Volumes spanned by k point configurations in \mathbb{R}^d
VVolumes spanned by k -point configurations in R d Belmiro Galo and Alex McDonaldFebruary 5, 2021
Abstract
Given a k -point configuration x P p R d q k , we consider the ` kd ˘ -vector of volumes determined by choosingany d points of x . We prove that a compact set E Ă R d determines a positive measure of such volumetypes if the Hausdorff dimension of E is greater than d ´ d ´ k ´ d . This generalizes results of Greenleaf,Iosevich, and Mourgoglou [12], Greenleaf, Iosevich, and Taylor [13], and the second listed author [19]. A recurrent theme throughout mathematics is to show that if one has a set which is sufficiently structuredin some way and applies a non-trivial map, the image is also structured. A classic example of this themeis the Falconer distance problem, which is one of the most important and interesting problems in geometricmeasure theory. Given a set E Ă R d , define its distance set to be∆ p E q “ t| x ´ y | : x, y P E u . The Falconer distance problem asks how large the Hausdorff dimension of a compact set E must be toensure that ∆ p E q has positive Lebesgue measure. Falconer [9] proved that dim E ą d ` implies ∆ p E q haspositive measure, where here and throughout dim E denotes the Hausdorff dimension of the set E . He alsofound a family of examples t E s u such that for any s ă d , one has dim E s ą s and L p ∆ p E qq “
0. This sug-gests what is now known as the Falconer distance problem, which asks for the smallest s such that dim E ą s implies L p ∆ p E qq ą
0. Falconer’s work implies this threshold is between d and d ` , and it is conjecturedthat d is in fact the correct threshold. The first major results were due to Wolff [20] and Erdogan [8], provingthe threshold d ` in the case d “ d ě
3, respectively. These were the best results until recently,when a number of improvements were made using the decoupling theorem of Bourgain and Demeter [2]. Thebest results currently state that for compact E Ă R d , the distance set ∆ p E q has positive Lebesgue measureif dim E ą s d where s “ { s “ { s d “ d ` when d ě s d “ d ` ` p d ´ q when d ě x P p R d q k we have x “ p x , ¨ ¨ ¨ , x k q whereeach x j P R d has components x j “ p x j , ¨ ¨ ¨ , x jd q . The most direct generalization of the Falconer distanceproblem in this context is the problem of congruence classes of such configurations. For k ď d , the congru-ence class of x P p R d q k ` is determined by the ` k ` ˘ -tuple of distances | x i ´ x j | . Define ∆ k p E q to be theset of vectors t| x i ´ x j |u ď i ă j ă k ` u with x i P E for all i . Note that the set ∆ p E q coincides with ∆ p E q defined above. Greenleaf, Iosevich, Liu, and Palsson [11] proved that ∆ k p E q has positive ` k ` ˘ dimensionalLebesgue measure if dim E ą d ´ d ´ k ` . The proof strategy was built on the fact that two configurations arecongruent if and only if there is an isometry mapping one to the other, which allowed the authors to studythe problem in terms of the group action. The group action framework was instrumental in the proof of thediscrete predecessor of the Falconer distance problem, known as the Erdos distinct distance problem, whichasks for the minimum number of distances determined by a set of N points in R d . In that context the groupaction framework was introduced by Elekes and Sharir [7] and ultimately used by Guth and Katz to resolve1 a r X i v : . [ m a t h . C A ] F e b he problem in the plane, obtaining the bound N { log N which is optimal up to powers of log [15].The configuration congruence problem becomes more subtle when k ą d . This is because the system ofdistance equations becomes overdetermined, and the space of congruence classes can no longer be identifiedwith the space of distance vectors R p k ` q . Invoking the group action framework again, one would expectheuristically that the space of congruence classes should have dimension d p k ` q ´ ` d ` ˘ , since the space ofconfigurations has dimension d p k ` q and the space of isometries has dimension ` d ` ˘ . Chatziconstantinou,Iosevich, Mkrtchyan, and Pakianathan [3] proved that in fact this heuristic is correct, and obtained a non-trivial dimensional threshold. Their proof used the theory of combinatorial rigidity. Given a p k ` q -pointconfiguration, they proved that the congruence class was determined (up to finitely many choices) if onefixes d p k ` q ´ ` d ` ˘ strategically chosen distances. They then used the group action framework to provethat ∆ k p E q has positive d p k ` q ´ ` d ` ˘ dimensional measure if dim E ą d ´ k ` .The key to the results in [11] and [3] is the fact that the congruence relation can be described in termsof action of the isometry group on the space of configurations. It is therefore natural to study other pointconfiguration problems where congruence is replaced by other geometric relations with a corresponding groupaction invariance. One such problem occurs by considering the volumes which are obtained by choosing any d points of a configuration. More precisely, we make the following definition. Definition 1.
The volume type of x P p R d q k is the vector t det p x j , ¨ ¨ ¨ , x j d qu ď j 㨨¨ă j d ď k P R p kd q . For a set E Ă R d , let V k,d p E q “ tt det p x j , ¨ ¨ ¨ , x j d qu ď j 㨨¨ă j d ď k : x , ..., x k P E u be the set of volume types determined by points in E . Finally, let V k,d “ V k,d p R d q be the space of allvolume types of k -point configurations in R d . Thus, the volume type of a k -point configuration x P p R d q k encodes all volumes obtained by choosingany d points from x (see figure 1).Figure 1: 5-point configuration x P ` R ˘ .2hen k “ d , the space of volume types is simply V d,d “ R , which we may equip with the Lebesguemeasure. In the case k “ d “
3, Greenleaf, Iosevich, and Mourgoglou [12] proved that V , p E q has positivemeasure if dim E ą {
5. This threshold was later improved and generalized to higher dimension by Green-leaf, Iosevich, and Taylor [13] who considered the case k “ d for any d ě V d,d p E q has positivemeasure if dim E ą d ´ ` d . Notice in the case d “ { {
3. When k islarge, the problem is overdetermined and hence one needs to define an appropriate measure on the space ofvolume types. The second listed author [19] proved that V k ` , may be identified with a space of dimension2 k ´ p k ` q -pointconfigurations has dimension 2 k ` p R q has dimension 3) and that V k ` , has positivemeasure if dim E ą ´ k . The second author also obtained a non-trivial result in the two dimensionalproblem over finite fields and rings of the form Z { p (cid:96) Z [18].Our first goal is to generalize these results to the case where k, d are natural numbers satisfying k ě d ě V k,d should be d p k ´ d q `
1. Ourfirst theorem shows that this is indeed the case.
Theorem 1.1.
The set V k,d is an embedded submanifold in R p kd q of dimension d p k ´ d q ` . This will be proved in Section 2. It follows that V k,d is equipped with p d p k ´ d q ` q -dimensional Lebesguemeasure, which we will denote by L d p k ´ d q` . It also follows that if E is compact, V k,d p E q is a compact subsetof V k,d .With this result, we are now ready to state our first main theorem. Theorem 1.2.
Let k ě d ě and let E Ă R d be a compact set with Hausdorff dimension greater than d ´ d ´ k ´ d . Then, L d p k ´ d q` p V k,d p E qq ą . We shall remark here that our decision to work with signed volume, rather than unsigned volume, is anarbitrary one. One can immediately deduce an unsigned version of Theorem 1.2 by decomposing the set V k,d p E q into 2 p kd q pieces according to the sign of each component and applying the pigeonhole principle. Wealso note that in the case k “ d our threshold is the same as the one in [13]. The general case is proved byreducing to the k “ d case, so a better exponent in that case would yield better general results.Another classic object of study in the distance problem is chains of distances determined by a set. Aconfiguration x P p R d q k determines a k ´ | x ´ x | , | x ´ x | , ..., | x k ´ ´ x k | . Bennett,Iosevich, and Taylor [1] proved that if dim E ą d ` then the set of distance chains determined by E haspositive measure. This result was later generalized by Iosevich and Taylor [16] to apply to all trees.Our other main theorem will pertain to chains of volumes. Since a volume is determined by d pointsrather than 2, we will consider chains in the sense of hypergraphs. Recall an r -regular hypergraph is a set ofvertices and hyperedges, where each hyperedge connects r vertices (so, in particular, a 2-regular hypergraphis just a graph). A chain in a hypergraph is a seqeunce of vertices where each shares some hyperedge withthe next.Given a d -uniform hypergraph on vertices t , ..., k u and a configuration x P p R d q k , we may considervolumes determined by points x j , ..., x j d such that p j , ..., j d q forms a hyperedge. In this framework, Theorem1.2 gives a result in the case where the hypergraph is complete. Our methods also allow us to obtain a resultin the case of a chain. This is our next theorem. Theorem 1.3.
Let E Ă R d be compact, and let C k,d “ tt det p x j , x j ` , ¨ ¨ ¨ , x j ` d ´ qu ď j ď k ` ´ d : x , ..., x k P E u . If dim E ą d ´ ` d , then L p k ` ´ d q p C k,d p E qq . Here, we pause to make a couple remarks. First, note that if our set E is contained in a hyperplanethrough the origin it cannot determine any non-zero volume, so the optimal threshold cannot be smaller3han d ´
1. Second, it is interesting to note that the threshold in Theorem 1.3 does not depend on k whereasthe threshold in Theorem 1.2 tends to d as k Ñ 8 . Our final theorem shows that this cannot be avoided.
Theorem 1.4 (Sharpness) . For any k ě d ě and any s k,d ă d ´ d p d ´ q d p k ´ q ` , there exists a compact set E k,d Ă R d such that dim E k,d ą s k,d and L d p k ´ d q` p V k,d p E k qq “ . We start by examining the relationship between volume types and the action of SL d p R q on the space ofconfigurations. Generically, the property of two configurations having the same volume type is equivalentto those configurations lying in the same orbit of this action. However, this equivalence breaks down forconfigurations which do not span R d . This leads to the following definition. Definition 2.
A configuration x P p R d q k is called degenerate if t x , ¨ ¨ ¨ , x d u is linearly dependent, and non-degenerate otherwise. We remark that we could broaden this notion of non-degeneracy to include configurations where any d points span R d , not just the first d points. However, in either case the set of degenerate configurations arenegligible so we have chosen this definition to simplify our proofs and notation.With our definition in place, we have the following lemma. Lemma 2.1.
Let x, y
P p R d q k be non-degenerate. Then x and y have the same volume type if and only ifthere exists a unique g P SL d p R q such that y “ gx (i.e., for each j we have y j “ gx j ).Proof. First, suppose x and y have the same volume types. Because x and y are non-degenerate, D : “ det p x , ¨ ¨ ¨ , x d q “ det p y , ¨ ¨ ¨ , y d q ‰ . Equivalently, the d ˆ d matrix with columns x ¨ ¨ ¨ x d is non-singular, same as y ¨ ¨ ¨ y d . We denote thesematrices by p x ¨ ¨ ¨ x d q and p y ¨ ¨ ¨ y d q , respectively. Let g “ p y ¨ ¨ ¨ y d qp x ¨ ¨ ¨ x d q ´ . This equation means that p gx ¨ ¨ ¨ gx d q “ p y ¨ ¨ ¨ y d q , so gx n “ y n for every 1 ď n ď d . Let i be anyindex, and write x i “ d ÿ n “ a n x n , y i “ d ÿ n “ b n y n . Observe thatdet p x , ¨ ¨ ¨ , x d ´ , x i q “ det ˜ x , ¨ ¨ ¨ , x d ´ , d ÿ n “ a n x n ¸ “ d ÿ n “ det ` x , ¨ ¨ ¨ , x d ´ , a n x n ˘ since the determinant behaves like a linear function on the rows of the matrix. Therefore,det p x , ¨ ¨ ¨ , x d ´ , x i q “ det ` x , ¨ ¨ ¨ , x d ´ , a d x d ˘ “ a d D. The same conclusion holds fordet p y , ¨ ¨ ¨ , y d ´ , y i q “ det ˜ y , ¨ ¨ ¨ , y d ´ , d ÿ n “ b n y n ¸ “ d ÿ n “ det ` y , ¨ ¨ ¨ , y d ´ , b n y n ˘ “ b d D.
4y assumption x and y have the same volume type so we conclude a d “ b d . An argument consideringdet p x , ¨ ¨ ¨ , x n ´ , x n ` , ¨ ¨ ¨ , x d , x i q similarly shows that a n “ b n for every 1 ď n ď d . Thus, gx i “ g d ÿ n “ a n x n “ d ÿ n “ a n gx n “ d ÿ n “ a n y n “ y i . Note that g P SL d p R q , since det g “ det pp y , ¨ ¨ ¨ , y d qp x , ¨ ¨ ¨ , x d q ´ q “ det p y , ¨ ¨ ¨ , y d q det p x , ¨ ¨ ¨ , x d q ´ “
1. This proves existence. Uniqueness follows from the fact that the configuration contains a basis, so g isdetermined by its action on the configuration. The converse follows from the matrix equation g p x , ¨ ¨ ¨ , x d q “ p y , ¨ ¨ ¨ , y d q and the fact that g has determinant 1.We conclude this section by proving Theorem 1.1. Given manifolds M and N , a smooth map Φ : M Ñ N is an immersion if the derivative D Φ has full rank everywhere. A smooth embedding is an injective immersionwhich is also a topological embedding, i.e. a homeomorphism from M to Φ p M q . A thorough treatment canbe found in chapter 5 of [17]. In particular, we will use the following theorem. Theorem 2.2 ([17], Theorem 5.31) . The image of a smooth embedding is an embedded submanifold.Proof of Theorem 1.1.
Let M be the subset of p R d q k consisting of configurations of the form p e , ..., e d ´ , te d , z d ` , ..., z k q with t P R zt u , z i P R d , where e i is the i -th standard basis vector in R d . We claim M has a unique rep-resentative of every non-degenerate volume type. To prove every volume type is represented, let x P p R d q k be non-degenerate. Let t “ det p x , ..., x d q and let g P SL d p R q be such that g p x , ..., x d q “ p e , ..., te d q . For i ą d , let z i “ gx i . This choice of t and z i produces an element of M with the same volume type as x . Toshow this representation is unique, suppose p e , ..., e d ´ , te d , z d ` , ..., z k q and p e , ..., e d ´ , t e d , w d ` , ..., w k q have the same volume type. Considering the volumes of the first d points, it is easy to see t “ t . If g isthe element of SL d p R q mapping the first configuration to the second, it follows that g fixes a basis and istherefore the identity. M is a manifold of dimension d p k ´ d q `
1, and we can take t, z d ` , ..., z k as the coordinates of the point p e , ..., e d ´ , te d , z d ` , ..., z k q . If Φ p t, z d ` , ..., z k q is the volume type of p e , ..., e d ´ , te d , z d ` , ..., z k q , then wehave a smooth injective map Φ : M Ñ R p kd q . We have t “ det p e , ..., te d q , and tz ij “ det p e , ..., e j ´ , z i , e j ` , ..., te d q . Let R be the row of the matrix D Φ corresponding to the component det p e , ..., te d q , and for each i, j ą d let R i,j be the row corresponding to the component det p e , ..., e j ´ , z i , e j ` , ..., te d q . Then R has a 1 inthe column corresponding to B{B t and 0 elsewhere. The row R i,j has a t in the column corresponding to B{B z ij , a z ij in the column corresponding to B{B t , and 0 elsewhere. It is therefore clear that D Φ has fullrank, so Φ is an immersion. It is also clear that Φ and Φ ´ are smooth, so Φ is an embedding. It followsfrom Theorem 2.2 that the image V k,d is an embedded submanifold of R p kd q . The dimension of V k,d must bedim M “ d p k ´ d q ` To prove our theorems, we will employ the usual strategy of defining pushforward measures supported onour sets V k,d p E q and C k,d p E q , taking approximations to those measures, and obtaining a uniform L bound5n those approximations. This will reduce to using mapping properties of generalized Radon transforms,which we establish here. We will be following the framework introduced in [13].Let X and Y be open subsets of R d ˆp d ´ q and R d , respectively. A symbol of order m on X ˆ Y ˆ R isa smooth map a : X ˆ Y ˆ R Ñ R satisfying the bound ˇˇˇˇ B n B θ n a p x, y, θ q ˇˇˇˇ À p ` | θ |q m ´ n on compact subsets of X ˆ Y . Also, for smooth phase functions ϕ : X ˆ Y ˆ R Ñ R , define C ϕ “ " p x, ∇ x ϕ p x, y, θ q , y, ´ ∇ y ϕ p x, y, θ q : θ ‰ , BB θ ϕ p x, y, θ q “ * . We view C ϕ as a subset of p T ˚ X zt uq ˆ p T ˚ Y zt uq . Given any subset C Ă p T ˚ X zt uq ˆ p T ˚ Y zt uq andany order m P R , define the class of Fourier integral operators of order m and with canonical relation C ,denoted by I m p C q , to be those with Schwartz kernels which are locally finite sums of kernels of the form K p x, y q “ ż e iϕ p x,y,θ q a p x, y, θ q dθ where C ϕ is a relatively open subset of C and a is a symbol of order m ´ ` d . We will use the followingresult. Theorem 3.1 ([13], Theorem 3.1) . Let C be a canonical relation and let A P I r ´ d ´ d have compactlysupported Schwartz kernel. Suppose the projections from p T ˚ X zt uq ˆ p T ˚ Y zt uq to each factor, restrictedto C , have full rank (so the first is an immersion and the second is a submersion). Then A is a boundedoperator L p Y q Ñ L ´ r p X q . Let Φ , η : X ˆ Y Ñ R be smooth and let η be compactly supported. A generalized Radon transform is an operator of the form Af p x q “ ż Φ p x,y q“ f p y q η p x, y q dσ x p y q , where σ x is the induced surface measure on the surface defined by Φ p x, y q “
0. This can be written interms of the delta distribution (and its Fourier transform) as an oscillatory integral; we have Af p x q “ ż Φ p x,y q“ f p y q η p x, y q dσ x p y q“ ż δ p Φ p x, y qq f p y q η p x, y q dy “ ż ż e πi Φ p x,y q θ f p y q η p x, y q p θ q dθ dy Therefore, A is a Fourier integral operator with phase function 2 π Φ p x, y q θ and amplitude η p x, y q θ . Thesymbol η p x, y q θ has order 0, so our generalized radon transforms are Fourier integral operators of order ´ d .This means Theorem 3.1 applies with r “ ´ d ´ , assuming the condition on the canonical relation holds.The generalized radon transforms we will be interested in are those given by the determinant function.Throughout this paper, R t will denote the operator R t f p x , ¨ ¨ ¨ x d ´ q “ ż det p x , ¨¨¨ ,x d q“ t f p x d q η p x , ¨ ¨ ¨ , x d q dσ t,x , ¨¨¨ ,x d ´ p x d q where σ t,x , ¨¨¨ ,x d ´ is the surface measure. These operators are shown to satisfy the canonical relationhypothesis of Theorem 3.1 in [13], which implies the following Sobolev bound for R t . Theorem 3.2.
The generalized Radon transform R t defined above is a bounded operator L p R d q Ñ L d ´ pp R d q d ´ q . .2 Frostman measures and Littlewood-Paley projections The following theorem is frequently used to study the dimension of fractal sets; see, for example, [21].
Theorem 3.3 (Frostman’s Lemma) . Let E Ă R d be compact. For any s ă dim E , there is a Borel probabilitymeasure µ supported on E satisfying µ p B r p x qq À r s for all x P R d and all r ą . A measure µ as in the theorem is called a Frostman probability measure of exponent s .We will be interested in the Littlewood-Paley decomposition of Frostman measures. Let µ be a Frostmanprobability measure on R d with exponent s and compact support. Then µ j is the j -th Littlewood-Paleypiece of µ , defined by x µ j p ξ q “ ψ p ´ j ξ q p µ p ξ q where ψ is a Schwarz function supported in the range ď | ξ | ď ď | ξ | ď
2. We will use the following bounds.
Lemma 3.4.
Let µ be a compactly supported Frostman probability measure with exponent s , and let p f µ q j be the j -th Littlewood Paley piece of the measure f µ for a function f . Then }p f µ q j } L À j p d ´ s q } f } L p µ q and }p f µ q j } L À j p d ´ s q } f } L p µ q Proof.
Firstly, let us prove the L bound. Since }p f µ q j } L ď } f } L p µ q } µ j } L it suffices to prove the boundin the case f “
1. Observe that p f µ q j p x q “ dj q ψ p j ¨q ˚ f µ p x q Since ψ is a Schwarz function, we have ψ p x q À p ` | x |q ´ . Therefore, | µ j p x q| À dj ż p ` j | x ´ y |q ´ dµ p y q Splitting this integral into two parts: 2 j | x ´ y | ă j | x ´ y | ą
1. We have2 dj ż j | x ´ y |ă p ` j | x ´ y |q ´ dµ p y qÀ dj µ pt y : 2 j | x ´ y | ă uqÀ j p d ´ s q and 2 dj ż j | x ´ y |ą p ` j | x ´ y |q ´ dµ p y q“ dj ÿ i “ ż i ď j | x ´ y |ď i ` p ` j | x ´ y |q ´ dµ p y qÀ dj ÿ i “ ´ i µ pt y : 2 i ď j | x ´ y | ď i ` uqÀ j p d ´ s q 8 ÿ i “ i p s ´ q À j p d ´ s q L bound, we first observe that }p f µ q j } L “ } { p f µ q j } L “ ż | x f µ p ξ q| ψ j p ξ q dξ “ jd ż ż x ψ p j p x ´ y qq f p x q f p y q dµ p x q dµ p y q where we have used Fourier inversion in the last line. Break the integral into two parts correspondingto | x ´ y | ă ´ j and | x ´ y | ą ´ j , where C is a large constant. Since ψ is a Schwartz function, it sufficesto bound the first part. Let K j “ dj χ t| x ´ y |ă ´ j u and let T j f p x q “ ş K j p x, y q f p y q dµ p y q . Our goal is toprove x T j f, f y L p µ q À j p d ´ s q } f } L p µ q . By Cauchy-Schwarz, it suffices to show the norm of T j as an operator L p µ q Ñ L p µ q is bounded by 2 j p d ´ s q . This follows from Schur’s test, as ż K p x, y q dµ p x q “ ż K p x, y q dµ p y q À j p d ´ s q . The generalized Radon transform applied to µ j also has Fourier transform concentrated at scale 2 j .This together with Theorem 3.2 allows us to prove the following bounds. Here and throughout, given f , ..., f n : X Ñ R , the function f b ¨ ¨ ¨ b f n is the function X n Ñ R given by f b ¨ ¨ ¨ b f n p x , ..., x n q “ f p x q ¨ ¨ ¨ f n p x n q Lemma 3.5.
Let ϕ be a smooth function which is supported on r´ , s and equal to 1 on r´ { , { s , and let ϕ ε p t q “ ε ´ ϕ p ε ´ t q . Let η : p R d q d Ñ R be a smooth cutoff function supported in the region | x i ´ e i | ă c where e i is the i -th standard basis vector and c is a small positive constant. Finally, let R εt be the approximategeneralized Radon transform defined by R εt f p x , ..., x d ´ q “ ż f p x d q η p x , ..., x d q ϕ ε p det p x , ..., x d q ´ t q dx d . If c is sufficiently small, we have the following.(i) } R εt p f µ q j } L À j p ´ s q } f } L p µ q . (ii) If j, j , ..., j d ´ are any indices such that | j ´ j i | ą for any i , then for every number N and functions f, f , ..., f d ´ we have @ R εt p f µ q j , p f µ q j b ¨ ¨ ¨ b p f d ´ µ q j d ´ D À N ´ N ¨ max p j,j ,...,j d ´ q , where x¨ , ¨y is the inner product on L p R d ´ q .Proof. We first prove that the Fourier transform of R εt p f µ q j decays rapidly outside the region | x j | « j .After we prove this, both statements follow from Plancherel and Theorem 3.2. By Fourier inversion, we have R εt µ j p x , ..., x d ´ q “ ż ż ż e πiξ d ¨ x d e πiτ p det p x q´ t q { p f µ q j p x d q p ϕ p ετ q η p x q dx d dξ d dτ, and therefore z R εt µ j p ξ , ..., ξ d ´ q “ ż ż ż e πi p r ξ ¨ x ` τ p det p x q´ t qq { p f µ q j p ξ d q p ϕ p ετ q η p x q dx dξ d dτ where r ξ “ p ξ , ..., ξ d ´ , ´ ξ d q . This integral can be written8 ż { p f µ q j p ξ d q p ϕ p ετ q I p τ, ξ q dτ dξ d , where I p τ, ξ q “ ż e πi p r ξ ¨ x ` τ p det p x q´ t qq η p x q dx. This is an oscillatory integral with phase functionΦ τ,ξ p x q “ r ξ ¨ x ` τ p det p x q ´ t q . We observe ∇ Φ τ,ξ p x q “ r ξ ` τ ¨ ∇ det p x q . For x in the support of η , we have ă | ∇ x i det p x q ´ e i | ă c in the statement of thetheorem is sufficiently small. Therefore, if Φ τ,ξ has critical points then we must have | ξ i | ď τ ď | ξ i | forall i . If 2 j ´ ă | ξ d | ă j ` and 2 j i ´ ă | ξ i | ă j i ` with | j ´ j i | ą
5, then Φ τ,ξ has no critical points and bynon-stationary phase (for example [21], proposition 6.1) we have I p τ, ξ q À N ´ N ¨ max p j,j ,...,j d ´ q . It follows from this and Lemma 3.4 that @ R εt p f µ q j , p f µ q j b ¨ ¨ ¨ b p f d ´ µ q j d ´ D “ A { R εt p f µ q j , { p f µ q j b ¨ ¨ ¨ b { p f d ´ µ q j d ´ E “ ż ż { p f µ q j p ξ q ¨ ¨ ¨ { p f d ´ µ q j d ´ p ξ d ´ q { p f µ q j p ξ d q p ϕ p ετ q I p τ, ξ q dτ dξ À N ´ N ¨ max p j,j ,...,j d ´ q . It also follows that } R εt p f µ q j } L “ } { R εt p f µ q j } L À ´ j p d ´ q ż | ξ |« j | ξ | d ´ { R εt p f µ q j p ξ q dξ “ ´ j p d ´ q } R εt p f µ q j } L d ´ À j p ´ s q } f } L p µ q Many Falconer type problems can be attacked by defining an appropriate pushforward measure and provingit is in L . The following lemma establishes this framework. Lemma 4.1.
Let M be an n -dimensional submanifold of R m equipped with n -dimensional Lebesgue measure L n and consider a map Φ : p R d q k Ñ M . For E Ă R d , let ∆ Φ p E q “ t Φ p x q : x P E k u . If µ is a probability measure supported on a compact set E and ´ n ż ¨ ¨ ¨ ż | Φ p x q´ Φ p y q|À ε dµ k p x q dµ k p y q À , then L n p ∆ Φ p E qq ą .Proof. Define a probability measure ν on M by the relation ż f p t q dν p t q “ ż f p Φ p x qq dµ k p x q . It suffices to prove ν is absolutely continuous with respect to L n . Let ϕ be a symmetric Schwartz functionon R m supported on the ball of radius 2 and equal to 1 on the unit ball. Let ϕ ε p x q “ ε ´ n ϕ p x { ε q and let ν ε “ ϕ ε ˚ ν . Then ż A ν ε p t q dt ď L n p A q { } ν ε } L , where dt denotes integration with respect to n -dimensional Lebesgue measure. This reduces matters toproving an upper bound on } ν ε } L which is uniform in ε . We have ν ε p t q “ ż ϕ ε p t ´ t q dν p t q“ ż ϕ ε p Φ p x q ´ t q dµ k p x q« ε ´ n ż | Φ p x q´ t |ď ε { dµ k p x q . Thus, || ν ε || L « ε ´ n ż ˜ż ¨ ¨ ¨ ż | Φ p x q´ t |ď ε { dµ k p x q dµ k p y q ¸ dt “ ε ´ n ż ¨ ¨ ¨ ż | Φ p x q´ Φ p y q|ď ε ˜ż | φ p x q´ t |ď ε { dt ¸ dµ k p x q dµ k p y q« ε ´ n ż ¨ ¨ ¨ ż | φ p x q´ φ p y q|ď ε dµ k p x q dµ k p y qÀ E Ă R d hassome additional structure. Lemma 4.2.
Let k ě d and let E Ă R d be a compact set with Hausdorff dimension dim E ą d ´ . Thenthere exist subsets E , ..., E k Ă E and a constant c with dim E j “ dim E and the property that for any choiceof d points x , ..., x d in different cells E j , we have det p x , ¨ ¨ ¨ , x d q ą c .Proof. Let µ be a Frostman probability measure on E with exponent s ą d ´ N be a large integer tobe determined later. The idea of the proof is that the 2 ´ N -neighborhood of a compact piece of a hyperplanehas negligible µ -measure, so we can construct our sets E j recursively by throwing away bad parts of E .Given a point x P R d , let B p x q denote the ball of radius 2 ´ N centered at x . Let S be a finite set such that t B p x q : x P S covers E , and let S Ă S be the subset obtained by discarding any x such that µ p B p x qq “ x , x P S be arbitrary points such that the balls B p x q and B p x q have distance ą ´ N . For2 ď j ď d ´
1, suppose x , ..., x j have been defined and are linearly independent. Let X denote the 2 ´ N ` -neighborhood of span p x , ..., x j q intersected with the ball of radius sup E . Then µ p X q À ´ N p s ´ j q . Since s ą j , for large N this is small, so we can choose x j ` P E z X . It follows that B p x j ` q does not intersectthe 2 ´ N neighborhood of span p x , ..., x j q . For d ď j ă k , suppose x , ..., x j have been defined and have theproperty that for any j , ..., j d ď j , B p x j q does not intersect the 2 ´ N -neighborhood of span p x j , ..., x j d ´ q .Again, if N is sufficiently large then the union of all ` jd ´ ˘ approximate hyplerplanes determined by any d ´ x , ..., x j has small µ measure, so we can choose x j ` to avoid all of them as well. It is clearthat the collection E j : “ B p x j q has the desired properties.To prove Theorem 1.2, by Lemmas 4.1 and 4.2 it suffices to bound ε ´ d p k ´ d q´ ż ż | Φ p x q´ Φ p y q|À ε dµ k p x q dµ k p y q (1)independent of ε . We follow the approach used in [11] and [19] to reduce matters to the k “ d case. Wefirst decompose the dµ k p y q factor into Littlewood-Paley pieces, reducing (1) to « ε ´ d p k ´ d q´ ÿ j ,...,j k ż ż | Φ p x q´ Φ p y q|À ε µ j p y q ¨ ¨ ¨ µ j k p y k q dy ¨ ¨ ¨ dy k dµ k p x q . (2)Here t µ j u are the Littlewood Paley pieces of µ , as defined in Section 3. Now that we have an integralin dy , we want to use the group action framework discussed in Section 2 to turn this into an integral overSL d p R q . The idea is that for fixed x , integrating over the region | Φ p x q ´ Φ p y q| ă ε is equivalent to integratingover y „ gx as g varies. If ε is sufficiently small then det p y , ¨ ¨ ¨ , y d q ‰ y in this region. Every such y has the same area type as a configuration of the form p x , ¨ ¨ ¨ , x dd ´ , t d , ¨ ¨ ¨ , t kd q . Moreover, there is an open set U d Ă R d ´ such that for every p g , ¨ ¨ ¨ , g dd ´ q P U d there exists a unique g P SL d p R q whose matrix has those entries, and the lower right entry is a rational function of the others.This gives a rational change of variables y “ g p x , ¨ ¨ ¨ , x dd ´ , t d , ¨ ¨ ¨ , t kd q , where g is viewed in terms of its coordinates. Since x lives in a fixed compact subset of configurationspace, the Jacobian determinant is « « ε ´ d p k ´ d q´ ż ż ż B ε ˜ ÿ j ,...,j k µ j b ¨ ¨ ¨ b µ j k ¸ p g p x , ¨ ¨ ¨ , x dd ´ , t d , ¨ ¨ ¨ , t kd qq dg dt dµ k p x q , (3)where the two inner integral signs represent integration over the first d ´ g and the d p k ´ d q` t t i u , respectively. The t i coordinates are integrated over the ball B ε raidus ε centeredat the last d p k ´ d q ` x . Taking the limit as ε Ñ
0, this is ÿ j ,...,j k ż ż ¨ ¨ ¨ ż µ j p gx q ¨ ¨ ¨ µ j k p gx k q dµ p x q ¨ ¨ ¨ dµ p x k q dg. (4)Here we make a couple simple reductions. First, µ j is a Schwarz function satisfying the L bound } µ j } L À j p d ´ s q (see for example [19], Lemma 3) which we use to reduce from general k ě d to the k “ d case. Moreover, the sum over j , ..., j k can be reduced to the sum over indices satisfying j ě ¨ ¨ ¨ ě j k ě O p q to the sum and other permutations of indices only change thesum by a multiplicative constant. Applying the L bound and running the sum in the indices j d ` , ..., j k ,it follows that (4) is 11 ÿ j 쨨¨ě j d j d p d ´ s qp k ´ d q ż ż ¨ ¨ ¨ ż µ j p gx q ¨ ¨ ¨ µ j d p gx d q dµ p x q ¨ ¨ ¨ dµ p x d q dg. (5)This reduces matters to the k “ d case. Using the same change of variables in the other direction, this is « ε ´ ÿ j 쨨¨ě j d j d p d ´ s qp k ´ d q ż ¨ ¨ ¨ ż | det p x ,...,x d q´ det p y ,...,y d q|ă ε µ j p y q ¨ ¨ ¨ µ j d p y d q dy dµ k p x q« ε ´ ÿ j 쨨¨ě j d j d p d ´ s qp k ´ d q ż ż ¨ ¨ ¨ ż | det p x , ¨¨¨ ,x d q´ t |ă ε | det p y , ¨¨¨ ,y d q´ t |ă ε µ j p y q ¨ ¨ ¨ µ j d p y d q dy dµ k p x q dt. « ÿ j 쨨¨ě j d j d p d ´ s qp k ´ d q ż ˜ ε ´ ż | det p x , ¨¨¨ ,x d q´ t |ă ε dµ k p x q ¸ x R εt µ j , µ j b ¨ ¨ ¨ b µ j d y dt (6)where x¨ , ¨y denotes the inner product on L pp R d q d ´ q and R εt is the approximation to the generalizedRadon transform discussed in Section 2. Let ν εk,d p t q “ ż | det p x i , ¨¨¨ ,x id q´ t |ă ε dµ k p x q . The quantity in (1) we are trying to bound is } ν εk,d } L , and we have proved. } ν εk,d } L À ÿ j 쨨¨ě j d j d p d ´ s qp k ´ d q ż ν εd,d p t q x R εt µ j , µ j b ¨ ¨ ¨ b µ j d y dt, Let S “ ÿ j 쨨¨ě j d ě j d p d ´ s qp k ´ d q sup t x R εt µ j , µ j b ¨ ¨ ¨ b µ j d y . If S is finite, we have } ν εk,d } L À } ν εd,d } L . Plugging in k “ d on the left, we have a uniform bound on } ν εd,d } L which in turn implies a uniformbound on } ν εk,d } L for all k ě d . So, it suffices to prove S is finite under the hypotheses of Theorem 1.2. ByLemma 3.5 it is clear that the part of the sum corresponding to indices with j d ă j ´ t x R εt µ j , µ j b ¨ ¨ ¨ b µ j y À j p ´ s `p d ´ s qp d ´ qq . Therefore, S À ÿ j ě j p p d ´ s qp k ´ d q` ´ s `p d ´ s qp d ´ q q . The sum will converge if s ą d ´ d ´ k ´ d , as claimed. To prove Theorem 1.3, it is enough to establish the following bound. The theorem then follows from Lemma4.1.
Lemma 4.3.
Let ϕ ε be an approximation to the identity on R , and let J εt,k “ ż ˜ k ` ´ d ź j “ ϕ ε p det p x j , ..., x j ` d ´ q ´ t j q ¸ dµ k p x q . For every k ě d there is a constant C k (which does not depend on t or ε ) such that J εt,k ď C k . roof. We first prove a bound in the case k “ d . Since J εt,d « ř j } R εt µ j } L p µ d ´ q , it is enough to prove } R εt µ j } L p µ d ´ q À ´ cj for some positive c . To accomplish this, fix t and let T εj f “ R εt p f µ q j . We wantto bound the norm of T εj as an operator L p µ q Ñ L p µ d ´ q . To do this, let g P L p µ d ´ q be given by g p x q “ g p x q ¨ ¨ ¨ g d ´ p x d ´ q with g i P L p µ q . Using Littlewood-Paley decomposition, Lemma 3.5, andCauchy-Schwarz we have @ T εj f, g D L p µ d ´ q À x R εt p f µ q j , p g µ q j b ¨ ¨ ¨ b p g d ´ µ q j yÀ j p ´ s `p d ´ s qp d ´ qq } f } L p µ q } g } L p µ d ´ q . It follows that the operator norm, and hence } R εt µ j } L p µ d ´ q , is bounded by 2 j p ´ s `p d ´ s qp d ´ qq , and thisseries converges when s ą d ´ ` d . This gives the desired bound in the case k “ d .For k ą d , let χ εt,k p x q “ k ` ´ d ź j “ ϕ ε p det p x j , ..., x j ` d ´ q ´ t j q . We have J εt,k “ ż χ εt,k p x q dµ k p x q“ ż χ ε ˜ t,k ´ p ˜ x q ϕ ε p det p x k ` ´ d , ..., x k q ´ t k ` ´ d q dµ k ´ p ˜ x q dµ p x k q« ÿ j ż χ ε ˜ t,k ´ p ˜ x q R εt k ` ´ d µ j p x k ` ´ d , ..., x k ´ q dµ k ´ p ˜ x qÀ p J ε ˜ t,k ´ q { ÿ j } R t k ` ´ d µ j } L p µ k ´ q À p J ε ˜ t,k ´ q { Let Φ p x q “ p det p x , ..., x d q , ..., det p x k ` ´ d , ..., x k qq . We have ε ´p k ` ´ d q ż ż | Φ p x q´ Φ p y q|ă ε dµ k p x q dµ k p y q À ż J ε Φ p x q ,k dµ k p x q À . Theorem 1.3 then follows from Lemma 4.1.
We conclude this paper by proving Theorem 1.4. Let Λ q,s be the q ´ ds -neighborhood of q ` Z d Ş ` r q , q s ˆ r , q s d ´ ˘˘ ,the right half of the lattice in the p d ´ q -dimensional unit cube with spacing q (see figure 2). By Theorem8.15 in [10] we can choose a sequence q n that increases sufficiently rapidly such thatdim ˜č n Λ q n ,s ¸ “ s Thus, for large q we may regard Λ q,s as an approximation to a set of Hausdorff dimension s . Let usmodify this situation to fit our problem. By Lemma 1.8 in [10] we have Lemma 4.4 ([10], Lemma 1.8) . Let ψ be Lipschitz and surjective, and let H s be the s-dimensional Hausdorffmeasure. Then H s p F q À H s p E q .
13s consequence of this lemma we have dim F ď dim E . If ψ is bijective and Lipschitz in both directions,then dim F “ dim E . Let E q,s (figure 3) be the image of Λ q,s under the spherical map ψ p x , x , ¨ ¨ ¨ , x d q “ x ˜ cos ´ πx ¯ , sin ´ πx ¯ cos ´ πx ¯ , ¨ ¨ ¨ , d ´ ź i “ sin ´ πx i ¯ cos ´ πx d ¯ , d ź i “ sin ´ πx i ¯¸ . is not hard to check this map is injective on r , sˆr , s d ´ and therefore bijective as a map Λ q,s Ñ E q,s .Moreover, let us fix a sequence q n such that dim ˜č n E q n ,s ¸ “ s and call E s “ Ş n E q n ,s . It remains toprove L d p k ´ d q` p V k,d p E s qq “ . Figure 2: Λ ,s for d “
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