T ψ ( p ) ∗ ) − . Then for any continuous functions a i , b i : G → R , i = 1 , · · · , k , we have the pointwiseidentities T ψ ( p ) − ( a , · · · , a n , b , · · · , b n ) · X i ψ ( p ) = a i T ψ ( p ) − ( a , · · · , a n , b , · · · , b n ) · X i X i ψ ( p ) = b i (3.4)and one has the explicit solution(3.5) φ explicit ( p ) := T ψ ( p ) − (0 , − F ( p ))to (3.3).The problem with this solution to (3.3) is that the solution φ explicit constructed in thismanner will have two degrees less regularity than ψ , which will be unsuitable for iterationpurposes. We will overcome this issue by applying the above procedure to the Littlewood-Paley components of ψ and F . Proposition 3.1 (Perturbation theorem, analogue of Proposition 7.1 of [15]) . Let M be areal number with M ≥ C − . Let m ∗ ≥ and < α < . Suppose we are given a C m ∗ ,α -map ψ : G → R D with thefollowing regularity properties:(1) (H¨older regularity at scale A ) We have (3.6) k∇ ψ k C m ∗− ,αA ≤ C A − . (2) (Non-degenerate first derivatives) For any p ∈ G , we have (3.7) C − M ≤ | X i ψ ( p ) | ≤ C M, i = 1 , · · · , k, (3) (Locally free embedding) For any p ∈ G , we have (3.8) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ^ i =1 X i ψ ( p ) ∧ k ^ i =1 X i X i ψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & C A − k M k . Let F : G → R k be a function with bounded C m ∗ − -norm: k F k C m ∗− < ∞ . Let ˜ φ : G → R D be a solution to the low-frequency equation (3.2) with bounded C m ∗ ,α -norm: k ˜ φ k C m ∗ ,α < ∞ . Then there exists a C m ∗ ,α -solution φ to (3.3) which is a small perturbationof ˜ φ : (3.9) k φ − ˜ φ k C m ∗ ,α . C A k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α , and which satisfies the variant estimate (3.10) k X i ( φ − ˜ φ ) · X j ψ k C . C k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α , i, j = 1 , · · · , k. Here, we treat α as a universal constant(we may take α = ), and we allow the constant C to depend on m ∗ . This will not contradict our hierarchy of constants, as m ∗ will be laterchosen to depend on G (more precisely, it will equal s + s + 1 ). Remark 3.1. One may imagine strengthening the theorem to obtain the stronger orthogo-nality statement X i φ · X j ψ = 0 , i, j = 1 · · · , k, but attempts to modify the Nash–Moser iteration scheme to accommodate this differencecauses the resulting infinite series to diverge(more specifically, we are then forced to placederivatives on P N ψ in (3.17) , which we must avoid in order to make the defining seriesconverge). The best one can achieve with the tools outlined in this section is (3.10) . Nev-ertheless, we will be able to choose ˜ φ with X i ˜ φ · X j P ≤ N ψ = 0 , i, j = 1 · · · , k, which, coupled with (3.10) and (2.9) , will allow us to establish that X i φ · X j ψ is sufficientlysmall for our purposes.Proof. It will suffice to find a function φ with the stated bounds solving the approximateequation(3.11) k B ( φ, ψ ) − F k C m ∗− ≤ A − m ∗ k F k C m ∗− + A − m ∗ k ˜ φ k C m ∗ ,α rather than the precise equation (3.3), while satisfying the following stronger versions of(3.9) and (3.10):(3.12) k φ − ˜ φ k C m ∗ ,α . C A k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α , and(3.13) k X i ( φ − ˜ φ ) · X j ψ k C . C k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α , i, j = 1 , · · · , k. One can then iteratively replace ( ˜ φ, F ) by the error term (0 , F − B ( φ, ψ )) and sum theresulting solutions to obtain an exact solution to (3.3). This is possible due to the linearityof this equation in φ .We will first construct a low-frequency solution φ ≤ N to the low-frequency equation B ( φ ≤ N , P ≤ N ψ ) = P ≤ N F NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 15 and then, given φ ≤ N/ for dyadic N > N by induction, we will add higher frequencycomponents φ N to obtain φ ≤ N := φ ≤ N/ + φ N , which approximately solves the higher-frequency equation B ( φ ≤ N , P ≤ N ψ ) ≈ P ≤ N F. More specifically, the construction goes as follows. We first construct the low-frequencycomponent as(3.14) φ ≤ N := ˜ φ + T − P ≤ N ψ (0 , − P ≤ N F ) . Then, from the fact that (3.5) solves (3.3), from (3.2), and the bilinearity of B , one has(3.15) B ( φ ≤ N , P ≤ N ψ ) = P ≤ N F. Next, for every dyadic N > N we recursively define φ N by the formula(3.16) φ N := T − P ≤ N ψ ( a N , · · · , a kN , b N , · · · , b kN )where ( a iN := − ( X i P ≤ N φ Abbreviating T = T P ≤ N ψ , recall the definition of the pseudoinverse T − = 1det( T T ∗ ) T ∗ adj( T T ∗ )where adj( A ) denotes the adjugate matrix of A . We then need to show the bounds (cid:13)(cid:13)(cid:13)(cid:13) ∇ j (cid:18) T T ∗ ) T ∗ adj( T T ∗ ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) C . C AB j for 0 ≤ j ≤ m ∗ − 1, where B j is the log-convex sequence B j := ( A − j ≤ j ≤ m ∗ − N j − m ∗ +2 A − m ∗ j > m ∗ − . . From (3.6), (3.7), and Theorem 2.1(iii), we have |∇ j X i P ≤ N ψ | . C M B j , i = 1 , · · · , k, ≤ j ≤ m ∗ − , and |∇ j X i X i P ≤ N ψ | . C A − B j , i = 1 , · · · , k, ≤ j ≤ m ∗ − . Thus, viewing T as a 2 k × D matrix, for any j -th order horizontal differential operator W j thefirst k rows of W j T have norm O C ( M B j ), and the bottom k have norm O C ( A − B j ). Bythe product rule, and noting that B i B j ≤ B i + j for all i, j ≥ k × k matrix W j ( T T ∗ ) has top left k × k block of size O C ( M B j ), top right and bottom left blocksof size O C ( A − M B j ), and bottom right block of size O C ( A − B j ). By the product ruleand cofactor expansion, W j adj( T T ∗ ) then has top left block of size O C ( M k − A − k B j ), topright and bottom left blocks of size O C ( M k − A − k +1 B j ), and bottom right block of size O C ( M k A − k +2 B j ). By the product rule, every row of the D × k matrix W j ( T ∗ adj( T T ∗ ))is of size O C ( M k A − k +1 B j ) (many are lower order than this).Similarly, ∇ j (det( T T ∗ )) has magnitude O C ( M k A − k B j ). Meanwhile, from (3.7), (3.8),(3.6), and using (2.9) to approximate P ≤ N ψ by ψ up to negligible error, we easily obtainthe wedge product lower bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ^ i =1 X i P ≤ N ψ ∧ k ^ i =1 X i X i P ≤ N ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & C M k A − k . From this and the Cauchy-Binet formula we have the matching lower bounddet( T T ∗ ) & C M k A − k for the determinant. Hence by the quotient rule, ∇ j (det( T T ∗ ) − ) has magnitude O C ( M − k A k B j ).The claim now follows from the product rule. (cid:3) Remark 3.2. Of course, Lemma 3.1 can be strengthened to guarantee not only C m ∗ − -regularity of T − but also all higher levels of regularity. We have stopped at C m ∗ − becausethis is only what we will need later. The rest of the proof of Proposition 3.1 follows mostly as in [15]. We have reproducedthe argument for completeness.From the above Lemma we have the estimate k T − P ≤ N ψ k C m ∗− . C A, NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 17 while from Theorem 2.1(iii)-(2.8) we have k P ≤ N F k C m ∗− . k F k C m ∗− and thus from (3.14),(3.19) k φ ≤ N − ˜ φ k C m ∗− . k T − P ≤ N ψ k C m ∗− k P ≤ N F k C m ∗− . C A k F k C m ∗− . Next, from Theorem 2.1(iii)-(2.9) we have, for N ≥ N dyadic, k∇ m P ≤ N φ 2, we have(3.22) k φ ≤ N k C m ∗ . C A k F k C m ∗− + k ˜ φ k C m ∗ ,α , N ≥ N . Inserting this back into (3.20) and (3.21), and noting again that m ∗ ≥ 2, we obtain k φ N k C m ∗ +11 /N . C A − m ∗ − α N − m ∗ − α k ˜ φ k C m ∗ ,α + ( A − m ∗ − α N − m ∗ − α + AN − m ∗ +1 ) k F k C m ∗− ≤ A − m ∗ − α N − m ∗ − α k ˜ φ k C m ∗ ,α + AN − m ∗ − α k F k C m ∗− (3.23)and k φ N k C m ∗ ≤ N m ∗ k φ N k C m ∗ +11 /N . C A − m ∗ − α N − α k ˜ φ k C m ∗ ,α + AN − α k F k C m ∗− . (3.24)We thus conclude that the sum φ := φ ≤ N + X N>N φ N converges in the C m ∗ norm(and consequently also the C norm, as m ∗ ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X N>N φ N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C m ∗ ,α . C A k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α . As noted above, from (3.24) we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X N>N φ N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C m ∗ ≤ X N>N k φ N k C m ∗ . C A k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α , so it remains to show H¨older regularity:(3.25) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ m ∗ X N>N φ N ( p ) − ∇ m ∗ X N>N φ N ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . C ( A k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α ) d ( p, q ) α for any p, q ∈ G . By the triangle inequality, it is enough to show X N>N (cid:12)(cid:12)(cid:12) ∇ m ∗ φ N ( p ) − ∇ m ∗ φ N ( q ) (cid:12)(cid:12)(cid:12) . C ( A k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α ) d ( p, q ) α On one hand, we may bound |∇ m ∗ φ N ( p ) − ∇ m ∗ φ N ( q ) | . k∇ m ∗ φ N k C . k φ N k C m ∗ (3.24) . C A − m ∗ − α N − α k ˜ φ k C m ∗ ,α + AN − α k F k C m ∗− = ( A − m ∗ − α k ˜ φ k C m ∗ ,α + A k F k C m ∗− ) N − α . On the other hand, one has |∇ m ∗ φ N ( p ) − ∇ m ∗ φ N ( q ) | . k∇ m ∗ +1 φ N k C d ( p, q ) . N m ∗ +1 k φ N k C m ∗ +11 /N d ( p, q ) (3.23) . C ( A − m ∗ − α N − α k ˜ φ k C m ∗ ,α + AN − α k F k C m ∗− )( N d ( p, q )) . C ( A − m ∗ − α k ˜ φ k C m ∗ ,α + A k F k C m ∗− ) N − α ( N d ( p, q )) . NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 19 Thus the left-hand side of (3.25) is bounded by . C ( A − m ∗ − α k ˜ φ k C m ∗ ,α + A k F k C m ∗− ) X N N − α min(1 , N d ( p, q ))and the claim (3.12) follows by summing the double-ended geometric series P N N − α min(1 , N d ( p, q ))using the hypothesis 0 < α < φ ≤ N converges in C to φ as N → ∞ , and P ≤ N ψ converges in C to ψ , we may write B ( φ, ψ ) as the uniform limit of B ( φ ≤ N , P ≤ N ψ ). Using (3.15) and(3.18), we have the telescoping sum B ( φ, ψ ) = B ( φ ≤ N , P ≤ N ψ ) + X N>N ( B ( φ ≤ N , P ≤ N ψ ) − B ( φ 1, one has from Theorem 2.1(iii)-(2.9) and (3.24) that k∇ P >N φ 1, we have from Theorem 2.1(iii)-(2.9) and (3.6) that k∇ P N ψ k C j . N j +1 k P N ψ k C j /N . N j +1 N − m ∗ − α k∇ m ∗ ψ k ˙ C ,α . C N j +1 − m ∗ − α A − m ∗ − α , and thus k B ( φ, ψ ) − F k C m ∗− . C X N>N X j + j =2 m ∗ − N j +1 − m ∗ − α N j +1 − m ∗ − α A − m ∗ − α ( A − m ∗ − α k ˜ φ k C m ∗ ,α + A k F k C m ∗− ) . C X N>N N − α A − m ∗ − α ( A − m ∗ − α k ˜ φ k C m ∗ ,α + A k F k C m ∗− ) . C N − α A − m ∗ − α ( A − m ∗ − α k ˜ φ k C m ∗ ,α + A k F k C m ∗− )(we used the choice of < α < 1) which gives (3.11).Finally, we prove (3.13). Fix i, j = 1 , · · · , k . We need to establish k X i ( φ − ˜ φ ) · X j ψ k C . C k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α . By the Leibniz rule we have X i ( φ − ˜ φ ) · X j ψ = X i (( φ − ˜ φ ) · X j ψ ) − ( φ − ˜ φ ) · X i X j ψ and hence by the triangle inequality we have k X i ( φ − ˜ φ ) · X j ψ k C ≤ k ( φ − ˜ φ ) · X j ψ k C + k φ − ˜ φ k C k X i X j ψ k C . We bound the second term using (3.6) and(3.12): k φ − ˜ φ k C k X i X j ψ k C ≤ k φ − ˜ φ k C k∇ ψ k C . C ( A k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α ) A − , so it remains to show that k ( φ − ˜ φ ) · X j ψ k C . C k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α . By the triangle inequality, the left-hand side is at most k ( φ ≤ N − ˜ φ ) · X j P ≤ N ψ k C + k ( φ ≤ N − ˜ φ ) · X j P >N ψ k C + X N>N ( k φ N · X j P ≤ N ψ k C + k φ N · X j P >N ψ k C )= 0 + k ( φ ≤ N − ˜ φ ) · X j P >N ψ k C + X N>N ( k X j P ≤ N φ Finally, from Theorem 2.1(iii)-(2.9) and (3.23) one has k φ N · X j P >N ψ k C . k φ N k C k X j P >N ψ k C . N k φ N k C m ∗ +11 /N N k P >N ψ k C /N . C N ( A − m ∗ − α N − m ∗ − α k ˜ φ k C m ∗ ,α + AN − m ∗ − α k F k C m ∗− ) N − m ∗ − α k∇ m ∗ ψ k ˙ C ,α . C ( k F k C m ∗− + A − m ∗ − α k ˜ φ k C m ∗ ,α ) N − m ∗ − α A − m ∗ − α . Inserting all these estimates, and recalling m ∗ ≥ 2, we obtain the claim. (cid:3) We state Proposition 3.1 for the case F = 0, which is the form that will be used later. Corollary 3.1. Let M be a real number with M ≥ C − . Let m ∗ ≥ and < α < . Suppose we are given a C m ∗ ,α -map ψ : G → R D with thefollowing regularity properties:(1) (H¨older regularity at scale A ) We have k∇ ψ k C m ∗− ,αA ≤ C A − . (2) (Non-degenerate first derivatives) For any p ∈ G , we have C − M ≤ | X i ψ ( p ) | ≤ C M, i = 1 , · · · , k, (3) (Locally free embedding) For any p ∈ G , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ^ i =1 X i ψ ( p ) ∧ k ^ i =1 X i X i ψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & C A − k M k . Let ˜ φ : G → R D be a C m ∗ ,α -solution to the low-frequency equation (3.2) with k ˜ φ k C m ∗ ,α < ∞ . Then there exists a C m ∗ ,α -solution φ to (3.3) which is a small perturbation of ˜ φ : k φ − ˜ φ k C m ∗ ,α ≤ A − m ∗ k ˜ φ k C m ∗ ,α , and which also satisfies the variant estimate k X i ( φ − ˜ φ ) · X j ψ k C ≤ A − m ∗ k ˜ φ k C m ∗ ,α , i, j = 1 , · · · , k. Again, we allow C to depend on α and m ∗ (this will not contradict our hierarchy ofconstants, as we can take α to be a universal constant, say α = , and we will take m ∗ = s + s + 1, where G is of step s ).4. Regular extensions of orthonormal systems Another tool that we will need is a certain result on extending orthonormal systems.In order to apply Corollary 3.1, we first need to construct a nontrivial solution ˜ φ to thelow-frequency equation (3.2), or more generally the stronger equation(4.1) X i ˜ φ · X j P ≤ N ψ = 0 , i, j = 1 , · · · , k as promised in Remark 3.1. One can see using the Leibniz rule that it is enough to solvethe system(4.2) ( ˜ φ · X i P ≤ N ψ = 0 , i = 1 , · · · , k, ˜ φ · X i X j P ≤ N ψ = 0 , i, j = 1 , · · · , k. However, the vectors { X i X j P ≤ N ψ } i,j =1 , ··· ,k may not be linearly independent, as { X i X j − X j X i ∈ V : 1 ≤ i < j ≤ k } may be linearly dependent, so the above system (4.2) may beoverdetermined. Instead, we will solve the equivalent system(4.3) ˜ φ · X i P ≤ N ψ = 0 , i = 1 , · · · , k, ˜ φ · X i X j P ≤ N ψ = 0 , i, j = 1 , · · · , k, i ≤ j, ˜ φ · X ,i P ≤ N ψ = 0 , i = 1 , · · · , k , where we recall that { X ,i } k i =1 is a basis of V (the equivalence of (4.2) and (4.3) follows from[ V , V ] = V ). The vectors { X i P ≤ N ψ } i =1 , ··· ,k ∪{ X i X j P ≤ N ψ } ≤ i ≤ j ≤ k ∪{ X ,i P ≤ N ψ } i =1 , ··· ,k can be made linearly independent if we require the stronger freeness property(4.4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ^ i =1 X i ψ ∧ ^ ≤ i ≤ j ≤ k X i X j ψ ∧ k ^ i =1 X ,i ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & C k Y i =1 | X i ψ | · Y ≤ i ≤ j ≤ k | X i X j ψ | · k Y i =1 | X ,i ψ | along with some regularity, say k∇ ψ k C A . C A − , for then we can apply Theorem 2.1(iii)-(2.9) to approximate the derivatives of P ≤ N ψ by the corresponding derivatives of ψ .Now suppose v , · · · , v k ( k +1)2 + k is the result of applying the Gram-Schmidt process tothe vectors { X i P ≤ N ψ } i =1 , ··· ,k ∪ { X i X j P ≤ N ψ } ≤ i ≤ j ≤ k ∪ { X ,i P ≤ N ψ } i =1 , ··· ,k . The freenessproperty (4.4) guarantees some regularity of the v i ’s, and solving (4.3) is equivalent tosolving(4.5) ˜ φ · v i = 0 , i = 1 , · · · , k ( k + 1)2 + k . Thus, the following general question arises: Question 4.1. Given a space X on which a function space ( F ( X ) , k · k F ) is defined, andgiven maps v , · · · , v m : X → S D − , D ≥ m +1 , which form a pointwise orthonormal systemand which have the uniform regularity bound k v i k F . , when can we extend the systemto include a new map v m +1 : X → S D − , such that v , · · · , v m , v m +1 forms a pointwiseorthonormal system and k v m +1 k F . X, F ( X ) ,m,D ? Under conditions that validate the above question, we would be able to simply take ˜ φ tobe v m +1 to solve (4.5), for then ˜ φ would have similar regularity as the v i ’s, i.e., it wouldhave bounded C m ∗ ,α norm(there is some loss of constant factors when applying Theorem2.1 (iii)-(2.8), but this is offset by the fact that there is a change of scale: we have controlon the C m ∗ ,αA norm of ψ , and we need only control the C m ∗ ,α norm of ˜ φ ).Actually, in order to also obtain a freeness property for ψ + φ , we will actually take ˜ φ tobe a linear combination of a larger extension v m +1 , · · · , v m + m ′ of v , · · · , v m with variablecoefficients; the coefficients will guarantee the freeness property in this case(see (5.24) and(5.27)). Obtaining a larger orthonormal extension will be possible simply by adding a newvector one by one.In [15, Section 8] Question 4.1 has been answered in the positive for the case X = H and k φ k F = k φ k C + R k∇ φ k C j for any j ≥ R ≥ 1. The proof in [15] used the fact that the Heisenberg group H admits a CW complex structure which is periodic with respect to its standard discretecocompact lattice. The methods of [15] can be generalized in a straightforward manner toprove the following theorem. NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 23 Theorem 4.1 (Corollary 8.4 of [15]) . Let G be a Carnot group that admits a cocompactlattice Γ and a CW structure whose cells can be obtained from left Γ -translation from afinite list of cells. Let ≤ m ≤ D − n − (where n is the topological dimension of G ), j ≥ ,and let { R i } ji =1 be a log-concave sequence of positive reals, i.e., R i R i ′ ≥ R i + i ′ whenever i + i ′ ≤ j . Let v , · · · , v m : G → S D − be functions that form an orthonormal system ateach point, with the uniform regularity bound j X k =1 R k k∇ k v i k C ≤ , i = 1 , · · · , m. Then there exists another function v m +1 : G → S D − such that v , · · · , v m along with v m +1 form an orthonormal system at each point, and (4.6) j X k =1 R k k∇ k v m +1 k C . G,D,j . In other words, we are given a bundle B over G , where for each p ∈ G the fibre of B over p is the collection of unit vectors v ∈ S D − which are perpendicular to v ( p ) , · · · , v m ( p )(so eachfibre is homeomorphic to S D − m − ), and we need to show that there is a section of this bundlewhich has the same level of control on the regularity of the bundle itself. In [15, Section8] this was achieved for the Heisenberg group H in the spirit of quantitative topology,by imposing a ‘uniform’ CW-structure on H as above and then inductively constructingthe section starting from low-dimensional skeleta. In the inductive step in [15], one hasto use the fact that the homotopy groups π i ( S n ) vanish for i < n , which necessitates the‘dimension gap’ m ≤ D − − v m +1 which isuniformly continuous, with the modulus of uniform continuity depending only on G , D and R . We can then obtain a section v m +1 with the stronger regularity property (4.6) simplyby mollifying ˜ v m +1 and then applying the Gram-Schmidt orthogonalization process; theregularity (4.6) will simply be a consequence of the algebra property for norms of the form k · k C + P jk =1 R k k∇ k · k C .General Carnot groups may not admit a cocompact lattice(as the structure constantsfor any basis may be irrational), and it is not clear whether G admits a “uniform” CWstructure that is amenable to the above proof method. We will avoid the need for a CWstructure by only using the fact that G is a doubling metric space. More precisely, we willfirst prove that the hardest part of Theorem 4.1, i.e., the case j = 1, can be done in thesetting of doubling metric spaces. We state this result separately in anticipation of futurework. Theorem 4.2. Let ( X, d ) be a K -doubling metric space( K ≥ ), and let m ≤ D − K − .If v , · · · , v m : X → S D − are 1-Lipschitz functions that form an orthonormal system ateach point, then there exists a K m ( m + 1) -Lipschitz function v m +1 : G → S D − suchthat v , · · · , v m along with v m +1 form an orthonormal system at each point.Proof. Take a maximal δ -net N δ of X , where δ = Km . By (2.5), |N δ ∩ B δ ( p ) | ≤ K and |N δ ∩ B δ ( p ) | ≤ K . Let Ω be a probability space on which independent random variables v ′ m +1 ( p ) ∈ S D − aredefined so that for each ω ∈ Ω and p ∈ X , v ′ m +1 ( p )( ω ) forms an orthonormal set along with v ( p )( ω ) , · · · , v m ( p )( ω ). Let ǫ = K , and for each p ∈ N δ , let us define the event A p = { ω ∈ Ω : ∃ q ∈ N δ ∩ ( B δ ( p ) \ { p } ) | v ′ m +1 ( p )( ω ) · v ′ m +1 ( q )( ω ) | > ǫ } . Note that for any p, q ∈ N δ distinct, we may computePr( | v ′ m +1 ( p ) · v ′ m +1 ( q ) | > ǫ ) = E p Pr q ( | v ′ m +1 ( p ) · v ′ m +1 ( q ) | > ǫ ) ≤ r π (cid:18) − ǫ D − m − (cid:19) , where the last inequality follows from a standard computation on the area of caps on thesphere(see [11, Chapter 2] for instance). Therefore we may estimate the probability of each A p using a union bound:Pr( A p ) ≤ X q ∈N δ ∩ ( B δ ( p ) \{ p } ) Pr( | v ′ m +1 ( p ) · v ′ m +1 ( q ) | > ǫ ) ≤ |N δ ∩ ( B δ ( p ) \ { p } ) | · r π (cid:18) − ǫ D − m − (cid:19) ≤ K · r π (cid:18) − ǫ D − m − (cid:19) . Also note that for each p ∈ N δ , A p is mutually independent with the collection of events { A q : q ∈ N δ \ B δ ( p ) } , which are all of the A q ’s except possibly |N δ ∩ B δ ( p ) | ≤ K ofthem. By the Lov´asz local lemma, we see that if(4.7) e · K · K · r π (cid:18) − ǫ D − m − (cid:19) < , then for any finite subcollection S ⊂ N δ we have Pr( T p ∈ S A c p ) > 0. But by our choice ofparameters D − m − ≥ K and ǫ = K , the condition (4.7) is indeed satisfied, sincethe LHS is bounded by e · K · r π (cid:18) − ǫ D − m − (cid:19) < e . K · exp (cid:18) − K (80 K ) (cid:19) = 1 ( ∵ . x − x − − log r π > x > S ⊂ N δ we have Pr( T p ∈ S A c p ) > 0, and in particular T p ∈ S A c p = ∅ . We can thus find an assignment { v ′ m +1 ,S ( p ) } p ∈ S such that for any distinct p, q ∈ S with d ( p, q ) < δ we have | v ′ m +1 ,S ( p ) · v ′ m +1 ,S ( q ) | ≤ ǫ . By taking an arbitraryenumeration of N δ , taking a monotone increasing sequence of S ’s that cover N δ and passingto a limit along a nonprincipal ultrafilter, we conclude the existence of an assignment { v ′ m +1 ( p ) } p ∈N δ such that for any distinct p, q ∈ N δ with d ( p, q ) < δ we have | v ′ m +1 ( p ) · v ′ m +1 ( q ) | ≤ ǫ .We now ‘interpolate’ the discrete vector field { v ′ m +1 ( p ) } p ∈N δ to produce a vector field { ˜ v m +1 ( p ) } p ∈ X defined on the entirety of X , which nearly has the desired properties. Wefirst construct a ‘quadratic’ partition of unity { φ q } q ∈N δ , i.e., functions φ q : X → [0 , q ∈ N δ such that • supp φ q ⊂ B δ ( q ), q ∈ N δ , • P q ∈N δ φ q = 1 on X , • k φ q k Lip ≤ K δ − , q ∈ N δ . NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 25 Indeed, we start by defining for each q ∈ N δ the function˜ φ q ( p ) = , d ( p, q ) ≤ δ, − d ( p,q ) δ , δ < d ( p, q ) ≤ δ, , d ( p, q ) > δ, and then define φ q := ˜ φ q qP r ∈N δ ˜ φ r . This obviously satisfies the first two properties, and it remains to compute k φ q k Lip . Weclearly have k ˜ φ q k C ≤ k ˜ φ q k Lip ≤ δ − .We first observe that X r ∈N δ ˜ φ r ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X r ∈N δ ˜ φ r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Lip ≤ (4 K − δ − . Indeed, the first follows from the fact that N δ is a maximal δ -net. For the second property,fix any p, p ′ ∈ X with p = p ′ . If B δ ( p ) ∩ B δ ( p ′ ) ∩ N δ = ∅ then we must have d ( p, p ′ ) ≥ δ ,because otherwise B δ ( p ) ∩ B δ ( p ′ ) ∩ N δ ⊇ B δ ( p ) ∩ N δ = ∅ , by maximality of N δ . We have1 ≤ X r ∈N δ ˜ φ r ( p ) = X r ∈ B δ ( p ) ∩N δ ˜ φ r ( p ) ≤ | B δ ( p ) ∩ N δ | ≤ K , and similarly 1 ≤ X r ∈N δ ˜ φ r ( p ′ ) ≤ K . Therefore, in this case, (cid:12)(cid:12)(cid:12)P r ∈N δ ˜ φ r ( p ) − P r ∈N δ ˜ φ r ( p ′ ) (cid:12)(cid:12)(cid:12) d ( p, p ′ ) ≤ ( K − δ − ≤ (4 K − δ − . On the other hand, if B δ ( p ) ∩ B δ ( p ′ ) ∩ N δ = ∅ then clearly | B δ ( p ) ∩ B δ ( p ′ ) ∩ N δ | ≤| B δ ( p ) ∩ N δ | + | B δ ( p ′ ) ∩ N δ | − ≤ K − 1. We have X r ∈N δ ˜ φ r ( p ) = X r ∈ B δ ( p ) ∩N δ ˜ φ r ( p ) = X r ∈ ( B δ ( p ) ∪ B δ ( p ′ )) ∩N δ ˜ φ r ( p ) and similarly X r ∈N δ ˜ φ r ( p ′ ) = X r ∈ ( B δ ( p ) ∪ B δ ( p ′ )) ∩N δ ˜ φ r ( p ′ ) , so, noting that | ˜ φ r ( p ) − ˜ φ r ( p ′ ) | ≤ | ˜ φ r ( p ) − ˜ φ r ( p ′ ) | ≤ k ˜ φ r k Lip d ( p, p ′ ) ≤ δ − d ( p, p ′ ), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X r ∈N δ ˜ φ r ( p ) − X r ∈N δ ˜ φ r ( p ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X r ∈ ( B δ ( p ) ∪ B δ ( p ′ )) ∩N δ | ˜ φ r ( p ) − ˜ φ r ( p ′ ) |≤ | ( B δ ( p ) ∪ B δ ( p ′ )) ∩ N δ | · δ − d ( p, p ′ ) ≤ (2 K − · δ − d ( p, p ′ ) . This finishes the verification that (cid:13)(cid:13)(cid:13)P r ∈N δ ˜ φ r (cid:13)(cid:13)(cid:13) Lip ≤ (4 K − δ − . By (2.2) and (2.3), we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) qP r ∈N δ ˜ φ r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Lip ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)s X r ∈N δ ˜ φ r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Lip ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X r ∈N δ ˜ φ r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Lip ≤ (2 K − δ − . Thus, by the definition of φ q and (2.1), we have k φ q k Lip ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) qP r ∈N δ ˜ φ r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C (cid:13)(cid:13)(cid:13) ˜ φ q (cid:13)(cid:13)(cid:13) Lip + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) qP r ∈N δ ˜ φ r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Lip (cid:13)(cid:13)(cid:13) ˜ φ q (cid:13)(cid:13)(cid:13) C ≤ · δ − + (2 K − δ − · ≤ K δ − . We now interpolate the vectors { v ′ m +1 ( q ) } q ∈N δ using the quadratic partition of unity { φ q } q ∈N δ :(4.8) ˜ v m +1 ( p ) := X q ∈N δ φ q ( p ) v ′ m +1 ( q ) , p ∈ X. The idea is that this interpolates nearby v ′ m +1 ’s, which are mutually almost orthogonal, so˜ v m +1 should nearly be a unit vector. Moreover, since the 1-Lipschitz functions v , · · · , v m vary slowly over distance δ , this ˜ v m +1 should also be nearly orthogonal to v , · · · , v m . This˜ v m +1 will oscillate much quicker than v , · · · , v m , but its Lipschitz constant will be controlledby K and δ .More precisely, we claim that ˜ v m +1 satisfies the following quantitative estimates:(1) ≤ | ˜ v m +1 ( p ) | ≤ for each p ∈ X ,(2) | ˜ v m +1 ( p ) · v i ( p ) | ≤ m for each p ∈ X and i = 1 , · · · , m ,(3) k ˜ v m +1 k Lip ≤ K (2 K − δ − .Indeed, by the support property of φ q , we see that the summation in (4.8) is locally finite:˜ v m +1 ( p ) = X q ∈N δ ∩ B δ ( p ) φ q ( p ) v ′ m +1 ( q ) . We can verify property (1) using the near-orthogonality and the fact that the sum of squaresof the φ q ’s is 1: (cid:12)(cid:12) | ˜ v m +1 ( p ) | − (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X q,q ′ ∈N δ ∩ B δ ( p ) q = q ′ φ q ( p ) φ q ′ ( p ) v ′ m +1 ( q ) · v ′ m +1 ( q ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X q,q ′ ∈N δ ∩ B δ ( p ) q = q ′ φ q ( p ) φ q ′ ( p ) (cid:12)(cid:12) v ′ m +1 ( q ) · v ′ m +1 ( q ′ ) (cid:12)(cid:12) ≤ X q,q ′ ∈N δ ∩ B δ ( p ) q = q ′ φ q ( p ) + φ q ′ ( p ) ǫ ≤ |N δ ∩ B δ ( p ) | ǫ ≤ K ǫ = 14 . NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 27 To verify property (2), we see that for each p ∈ X and i = 1 , · · · , m ,˜ v m +1 ( p ) · v i ( p ) = X q ∈N δ ∩ B δ ( p ) φ q ( p ) v ′ m +1 ( q ) · v i ( p ) . We observe that for each q ∈ N δ ∩ B δ ( p ), we can estimate | v ′ m +1 ( q ) · v i ( p ) | ≤ | v ′ m +1 ( q ) · v i ( q ) | + | v ′ m +1 ( q ) · ( v i ( p ) − v i ( q )) |≤ | v i ( p ) − v i ( q ) | ≤ δ, where we have used the orthogonality of v ′ m +1 ( q ) against v i ( q ) and the fact that k v i k Lip ≤ | ˜ v m +1 ( p ) · v i ( p ) | ≤ X q ∈N δ ∩ B δ ( p ) φ q ( p ) · δ ≤ |N δ ∩ B δ ( p ) | / · δ ≤ K · δ = 14 m . To verify (3), we first fix p, p ′ ∈ X with p = p ′ . If B δ ( p ) ∩ B δ ( p ′ ) ∩ N δ = ∅ then wemust have d ( p, p ′ ) ≥ δ , because otherwise B δ ( p ) ∩ B δ ( p ′ ) ∩ N δ ⊇ B δ ( p ) ∩ N δ = ∅ . By (1),we have √ ≤ | ˜ v m +1 ( p ) | , | ˜ v m +1 ( p ′ ) | ≤ √ , so | ˜ v m +1 ( p ) − ˜ v m +1 ( p ′ ) | d ( p, p ′ ) ≤ √ − √ δ − ≤ K (2 K − δ − . On the other hand, if B δ ( p ) ∩ B δ ( p ′ ) ∩ N δ = ∅ then clearly | B δ ( p ) ∩ B δ ( p ′ ) ∩ N δ | ≤| B δ ( p ) ∩ N δ | + | B δ ( p ′ ) ∩ N δ | − ≤ K − 1. We have˜ v m +1 ( p ) = X r ∈ B δ ( p ) ∩N δ φ r ( p ) v m +1 ( r ) = X r ∈ ( B δ ( p ) ∪ B δ ( p ′ )) ∩N δ φ r ( p ) v m +1 ( r )and similarly ˜ v m +1 ( p ′ ) = X r ∈ ( B δ ( p ) ∪ B δ ( p ′ )) ∩N δ φ r ( p ′ ) v m +1 ( r ) , so (cid:12)(cid:12) ˜ v m +1 ( p ) − ˜ v m +1 ( p ′ ) (cid:12)(cid:12) ≤ X r ∈ ( B δ ( p ) ∪ B δ ( p ′ )) ∩N δ | φ r ( p ) − φ r ( p ′ ) | · | v m +1 ( r ) |≤ X r ∈ ( B δ ( p ) ∪ B δ ( p ′ )) ∩N δ k φ r k Lip d ( p, p ′ ) ≤ | ( B δ ( p ) ∪ B δ ( p ′ )) ∩ N δ | · K δ − d ( p, p ′ ) ≤ K (2 K − δ − d ( p, p ′ ) . This finishes the verification of properties (1) to (3).Properties (1)-(3) above finally allow us to use the Gram-Schmidt orthogonalization pro-cess to obtain a true v m +1 with the desired properties: v m +1 ( p ) := ˜ v m +1 ( p ) − P mi =1 ˜ v m +1 ( p ) · v i ( p ) | ˜ v m +1 ( p ) − P mi =1 ˜ v m +1 ( p ) · v i ( p ) | , p ∈ G. This is well-defined because | ˜ v m +1 ( p ) − m X i =1 ˜ v m +1 ( p ) · v i ( p ) | ≥ | ˜ v m +1 ( p ) | − m X i =1 | ˜ v m +1 ( p ) · v i ( p ) | ≥ √ − > , and clearly forms an orthonormal system along with v ( p ) , · · · , v m ( p ). We also note that | ˜ v m +1 ( p ) − m X i =1 ˜ v m +1 ( p ) · v i ( p ) | ≤ | ˜ v m +1 ( p ) | + m X i =1 | ˜ v m +1 ( p ) · v i ( p ) | ≤ √ 32 + 14 . To check the Lipschitz regularity of v m +1 we first compute(recalling δ − = 8 Km ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˜ v m +1 − m X i =1 ˜ v m +1 · v i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Lip ≤ k ˜ v m +1 k Lip + m X i =1 ( k ˜ v m +1 k C k v i k Lip + k ˜ v m +1 k Lip k v i k C ) ≤ K (2 K − δ − + m ( √ · K (2 K − δ − · K δ − (cid:16) K − √ K + 2 K m − m (cid:17) ≤ K δ − ( m + 1) = 32 K m ( m + 1) . By (2.4) we finally have k v m +1 k Lip ≤ (cid:16)(cid:0) √ − (cid:1) − + (cid:0) √ − (cid:1) − (cid:0) √ 32 + 14 (cid:1)(cid:17) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˜ v m +1 − m X i =1 ˜ v m +1 · v i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Lip ≤ (cid:16)(cid:0) √ − (cid:1) − + (cid:0) √ − (cid:1) − (cid:0) √ 32 + 14 (cid:1)(cid:17) · K m ( m + 1) ≤ K m ( m + 1) . (cid:3) Remark 4.1. In Theorem 4.1, the required dimension gap D − m − is precisely thetopological dimension of the given domain G , while in Theorem 4.2 the dimension gap is K , which is exponential in the ‘metric dimension’ log K of the given domain X . Perhapsone can hope to reduce the dimension gap in Theorem 4.2 to somewhere near Ω(log K ) , sayby using the machinery of random nets and partitions and their padding properties as in [12] . Note that a dimension gap of Ω(log K ) is necessary: take for example X = S n , D = 2 n + 1 , m = 1 , and v : S n ֒ → R n +1 the standard inclusion. Then the log of thedoubling constant for X is a universal constant multiple of n , and the dimension gap is D − m − n − . However, the high-dimensional hairy ball theorem tells us that nocontinuous orthonormal extension is possible, let alone a Lipschitz orthonormal extension. We now state and prove a version of Theorem 4.1 but for general Carnot groups, usingthe idea of proof of Theorem 4.2. Theorem 4.3. Let ≤ m ≤ D − n h +7 n h − , j ≥ , and let { R i } ji =1 be a log-concavesequence of positive reals. Let v , · · · , v m : G → S D − form an orthonormal system at eachpoint, with the uniform regularity bound j X k =1 R k k∇ k v i k C ≤ , i = 1 , · · · , m. NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 29 Then there exists v m +1 : G → S D − such that v , · · · , v m along with v m +1 form an orthonor-mal system at each point, and j X k =1 R k k∇ k v m +1 k C . G,m,j . Proof. By a simple rescaling argument using the scaling map δ R , we may assume R = 1.We repeat the proof of Theorem 4.2 with a slight variation.Take a maximal δ -net N δ of G , where δ = · nh m . Let ǫ = nh +1 . We repeat the firstpart of the proof of Theorem 4.2, but instead using the estimates |N δ ∩ B . δ ( p ) | ≤ n h , |N δ ∩ B δ ( p ) | ≤ n h which follow from (2.7). We again define the probability space Ω and random variables { v ′ m +1 ( p ) ∈ S D − : p ∈ N δ } , and for each p ∈ N δ , we define the slightly different event A p = { ω ∈ Ω : ∃ q ∈ N δ ∩ ( B . δ ( p ) \ { p } ) | v ′ m +1 ( p )( ω ) · v ′ m +1 ( q )( ω ) | > ǫ } . By the same computations as before we have that for any p, q ∈ N δ distinct,Pr( | v ′ m +1 ( p ) · v ′ m +1 ( q ) | > ǫ ) ≤ r π (cid:18) − ǫ D − m − (cid:19) , so we again estimate the probability of each A p using a union bound:Pr( A p ) ≤ |N δ ∩ ( B . δ ( p ) \ { p } ) | · r π (cid:18) − ǫ D − m − (cid:19) ≤ n h · r π (cid:18) − ǫ D − m − (cid:19) . Observing that for each p ∈ N δ , A p is mutually independent with the collection of events { A q : q ∈ N δ \ B δ ( p ) } , which are all of the A q ’s except possibly |N δ ∩ B δ ( p ) | ≤ n h ofthem, we see that our choice of parameters D − m − ≥ n h +7 n h and ǫ = nh +1 allows usto apply the Lov´asz local lemma because e · n h · n h · r π (cid:18) − ǫ D − m − (cid:19) < e · n h · r π (cid:18) − ǫ D − m − (cid:19) < e n h · exp (cid:18) − n h +5 (2 n h +7 n h ) (cid:19) = 1( ∵ x − log(28) x − − 12 log π > x ≥ . By the same limiting argument, we conclude the existence of an assignment { v ′ m +1 ( p ) } p ∈N δ such that for any distinct p, q ∈ N δ with d ( p, q ) < . δ we have | v ′ m +1 ( p ) · v ′ m +1 ( q ) | ≤ ǫ .As in the previous proof, we now ‘interpolate’ the discrete vector field { v ′ m +1 ( p ) } p ∈N δ to produce a vector field { ˜ v m +1 ( p ) } p ∈ G defined on the entirety of G , which nearly has thedesired properties. It is not difficult to define a ‘quadratic’ partition of unity { φ q } q ∈N δ , i.e.,functions φ q : G → [0 , 1] defined for each q ∈ N δ such that • supp φ q ⊂ B . δ ( q ), q ∈ N δ , • P q ∈N δ φ q = 1 on G , • k φ q k C + P jk =1 R k k∇ k φ q k C ≤ k φ q k C j +1 . G,m 1, uniformly over q ∈ N δ .We now interpolate the { v ′ m +1 ( p ) } p ∈N δ using { φ q } q ∈N δ :(4.9) ˜ v m +1 ( p ) = X q ∈N δ φ q ( p ) v ′ m +1 ( q ) , p ∈ G. We claim that ˜ v m +1 nearly satisfies the required properties, namely it satisfies(1) ≤ | ˜ v m +1 ( p ) | ≤ for each p ∈ G ,(2) | ˜ v m +1 ( p ) · v i ( p ) | ≤ m for each p ∈ G and i = 1 , · · · , m ,(3) k ˜ v m +1 k C + P jk =1 R k k∇ k ˜ v m +1 k C . G,j,m φ q , we see that the summation in (4.9) is locally finite:˜ v m +1 ( p ) = X q ∈N δ ∩ B . δ ( p ) φ q ( p ) v ′ m +1 ( q ) . From this, property (3) above immediately follows. We can verify property (1) using thenear-orthogonality and the fact that the sum of squares of the φ q ’s is 1: (cid:12)(cid:12) | ˜ v m +1 ( p ) | − (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X q,q ′ ∈N δ ∩ B . δ ( p ) q = q ′ φ q ( p ) φ q ′ ( p ) v ′ m +1 ( q ) · v ′ m +1 ( q ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X q,q ′ ∈N δ ∩ B . δ ( p ) q = q ′ φ q ( p ) φ q ′ ( p ) (cid:12)(cid:12) v ′ m +1 ( q ) · v ′ m +1 ( q ′ ) (cid:12)(cid:12) ≤ X q,q ′ ∈N δ ∩ B . δ ( p ) q = q ′ φ q ( p ) + φ q ′ ( p ) ǫ ≤ |N δ ∩ B . δ ( p ) | ǫ ≤ n h ǫ = 14 . Finally, we verify property (2). We have, for each p ∈ G and i = 1 , · · · , m ,˜ v m +1 ( p ) · v i ( p ) = X q ∈N δ ∩ B . δ ( p ) φ q ( p ) v ′ m +1 ( q ) · v i ( p ) . We observe that for each q ∈ N δ ∩ B . δ ( p ), we can estimate | v ′ m +1 ( q ) · v i ( p ) | ≤ | v ′ m +1 ( q ) · v i ( q ) | + | v ′ m +1 ( q ) · ( v i ( p ) − v i ( q )) |≤ | v i ( p ) − v i ( q ) | ≤ . δ, where we have used the orthogonality of v ′ m +1 ( q ) against v i ( q ) and the fact that k∇ v i k C ≤ 1. From these facts and Cauchy-Schwarz, we have the bound | ˜ v m +1 ( p ) · v i ( p ) | ≤ X q ∈N δ ∩ B . δ ( p ) φ q ( p ) · . δ ≤ |N δ ∩ B . δ ( p ) | / · . δ ≤ n h · . δ = 14 m . This finishes the verification of properties (1) to (3). Now we can use Gram-Schmidtorthogonalization to obtain a true v m +1 : v m +1 ( p ) := ˜ v m +1 ( p ) − P mi =1 ˜ v m +1 ( p ) · v i ( p ) | ˜ v m +1 ( p ) − P mi =1 ˜ v m +1 ( p ) · v i ( p ) | , p ∈ G. NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 31 This is well-defined because | ˜ v m +1 ( p ) − m X i =1 ˜ v m +1 ( p ) · v i ( p ) | ≥ | ˜ v m +1 ( p ) | − m X i =1 | ˜ v m +1 ( p ) · v i ( p ) | ≥ √ − > , and clearly forms an orthonormal system along with v ( p ) , · · · , v m ( p ). The required reg-ularity of v m +1 follows from the algebra property for norms defined by log-concave se-quences. (cid:3) By applying Theorem 4.3 several times, we obtain the following. Corollary 4.1. Let ≤ m ≤ D − n h +7 n h − m ′ , j ≥ , and let { R i } ji =1 be a log-concavesequence of positive reals. Let v , · · · , v m : G → S D − form an orthonormal system at eachpoint, with the uniform regularity bound j X k =1 R k k∇ k v i k C ≤ , i = 1 , · · · , m. Then there exist v m +1 , · · · , v m + m ′ : G → S D − such that v , · · · , v m along with v m +1 , · · · , v m + m ′ form an orthonormal system at each point, and j X k =1 R k k∇ k v i k C . G,m,m ′ ,j , i = m + 1 , · · · , m + m ′ . Remark 4.2. The need for a normal field in embedding problems dates back to [13] , wherethe existence of a section was demonstrated using a homotopy argument from [14] based onthe fact that the base space is contractible. However, such an argument in this situation willfail to control the regularity of the section at points of the base far from the contraction point.In [15, Section 8] this was achieved for the Heisenberg group H by imposing a ‘uniform’CW-structure on H and then inductively defining the section starting from low-dimensionalskeleta. In the inductive step in [15] , one has to use the fact that the homotopy groups π i ( S n ) vanish for i < n , which necessitates the ‘dimension gap’ D − m − ≥ . Here we havechosen the section over a 0-skeleton such that the section is locally roughly an orthonormalset, and obtained a section by directly interpolating. This allows us to avoid the need for aCW structure by only using the doubling property of the base space, but have thus increasedthe dimension gap exponentially. Main Iteration Lemma The starting point of the iterative construction is a function that oscillates at a fixedscale while satisfying a good freeness property. Incidentally, this will also be an ingredientin the inductive step of the iterative construction, when we pass from a larger scale A m +1 down to a smaller scale A m . We begin by constructing this single oscillating function onthe Carnot group G . (See also Proposition 5.2 of [15] for an analogous statement.) Proposition 5.1. There exists a smooth map φ : G → R nh with the following properties.(1) (Smoothness) For all j ≥ , k φ k C j . G,j . (2) (Locally free embedding) For all p ∈ G , we have (5.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W φ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ . Proof. Inspired by the Veronese-type embedding used in [15], we first begin with the function ϕ : G → L sr =1 ⊗ r R n = R P sr =1 ( nr ), ϕ (exp( x )) = s M r =1 r ! ⊗ r x, x ∈ g , where in the image we identify g with R n via P sr =1 P k r i =1 x r,i X r,i ↔ P sr =1 P k r i =1 x r,i f r,i . Inthese coordinates, we would have ϕ exp s X r =1 k r X i =1 x r,i X r,i !! = s X r =1 r ! ⊗ r s X r =1 k r X i =1 x r,i f r,i ! , x r,i ∈ R . Recall that X r,i = ∂∂x r,i + s X r ′ >r k r ′ X j =1 (polynomial in { x r ′′ ,i ′ } r ′′ 1] which is identically 1 on the unit ball B andwhich vanishes on B c . . Then the function ϕ : G → R P sr =1 ( nr ) defined by ϕ = ηϕ has NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 33 bounded C j norm for all j ≥ 1, and satisfies (cid:12)(cid:12)(cid:12)V W = X r ,j , ··· ,X rm,jm W ϕ (cid:12)(cid:12)(cid:12) = 1 on B . Nowtake a maximal 1-net N of G . We claim that we can decompose N = nh G a =1 N a , where n h is the Hausdorff dimension of G , and each N a is a 3-net of G . (This is a standardcoloring argument.) Indeed, each point g ∈ N has at most 7 n h − N in its3-neighborhood by (2.7); having inductively assigned a finite number of points of N inone of the 7 n h sets {N a } nh a =1 ’s, any other point of N can be assigned to one of them in aconsistent manner.We now define φ : G → R nh P sr =1 ( nr ) as φ ( p ) := nh M a =1 X g ∈N a ϕ ( g − p ) , p ∈ G. Then this satisfies the given properties because for each a = 1 , · · · , n h , the function P g ∈N a ϕ ( g − p ) is a sum of smooth compactly supported functions whose supports gB . are disjoint, so the sum P g ∈N a ϕ ( g − p ) has bounded C j norm and we have the wedgeproduct bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W X g ∈N a ϕ ( g − p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ p ∈ N a B . As {N a B } nh a =1 covers G , we see that ϕ satisfies (5.1). As P sr =1 (cid:0) nr (cid:1) ≤ n ≤ n h we aredone by composing φ with an embedding R nh P sr =1 ( nr ) → R nh . (cid:3) Remark 5.1. The method of proof of Proposition 5.1 in [15] for the case G = H , ormore generally for the case when G admits a cocompact lattice Γ , is the following. Sincethe nilmanifold G/ Γ is a smooth compact n -dimensional manifold, by the strong Whitneyimmersion theorem [16] there exists a smooth immersion G/ Γ → R n − . By pre-composingwith the projection G → G/ Γ , one obtains a map ϕ : G → R n − that satisfies the weakerfreeness property (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s ^ r =1 k r ^ i =1 X r,i ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & while having bounded C j norms due to the compactness of G/ Γ . We can then obtain thestronger freeness property (5.1) by composing ϕ with a Veronese-type embedding, say φ : G → L sr =1 ⊗ r R n − = R P sr =1 ( n − r ) , where φ = s M r =1 r ! ( ϕ ) ⊗ r . (One can indeed prove the stronger freeness property by using a simple change of coordinatesargument.) There is an exponential savings in the target dimension: the target dimensionin this case is polynomial in the topological dimension of G , whereas the target dimension in Proposition 5.1 is exponential in the Hausdorff dimension of G . Perhaps one could improvethe target dimension in Proposition 5.1 to be polynomial in the Hausdorff dimension of G ,say by using random nets and partitions as in [12] and being more careful about how wepaste the different embeddings. Now we state the inductive step. Having constructed a map ψ : G → R D which ‘rep-resents’ the geometry of G at scale A m +1 and above, we need to construct a correction φ : G → R D which oscillates at scale A m such that ψ + φ represents the geometry of G at scale A m . When we say that a mapping represents the geometry of G at scale A m , wemean that its C m ∗ , / A m norm is controlled and that it has good freeness properties. To makea viable induction argument, we need to show that the quantitative controls on the C m ∗ , / A m norm and the freeness properties are preserved when we pass from ψ to ψ + φ . By rescaling,we may simply assume m = 0. The precise statement for the inductive step is as follows. Proposition 5.2 (Main iteration step) . Let M be a real number with M ≥ C − , and let m ∗ ≥ max { , s } . Suppose a map ψ : G → R · nh obeys the following estimates:(1) (Non-degenerate first derivatives) For any p ∈ G , we have (5.2) C − M ≤ | X i ψ ( p ) | ≤ C M, i = 1 , · · · , k, and (5.3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =1 X j m ψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − i − i +20 i Y m =1 | X j m ψ ( p ) | for ≤ i ≤ k and ≤ j < · · · < j i ≤ k .(2) (Locally free embedding) For any p ∈ G , we have (5.4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W ψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +20 A − P sj =2 ( j − ( n + j − j ) i Y m =1 | X j m ψ ( p ) | . (3) (H¨older regularity at scale A ) We have (5.5) k∇ ψ k C m ∗− , / A ≤ C A − . Then there exists a map φ : G → R · nh such that φ obeys the following estimates:(1) (Regularity at scale 1) We have (5.6) k φ k C m ∗ , / . G . (2) (Orthogonality) We have (5.7) B ( φ, ψ ) = 0 . (3) (Non-degenerate first derivatives) We have (5.8) | X i φ | & G , i = 1 , · · · , k. and the sum ψ + φ obeys the following regularity estimates: NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 35 (1) (Non-degenerate first derivatives) For any p ∈ G , we have (5.9) C − p M + 1 ≤ | X i ( ψ + φ )( p ) | ≤ C p M + 1 , i = 1 , · · · , k, and (5.10) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =1 X j m ( ψ + φ )( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − i − i +20 i Y m =1 | X j m ( ψ + φ )( p ) | for ≤ i ≤ k and ≤ j < · · · < j i ≤ k .(2) (Locally free embedding) For any p ∈ G , we have (5.11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W ( ψ + φ )( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +30 A − P sj =2 ( j − ( n + j − j ) i Y m =1 | X j m ( ψ + φ )( p ) | . (3) (H¨older regularity at scale ) We have (5.12) k∇ ( ψ + φ ) k C m ∗− , / ≤ C . Remark 5.2. The dimension · n h will result from Proposition 5.1 along with Corollary4.1; see Lemma 5.2 below. Remark 5.3. Proposition 5.2 shows that if ψ is a map with the regularity and freenessproperties (5.2) - (5.5) at scale A , then we can find a correction φ so that ψ + φ is a mapwith the same regularity and freeness properties (5.9) - (5.12) but at scale . In particular,the constants in the freeness properties are the same. This will be made possible by makinga ‘hierarchy’ of freeness properties: the freeness property (5.10) for i -fold wedge products ofhorizontal derivatives of ψ + φ will be proven by the freeness property (5.3) for ( i − -foldwedge products of horizontal derivatives of ψ , and the freeness property (5.11) for the wedgeproduct of up to s -order derivatives of ψ + φ will be proven by the the freeness property (5.3) for k -fold wedge products of horizontal derivatives of ψ . Thus, we do not lose constantswhen passing from ψ to ψ + φ , and this will allow us to close the iteration. Proposition 5.2 will be a consequence of the following lemma, which is a generalizationof Proposition 5.1 of [15] for the case G = H . Lemma 5.1 (Main iteration lemma) . Let M be a real number with M ≥ C − , and let m ∗ ≥ max { , s } . Suppose a map ψ : G → R · nh obeys the following estimates:(1) (Non-degenerate first derivatives) For any p ∈ G , we have (5.13) C − M ≤ | X i ψ ( p ) | ≤ C M, i = 1 , · · · , k, and (5.14) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =1 X j m ψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − i − i +20 M i , for ≤ i ≤ k and ≤ j < · · · < j i ≤ k . (2) (Locally free embedding) For any p ∈ G , we have (5.15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W ψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +20 A − P sj =2 ( j − ( n + j − j ) M k . (3) (H¨older regularity at scale A ) We have (5.16) k∇ ψ k C m ∗− ,αA ≤ C A − . Then there exists a map φ : G → R · nh obeying the following estimates.(1) (Non-degenerate first derivatives) For any p ∈ G , we have (5.17) | X i φ ( p ) | & G and (5.18) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =1 X j m ( ψ + φ )( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =1 X j m ψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − i +50 M i − , for ≤ i ≤ k and ≤ j < · · · < j i ≤ k .(2) (Locally free embedding) For any p ∈ G , we have (5.19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W ( ψ + φ )( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +30 M k . (3) (H¨older regularity at scale 1) We have (5.20) k φ k C m ∗ ,α . G . (4) (Orthogonality) We have (5.21) B ( φ, ψ ) = 0 . We now show why Proposition 5.2 follows from Lemma 5.1. proof of Proposition 5.2 assuming Lemma 5.1. Let ψ be as in Proposition 5.2. One caneasily verify the hypotheses of Lemma 5.1 for ψ , as (5.14) and (5.15) each follow from (5.3)and (5.4) combined with (5.2). Thus, by Lemma 5.1, there exists a function φ : G → R · nh that satisfies (5.6)-(5.8) and the following:(1) (Non-degenerate first derivatives) For p ∈ G , 2 ≤ i ≤ k and 1 ≤ j < · · · < j i ≤ k ,(5.22) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ q =1 X j q ( ψ + φ )( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ q =1 X j q ψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − i +50 M i − (2) (Locally free embedding) For any p ∈ G , we have(5.23) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W ( ψ + φ )( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +30 M k . NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 37 We now verify that the map ψ + φ satisfies the properties (5.9) to (5.12).(1) Verification of (5.9). By (5.7), we have | X i ( ψ + φ ) | = | X i ψ | + | X i φ | . But by (5.8)and (5.6), we have | X i φ | ∼ G 1, so we have C − ≤ | X i φ | ≤ C . Combining thesefacts with (5.2) we obtain (5.9).(2) Verification of (5.10). This follows from (5.22): (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ q =1 X j q ( ψ + φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ q =1 X j q ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + C − i +50 M i − ≥ C − i − i +40 i Y q =1 (cid:12)(cid:12) X j q ψ (cid:12)(cid:12) + C − i +50 M i − ≥ C − i − i +40 i Y q =1 (cid:16)(cid:12)(cid:12) X j q ψ (cid:12)(cid:12) + | X j q φ | (cid:17) = C − i − i +40 i Y q =1 | X j q ( ψ + φ ) | where in the third inequality we used | X j q ψ | ≤ C M and | X j q φ | . G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X ri,ji ≤ i ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r i ,j i ) W ( ψ + φ )( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +30 M k ≥ C − k − k +30 k Y i =1 | X i ( ψ + φ )( p ) | , where in the last inequality we used that | X i ( ψ + φ ) | = | X i ψ | + | X i φ | ≤ C M + O G (1) ≤ C M (recall that M ≥ C − ).(4) Verification of (5.12). This follows from (5.5) and (5.6): k∇ ( ψ + φ ) k C m ∗− , / ≤ k∇ ψ k C m ∗− , / A + k φ k C m ∗ , / ≤ C . (cid:3) For the rest of the section, we will prove Lemma 5.1.Suppose that ψ is as in Lemma 5.1. We will first construct a solution ˜ φ to the low-frequency equation (3.2) as(5.24) ˜ φ ( p ) = U ( p ) φ ( p ) , where φ : G → R nh is as in Proposition 5.1, and U : R nh → R · nh is a linearisometry with the following properties, which is constructed using Corollary 4.1. (See alsoLemma 9.1 of [15] for an analogous statement for H .) Lemma 5.2. There exists U : G → hom( R nh , R · nh ) such that(1) For each p ∈ G , U ( p ) ∈ hom( R nh , R · nh ) is an isometry. (2) For each p ∈ G , s ∈ R nh , we have (5.25) ( U ( p ) s ) · X i P ≤ N ψ ( p ) = 0 , ( U ( p ) s ) · X i X j P ≤ N ψ ( p ) = 0 , ≤ i, j ≤ k. (3) We have the smoothness k∇ U k C m ∗ . C A . Proof. Let W , · · · , W k ( k +3)2 + k denote the rescaled differential operators(( M − X i ) ki =1 , ( AX i X j ) ≤ i ≤ j ≤ k , ( AX ,i ) k i =1 ) . Let w i ( p ) := W i P ≤ N ψ ( p ) for 1 ≤ i ≤ k ( k +3)2 + k . Then, by (5.13) and Theorem 2.1 (3), w i = W i ψ + O (cid:18) AM (cid:19) = O ( C ) , i = 1 , · · · , k, and by (5.16) and Theorem 2.1 (3), w i = W i ψ + O (cid:18) A (cid:19) = O ( C ) , i = k + 1 , · · · , k ( k + 3)2 + k . On the other hand, (5.15) tells us that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ i =1 , ··· , k ( k +3)2 + k W i ψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & C , so by applying the triangle inequality and Cauchy-Schwarz inequality, we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ i =1 , ··· , k ( k +3)2 + k w i ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & C . By applying Cauchy-Schwarz again, we conclude that(5.26) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ i =1 , ··· ,j w i ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ C , j = 1 , · · · , k ( k + 3)2 + k . By more applications of (5.16) along with Theorem 2.1 (3), we can see that k w i k C + A k∇ w i k C m ∗ . C , along with (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ^ i =1 , ··· ,j w i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C + A (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ^ i =1 , ··· ,j ∇ w i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C m ∗ . C . Note that this norm is of the form (2.6) and thus we can apply the product rule to thisnorm. Now let us consider the orthonormal system v , · · · , v k ( k +3)2 + k formed by applyingthe Gram-Schmidt process to the vectors w i , i.e., we inductively define v i := | V j
By (5.26) this is well-defined, and by a repeated application of the product rule, one candeduce the smoothness k v i k C + A k∇ v i k C m ∗ . C , i = 1 , · · · , k ( k + 3)2 + k . We now apply Corollary 4.1 with m = k ( k +3)2 , m ′ = 14 n h , j = m ∗ + 1, R i = A to theabove v i ’s. This is possible because k ( k + 3)2 + k + 14 n h + 2 n h +7 n h ≤ n h + 32 n h + 14 n h + 2 n h +7 n h ≤ · n h ( ∵ n h ≥ . Thus we have maps v k ( k +3)2 + k +1 , · · · , v k ( k +3)2 + k +14 nh : G → R · nh such that k v i k C + A k∇ v i k C m ∗ . C , i = 1 , · · · , k ( k + 3)2 + k + 14 n h and such that v ( p ) , · · · , v k ( k +3)2 + k +14 nh ( p ) are orthonormal for all p ∈ G .Now define U ( p ) to be the map U ( p )( s ) = nh X i =1 s i v k ( k +3)2 + k + i ( p ) . This clearly has the properties (1) and (3) asserted above, and we can also deduce property(2) once we note that X i X j − X j X i ∈ span { X , , · · · , X ,k } . (cid:3) Now let us define ˜ φ as in (5.24). By Lemma 5.2 and Proposition 5.1 (2), we have k ˜ φ k C m ∗ , / . k ˜ φ k C m ∗ +1 . G 1, where we take α = . Also, by (5.25) and the Leibniz rule,it is clear that ˜ φ satisfies (4.1) and a fortiori solves the low-frequency equation (3.2). It isalso clear that ψ satisfies the hypothesis of Corollary 3.1. By applying Corollary 3.1, thereexists a C m ∗ , / -function φ : G → R · nh such that B ( φ, ψ ) = 0 , k φ − ˜ φ k C m ∗ , / . C A − m ∗ , and k X i ( φ − ˜ φ ) · X j ψ k C . C A − m ∗ , ≤ i, j ≤ k. It remains to verify conditions (5.17)-(5.21). Conditions (5.20) and (5.21) are immediatefrom the construction. For later use, we note that(5.27) X i φ = X i ˜ φ + X i ( φ − ˜ φ ) = U ( X i φ ) + ( X i U ) φ + O C (cid:16) A − m ∗ (cid:17) = U ( X i φ ) + O C (cid:0) A − (cid:1) for i = 1 , · · · , k . From this and Proposition 5.1, we immediately have | X i φ | ∼ G , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ^ i =1 X i φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ G , so in particular (5.17) immediately follows. It remains to verify (5.18) and (5.19). (1) Verification of (5.18). We first estimate, for any 1 ≤ i, j ≤ k , i = j , that X i φ · X j ψ = X i ( φ − ˜ φ ) · X j ψ + X i ˜ φ · X j P >N ψ + X i ˜ φ · X j P ≤ N ψ = X i ( φ − ˜ φ ) · X j ψ + X i ˜ φ · X j P >N ψ + X i ( ˜ φ · X j P ≤ N ψ ) − ˜ φ · X i X j P ≤ N ψ Lemma . X i ( φ − ˜ φ ) · X j ψ + X i ˜ φ · X j P >N ψ + 0 − O C (cid:16) A − m ∗ (cid:17) + O (1) · O G (cid:0) A − (cid:1) = O G (cid:0) A − (cid:1) . (5.28) Now we observe that for 1 ≤ j < · · · < j i ≤ k we have the expansion i ^ m =1 X j m ( ψ + φ ) = i ^ m =1 X j m ψ + i − X n =1 i ^ m =1 X j m f nm , for some sequence { f nm } n =1 , ··· , i − , m =1 , ··· ,i of functions, each being either φ or ψ .Note that for each n there must exist some m such that f nm = φ . This expansionimplies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =1 X j m ( ψ + φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =1 X j m ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − X n =1 i ^ m =1 X j m f nm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 i − X n =1 (cid:28) i ^ m =1 X j m ψ, i ^ m =1 X j m f nm (cid:29) . For each fixed n , the depolarised Cauchy-Binet formula shows that (cid:28) V im =1 X j m ψ, V im =1 X j m f nm (cid:29) can be represented as the determinant of an i × i matrix, whose entries are each ofsize O C ( M ) and one of whose columns(the m -th column, where m is such that f nm = φ ) consists of entries of size O C ( A − ). Thus the determinant of this i × i matrix is of size O C ( A − M i − ), or equivalently (cid:28) V im =1 X j m ψ, V im =1 X j m f nm (cid:29) = O C ( A − M i − ). Summing over all n , we obtain2 i − X n =1 (cid:28) i ^ m =1 X j m ψ, i ^ m =1 X j m f nm (cid:29) = O C ( A − M i − ) , and so it is enough to show that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − X n =1 i ^ m =1 X j m f nm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − i +5 . M i − . But by Cauchy-Schwarz, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − X n =1 i ^ m =1 X j m f nm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≍ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − X n =1 i ^ m =1 X j m f nm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =2 X j m φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − X n =1 i ^ m =1 X j m f nm ∧ i ^ m =2 X j m φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j φ ∧ i ^ m =2 X j m ψ ! ∧ i ^ m =2 X j m φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =2 X j m ψ ∧ i ^ m =1 X j m φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 41 By the Cauchy-Binet formula, this is the determinant of a certain (2 i − × (2 i − i − × ( i − 1) block consists of entries of size O C ( M ), the lower-right i × i block consistsof entries of size O C (1), while the off-block-diagonal entries are of size O C ( A − ).Therefore, we may estimate the total determinant with the determinant of the blockdiagonal approximation, with error O C ( A − M i − ): (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =2 X j m ψ ∧ i ^ m =1 X j m φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =2 X j m ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =1 X j m φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O C ( A − M i − ) . But by (5.14), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =2 X j m ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − i +60 M i − . This completes the verification of (5.18).(2) Verification of (5.19). We first observe that for differential operators W of degree atleast 2, W φ dominates W ψ , and so we may approximate V W W ( ψ + φ ) by V W W φ ,where W ranges over such operators.More precisely, for W = X r ,j · · · X r m ,j m , where either m = 1 and r ≥ ≤ m ≤ s , we have W ( ψ + φ ) = W ˜ φ + W ( φ − ˜ φ ) + W ψ = W (cid:16) U ( φ ) (cid:17) + O C ( A − m ∗ ) + O C ( A − )= W (cid:16) U ( φ ) (cid:17) + O C ( A − ) . (Note that our choice of m ∗ ≥ s allows us to use our bounds on k φ − ˜ φ k C m ∗ and k∇ ψ k C m ∗− A above). Many applications of the Leibniz rule tell us that W (cid:16) U ( φ ) (cid:17) − U ( W φ ) is a linear combination of derivatives of U times φ or derivatives of φ , sofrom k∇ U k C m ∗ . C A − , we have W (cid:16) U ( φ ) (cid:17) − U ( W φ ) = O C ( A − )and consequently W ( ψ + φ ) = U ( W φ ) + O C ( A − ) . Therefore we have ^ W = X r ,j ··· ,X rm,jm m =1 and r ≥ , or ≤ m ≤ s W ( ψ + φ ) = ω + O C ( A − ) , where ω := ^ W = X r ,j ··· ,X rm,jm m =1 and r ≥ , or ≤ m ≤ s U ( W φ ) . Since U is an isometry, and φ has the freeness property (5.1), we have | ω | = (cid:12)(cid:12)V W as above W φ (cid:12)(cid:12) ≍ G 1. To verify (5.19), it is enough to verify (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ^ i =1 X i ( ψ + φ ) ∧ ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & C − k − k +3 . M k . By Cauchy-Schwarz, and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ^ i =1 X i P ≤ N ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k Y i =1 | X i P ≤ N ψ | . C k Y i =1 | X i ψ | ≤ C k M k , we see that it is enough to verify (cid:28) k ^ i =1 X i P ≤ N ψ ∧ ω, k ^ i =1 X i ( ψ + φ ) ∧ ω (cid:29) & C − k − k +3 . M k . Since all of the components of ω are orthogonal to the vectors X i P ≤ N ψ , we can useCauchy-Binet twice to see that the left hand side is equal to (cid:28) k ^ i =1 X i P ≤ N ψ, k ^ i =1 X i ( ψ + φ ) (cid:29) | ω | . Using that for 1 ≤ i, j ≤ k we have X i P ≤ N ψ · X j ( ψ + φ ) = X i ψ · X j ψ − X i P >N ψ · X j ( ψ + φ )+ X i ψ · X j φ = X i ψ · X j ψ + O ( A − M ) , we see that, using Cauchy-Binet twice, that (cid:28) k ^ i =1 X i P ≤ N ψ, k ^ i =1 X i ( ψ + φ ) (cid:29) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ^ i =1 X i ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O C ( A − M k − )and now, from (5.14), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ^ i =1 X i ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +40 M k , and the claim follows.This concludes the proof of Lemma 5.1.6. Construction of the embedding By repeating Proposition 5.2 a finite number of times, one obtains the following. (Seealso Claim 5.4 of [15] for an analogous statement for H .) Proposition 6.1 (Finite iteration) . Let < ε ≤ /A , and let M ≤ M be integers. Onecan find maps φ m : G → R · nh for M ≤ m ≤ M obeying the following bounds, where φ ≥ m : G → R · nh is defined by φ ≥ m := X m ≤ m ′ ≤ M A − ε ( m ′ − m ) φ m ′ . NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 43 (1) (Smoothness at scale A m ) For all M ≤ m ≤ M , we have (6.1) k φ m k C s s +1 Am ≤ C A m and (6.2) k∇ φ ≥ m k C s s − , / Am ≤ C A − m . (2) (Orthogonality) One has, for all M ≤ m ≤ M , (6.3) X m ′ >m A − ε ( m ′ − m ) B ( φ m , φ m ′ ) = 0 . (3) (Non-degeneracy) For all p ∈ G and M ≤ m ≤ M , we have the estimates (6.4) | X i φ m ( p ) | ≥ C − , i = 1 , · · · , k, (6.5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ q =1 X j q φ ≥ m ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − i − i +20 i Y q =1 | X j q φ ≥ m ( p ) | , ≤ i ≤ k, ≤ j < · · · < j i ≤ k, and (6.6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X ri,ji ≤ i ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r i ,j i ) W φ ≥ m ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +30 A − m P sj =2 ( j − ( n + j − j ) k Y i =1 | X i φ ≥ m ( p ) | . Proof. We prove this by induction on M − M . Note that the statement is invariant underrescaling, so we may assume M = 0. The case M = 0 follows directly from Proposition 5.1,so we may assume M > 0. By the inductive hypothesis applied to M ′ = 1 and M ′ = M ,we may find functions φ m , 1 ≤ m ≤ M , so that(1) (Smoothness at scale A n ) For all 1 ≤ m ≤ M , we have(6.7) k φ m k C s s +1 Am ≤ C A m and(6.8) k∇ φ ≥ m k C s s − , / Am ≤ C A − m . (2) (Orthogonality) One has, for all 1 ≤ m ≤ M ,(6.9) X m ′ >m A − ε ( m ′ − m ) B ( φ m , φ m ′ ) = 0 . (3) (Non-degeneracy) For all p ∈ G and 1 ≤ m ≤ M , we have the estimates(6.10) | X i φ m ( p ) | ≥ C − , i = 1 , · · · , k, (6.11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ q =1 X j q φ ≥ m ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − i − i +20 i Y q =1 | X j q φ ≥ m ( p ) | , ≤ i ≤ k, ≤ j < · · · < j i ≤ k, and(6.12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X ri,ji ≤ i ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r i ,j i ) W φ ≥ m ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +30 A − m P sj =2 ( j − ( n + j − j ) k Y i =1 | X i φ ≥ m ( p ) | . We now verify the hypotheses of Proposition 5.2 for ψ := A − ε φ ≥ = P ≤ m ≤ M A − εm φ m with M = (cid:16)P ≤ m ≤ M A − εm (cid:17) / , m ∗ = s + s + 1 and α = . We have M ≥ A − ε ≥ A − /A ≥ , so we have M ≥ C − .(1) Verification of (5.2). From (6.9) and (6.10) we have, for i = 1 , · · · , k , | X i ψ | = X ≤ m ≤ M A − εm | X i φ m | ≥ X ≤ m ≤ M A − εm C − = C − M , and from (6.7) we have, for i = 1 , · · · , k , | X i ψ | = X ≤ m ≤ M A − εm | X i φ m | ≤ X ≤ m ≤ M A − εm C = C M . (2) Verification of (5.3). From (6.11) we have, for 2 ≤ i ≤ k and 1 ≤ j < · · · < j i ≤ k , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ q =1 X j q ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = A − iε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ q =1 X j q φ ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ A − iε C − i − i +20 i Y q =1 (cid:12)(cid:12) X j q φ ≥ (cid:12)(cid:12) = C − i − i +20 i Y q =1 (cid:12)(cid:12) X j q ψ (cid:12)(cid:12) . (3) Verification of (5.4).From (6.12) we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W ψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = A − P sm =1 ( n + m − m ) ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W φ ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ A − P sm =1 ( n + m − m ) ε C − k − k +30 A − m P sj =2 ( j − ( n + j − j ) k Y i =1 | X i φ ≥ m ( p ) | = A − ( P sm =1 ( n + m − m ) − k ) ε C − k − k +30 A − m P sj =2 ( j − ( n + j − j ) k Y i =1 | X i ψ ( p ) |≥ C − k − k +20 A − m P sj =2 ( j − ( n + j − j ) k Y i =1 | X i ψ ( p ) | . (4) Verification of (5.5).From (6.8) we have k∇ ψ k C s s − , / A = A − ε k∇ φ ≥ k C s s − , / A ≤ A − ε C A − ≤ C A − . Hence ψ and M satisfy the assumptions of Proposition 5.2, and so there exists a function φ : G → R · nh that satisfies the following: NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 45 (1) (Regularity at scale 1) We have(6.13) k φ k C s s +1 , / . G . (2) (Orthogonality) We have(6.14) B ( φ , ψ ) = 0 . (3) (Non-degenerate first derivatives) For any p ∈ G , we have(6.15) | X i φ ( p ) | & G , i = 1 , · · · , k, (6.16) C − p M + 1 ≤ | X i ( ψ + φ )( p ) | ≤ C p M + 1 , i = 1 , · · · , k, and(6.17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i ^ m =1 X j m ( ψ + φ )( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − i − i +20 i Y m =1 | X j m ( ψ + φ )( p ) | for 2 ≤ i ≤ k and distinct 1 ≤ j < · · · < j i ≤ k .(4) (Locally free embedding) For any p ∈ G , we have(6.18) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W ( ψ + φ )( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +30 A − P sj =2 ( j − ( n + j − j ) i Y m =1 | X j m ( ψ + φ )( p ) | . (5) (H¨older regularity at scale 1) We have(6.19) k∇ ( ψ + φ ) k C s s − , / ≤ C . Note that φ ≥ = ψ + φ . To verify that the larger family of maps { φ m } ≤ m ≤ M satisfies theproperties (6.1) to (6.6), we need only verify these properties for m = 0. But, for m = 0,(6.1) follows directly from (6.13) and (6.4) follows from (6.15), while (6.2), (6.3), (6.5), and(6.6) are precisely conditions (6.19), (6.14), (6.17), and (6.18), respectively. (cid:3) By taking the limit M → −∞ , M → ∞ , we now obtain a full set of lacunary maps.(See also Theorem 4.1 of [15] for an analogous statement for H .) Theorem 6.1 (Maps oscillating at lacunary scales) . Let < ε ≤ /A . Then one can finda map φ m : G → R · nh for each integer m obeying the following bounds: • (Smoothness at scale A n ) For all integers m , one has k φ m k C s sAm . C A m . In particular, we have (6.20) X r,i φ m ( p ) = O C ( A − m ( r − ) , for all r, i. • (Orthogonality) For all integers m , one has X m ′ >m A − ε ( m ′ − m ) B ( φ m , φ m ′ ) = 0 identically on G . (By (6.20) this sum is absolutely convergent.) • (Non-degeneracy and immersion) For all integers m and all p ∈ G , one has | X i φ m ( p ) | & C and (6.21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X ri,ji ≤ i ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r i ,j i ) W φ ≥ m ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & C Y W = X r ,j ··· X ri,ji ≤ i ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r i ,j i ) | W φ ≥ m ( p ) | , where φ ≥ m ( p ) = X m ′ ≥ m A − ε ( m ′ − m ) φ m ′ . Proof. For each M ∈ N , we can apply Proposition 6.1 to find φ Mm : G → R · nh for − M ≤ m ≤ M obeying the following bounds, where φ M ≥ m : G → R · nh is defined by φ M ≥ m := X m ≤ m ′ ≤ M A − ε ( m ′ − m ) φ Mm ′ . (1) (Smoothness at scale A m ) For all − M ≤ m ≤ M , we have(6.22) k φ Mm k C s s +1 Am ≤ C A m . (2) (Orthogonality) One has, for all − M ≤ m ≤ M , X m ′ >m A − ε ( m ′ − m ) B ( φ Mm , φ Mm ′ ) = 0 . (3) (Non-degeneracy) For all p ∈ G and − M ≤ m ≤ M , we have the estimates | Xφ Mm ( p ) | ≥ C − , (6.23) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X ri,ji ≤ i ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r i ,j i ) W φ M ≥ m ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +30 A − m P sj =2 ( j − ( n + j − j ) k Y i =1 | X i φ M ≥ m ( p ) | . From (6.22) and (6.23) we see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X ri,ji ≤ i ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r i ,j i ) W φ M ≥ m ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C − k − k +3 − P sj =2 ( j − ( n + j − j ) Y W = X r ,j ··· X ri,ji ≤ i ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r i ,j i ) | W φ M ≥ m ( p ) | . For each m ∈ Z , the sequence { φ Mm } M ≥| m | is bounded in the C s + s +1 A m norm, so by theArzela-Ascoli theorem, one can find a subsequence of { M k } such that { φ M k m } M k ≥| m | locallyconverges in the C s + s topology, say to φ m , for every m ∈ Z . It now readily follows thatthese φ m satisfy the above properties. (cid:3) NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 47 Once we have this lacunary family guaranteed by Theorem 6.1, we can construct a func-tion Φ : G → R · nh byΦ ( p ) := ∞ X m = −∞ A − εm ( φ m ( p ) − φ m (0)) . Then Φ is ‘almost’ a bi-Lipschitz embedding of ( G, d − εG ) into R · nh : Proposition 6.2. The map Φ : G → R · nh satisfies the following estimates.(1) (Lipschitz upper bound) | Φ ( p ) − Φ ( p ′ ) | . A ε − / d G ( p, p ′ ) − ε , ∀ p, p ′ ∈ G. (2) (Partial Lipschitz lower bound) For p, p ′ ∈ G so that A n A − / ( s +1) ≤ d G ( p, p ′ ) ≤ A n A − / ( s +1) for some integer n , we have (6.24) | Φ ( p ) − Φ ( p ′ ) | & A d G ( p, p ′ ) − ε . Proof. (1) Let p, p ′ ∈ G . By translating and rescaling, we may assume p ′ = 0 and A − ≤ d G ( p, ≤ 1. We introduce the low-frequency componentΨ( q ) := ∞ X m =0 A − εm ( φ m ( q ) − φ m (0)) , q ∈ G. Then(6.25) Φ ( q ) = Ψ( q ) + O C ( A − ) , so it will be enough to show | Ψ( p ) | . A ε − / . As d G ( p, ≤ 1, there exists a horizontal curve γ in G from 0 to p of length ≤ | Ψ( p ) | = | Ψ( p ) − Ψ(0) | ≤ · k|∇ Ψ |k L ∞ ( G ) , But by the orthogonality statement of Theorem 6.1, we have | X i Ψ | = | ∞ X m =0 A − εm X i φ m | = ∞ X m =0 A − εm | X i φ m | ! / ≍ C M, i = 1 , · · · , k, where M := ∞ X n =0 A − εn ! / = (cid:18) − A − ε (cid:19) / ≍ √ ε log A . A ε − / , so we conclude | Ψ( p ) | ≤ k∇ Ψ k L ∞ ℓ . C M . A ε − / , as desired. (2) Let p, p ′ ∈ G be so that A n A − / ( s +1) ≤ d G ( p, p ′ ) ≤ A n A − / ( s +1) for some integer n . Again, by translating and rescaling, we may assume p ′ = 0 and n = 0, i.e. A − / ( s +1) ≤ d G ( p, ≤ A − / ( s +1) . Writing p = exp (cid:16)P sr =1 P k r i =1 x r,i X r,i (cid:17) , we have s X r =1 k r X j =1 | x r,j | /r ≍ G d G ( p, ≍ A − / ( s +1) . Equivalently, we have | x r,j | . G A − r/ ( s +1) for all r, j , and there exists some pair ( r, j )such that | x r,j | ≍ G A − r/ ( s +1) .By Taylor expansionΨ( p ) = Ψ(0) + s X m =1 m ! s X r =1 k r X j =1 x r,j X r,j m Ψ(0) + O C ( A − )(note that this is where we used the C s ( s +1) -regularity of Ψ), and in light of (6.25)and Ψ(0) = 0, we haveΦ ( p ) = s X m =1 m ! s X r =1 k r X j =1 x r,j X r,j m Ψ(0) + O C ( A − ) . This expression contains many terms of the form X r ,j · · · X r m ,j m Ψ, where the( r , j ) , · · · , ( r m , j m ) are arbitrary with m ≤ s and are often unordered. In or-der to use the freeness property (6.21), it would be necessary to modify the aboveTaylor expansion formula so that the only differential operators acting on Ψ are theones of the form X r ,j · · · X r m ,j m where ( r , j ) (cid:22) · · · (cid:22) ( r m , j m ).This modification is possible once we note that for any permutation π of { , · · · , m } we can express X r ,j · · · X r m ,j m − X r π (1) ,j π (1) · · · X r π ( m ) ,j π ( m ) as a linear combinationof differential operators X r ′ ,j ′ · · · X r ′ m ′ ,j ′ m ′ where P m ′ i =1 r ′ i = P mi =1 r i and ( r ′ , j ′ ) (cid:22)· · · (cid:22) ( r ′ m ′ , j ′ m ′ ). This can be proven by a simple induction argument on m using thefact that [ X r,j , X r ′ ,j ′ ] ∈ V r + r ′ is a linear combination of X r + r ′ ,i for i = 1 , · · · , k r + r ′ if r + r ′ ≤ s , and [ X r,j , X r ′ ,j ′ ] = 0 if r + r ′ > s . Applying this fact to the aboveTaylor expansion formula and keeping track of the degrees, we obtain the followingmodified Taylor expansion formula:Ψ( p ) = s X r =1 k r X j =1 ( x r,j + p r,j ) X r,j Ψ(0)+ s X m =2 X ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) p ( r ,j ) , ··· , ( r m ,j m ) X r ,j · · · X r m ,j m Ψ(0) + O C ( A − )(6.26) where p r,j is a homogeneous polynomial of degree r where each monomial is aproduct of at least two terms, each of the form x r ′ ,j ′ with r ′ < r (recall that wedefine the homogeneous degree by assigning weight r to x r,j ), and p ( r ,j ) , ··· , ( r m ,j m ) is a homogeneous polynomial of degree P mi =1 r i .By the freeness property (6.21), each term in (6.26)(except for the error term)may serve as a lower bound for the entire sum, up to multiplicative constants. Thusit suffices to show that there exists r, j such that x r,j + p r,j has non-negligible size. NOWFLAKES OF CARNOT GROUPS INTO EUCLIDEAN SPACE 49 More precisely, the freeness property (6.21) tells us that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ^ W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) W Ψ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≍ C Y W = X r ,j ··· X rm,jm ≤ m ≤ s, ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) | W Ψ( p ) | , which immediately gives us, for each ( r , j ), the following control on the main termof the Taylor expansion: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s X r =1 k r X j =1 ( x r,j + p r,j ) X r,j Ψ(0) + s X m =2 X ( r ,j ) (cid:22)···(cid:22) ( r m ,j m ) p ( r ,j ) , ··· , ( r m ,j m ) X r ,j · · · X r m ,j m Ψ(0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & C | ( x r ,j + p r ,j ) X r ,j Ψ(0) | . So it remains to single out a pair ( r , j ) such that the right-hand side is large enough.Indeed, recall that we have | x r,j | . G A − r/ ( s +1) for all r, j , and | x r,j | ≍ G A − r/ ( s +1) for some pair ( r, j ). Therefore, there exists some ( r , j ) such that | x r,j | < A − r − . for all r < r , ≤ j ≤ k r , | x r ,j | ≥ A − r − . . Then, as p r ,j is homogeneous of degree r and consist of monomials which arethe product of at least two terms, we must have | p r ,j | . G A − r − . Hence | x r ,j + p r ,j | ≥ | x r ,j |−| p r ,j | ≥ A − r − . and we see that | ( x r ,j + p r ,j ) X r ,j Ψ(0) | & C A − r − . . In conclusion, we have | Ψ( p ) | & C A − r − . , or | Ψ( p ) | & A 1. This com-pletes the proof. (cid:3) Now, with Proposition 6.2 in hand, we are ready to prove the main theorem. Proof. Given the map Φ as above, we define the map φ : G → R (128 · nh ) log A by φ ( p ) := (Φ ( δ m ( p ))) log A − m =0 . It remains to observe d G ( p, p ′ ) − ε . A | φ ( p ) − φ ( p ′ ) | . A ε − / d G ( p, p ′ ) − ε , p, p ′ ∈ G. The upper bound follows from Proposition 6.2 (i), and the lower bound follows by observingthat we may apply (6.24) to at least one of the functions Φ ( δ m ( p )) for any given pair p, p ′ ∈ G . (cid:3) References [1] Abraham, I., Bartal, Y., and Neiman, O. Embedding metric spaces in their intrinsic dimension. In Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms (2008), pp. 363–372.[2] Assouad, P. Plongements lipschitziens dans R n . Bull. Soc. Math. France 111 (1983), 429–448.[3] Bartal, Y., Recht, B., and Schulman, L. J. Dimensionality reduction: beyond the Johnson-Lindenstrauss bound. In Proceedings of the twenty-second annual ACM-SIAM symposium on DiscreteAlgorithms (2011), SIAM, pp. 868–887.[4] Gottlieb, L.-A., and Krauthgamer, R. A nonlinear approach to dimension reduction. DiscreteComput. Geom. 54 , 2 (2015), 291–315.[5] Gupta, A., Krauthgamer, R., and Lee, J. R. Bounded geometries, fractals, and low-distortionembeddings. In (2003), IEEE, pp. 534–543. [6] Har-Peled, S., and Mendel, M. Fast construction of nets in low-dimensional metrics and theirapplications. SIAM J. Comput. 35 , 5 (2006), 1148–1184.[7] H¨ormander, L. Hypoelliptic second order differential equations. Acta Math. 119 (1967), 147–171.[8] Hulanicki, A. A functional calculus for Rockland operators on nilpotent Lie groups. Studia Math. 78 (1984), 253–266.[9] Lang, U., and Plaut, C. Bilipschitz embeddings of metric spaces into space forms. Geom. Dedicata87 , 1-3 (2001), 285–307.[10] Le Donne, E. A primer on Carnot groups: homogenous groups, Carnot-Carath´eodory spaces, andregularity of their isometries. Anal. Geom. Metr. Spaces 5 , 1 (2017), 116–137.[11] Milman, V. D., and Schechtman, G. Asymptotic Theory of Finite Dimensional Normed Spaces .Springer-Verlag, Berlin, Heidelberg, 1986.[12] Naor, A., and Neiman, O. Assouad’s theorem with dimension independent of the snowflaking. Rev.Mat. Iberoam. 28(4) (2010), 1123–1142.[13] Nash, J. C1 isometric imbeddings. Ann. of Math. (1954), 383–396.[14] Steenrod, N. The topology of fibre bundles . Princeton University Press, 1999.[15] Tao, T. Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimaldistortion. arXiv preprint arXiv:1811.09223 (2018).[16] Whitney, H. The singularities of a smooth n-manifold in (2n-1)-space. Ann. of Math. (1944), 247–293.(Sang Woo Ryoo) Mathematics Department, Princeton University,Princeton, New Jersey 08544-1000, United States E-mail address ::