An improved result for Falconer's distance set problem in even dimensions
Xiumin Du, Alex Iosevich, Yumeng Ou, Hong Wang, Ruixiang Zhang
aa r X i v : . [ m a t h . C A ] J un AN IMPROVED RESULT FOR FALCONER’S DISTANCE SETPROBLEM IN EVEN DIMENSIONS
XIUMIN DU, ALEX IOSEVICH, YUMENG OU, HONG WANG, AND RUIXIANG ZHANG
Abstract.
We show that if compact set E ⊂ R d has Hausdorff dimension largerthan d + , where d ≥ E has positiveLebesgue measure. This improves the previously best known result towards Falconer’sdistance set conjecture in even dimensions. Introduction
Let E ⊂ R d be a compact set, its distance set ∆( E ) is defined by∆( E ) := {| x − y | : x, y ∈ E } . A classical question in geometric measure theory, introduced by Falconer in the early80s ([6]) is, how large does the Hausdorff dimension of a compact subset of R d , d ≥ Conjecture. [Falconer]
Let d ≥ and E ⊂ R d be a compact set. Then dim( E ) > d ⇒ | ∆( E ) | > . Here | · | denotes the Lebesgue measure and dim( · ) is the Hausdorff dimension. This conjecture, still open in all dimensions, has a famous predecessor in discretegeometry known as the Erd˝os distinct distance conjecture. It says that N points in R d , d ≥
2, determine at least C ǫ N d − ǫ , ǫ >
0, distinct Euclidean distances. The twodimensional case was solved by Guth and Katz [10] after more than half of centuryof partial results. The higher dimensional case is still open, with the best knownexponents obtained by Solymosi and Vu [20]. There are some intriguing connectionsbetween the Erd˝os and Falconer distance problem, the issue that we shall touch uponat the end of this paper.The main purpose of this paper is to improve the best known dimensional thresholdtowards the Falconer conjecture in even dimensions.
Theorem 1.1.
Let d ≥ be an even integer and E ⊂ R d be a compact set. Then dim( E ) > d ⇒ | ∆( E ) | > . Falconer’s conjecture has attracted a great amount of attention in the past decades,and different methods have been invented to lower the dimensional threshold thatis sufficient for the distance set to have positive Lebesgue measure. To name a few important landmarks in the study of the problem: in 1985, Falconer [6] showed that | ∆( E ) | > E ) > d + . The thresholds were improved by Wolff [21] to in thecase d = 2, and by Erdo˘gan [5] to d + when d ≥
3. These records were only veryrecently rewritten in 2018: , d = 2 , (Guth–Iosevich–Ou–Wang [9]) , d = 3 , (Du–Guth–Ou–Wang–Wilson–Zhang [3]) d d − , d ≥ , (Du–Zhang [4]) . Our main result in this paper further improves the thresholds in even dimensions d ≥ d + matches the two dimensional case. Similarly to [9], we in fact prove aslightly stronger version of the main theorem regarding the pinned distance set. Theorem 1.2.
Let d ≥ be an even integer and E ⊂ R d be a compact set. Supposethat dim ( E ) > d + , then there is a point x ∈ E such that its pinned distance set ∆ x ( E ) has positive Lebesgue measure, where ∆ x ( E ) := {| x − y | : y ∈ E } . Let E ⊂ R d be a compact set with positive α -dimensional Hausdorff measure. It isa standard result that there exists a probability measure µ supported on E such that µ ( B ( x, r )) . r α , ∀ x ∈ R d , ∀ r > . Such measure is usually referred to as Frostman measure. In the study of the Falconerproblem, a classical analytic approach due to Mattila [17] is to reduce the desired result( | ∆( E ) | >
0) to showing certain estimates of the decay rate of the Fourier transformof µ . This is also precisely the route taken in many prior works including [21, 5, 3, 4].However, this approach also has its limit. For instance, it is known that when d = 2,the best possible Falconer threshold it could imply is , which matches the resultof Wolff [21]. And when d = 3, would be the best possible (see [5]). In higherdimensions, there is no currently known example showing such constraint. Whereas, in[2] it shows that further constraints arise if the method employed does not distinguishthe spherical and parabolic decay rates.In [9], the authors studied the two dimensional Falconer problem, and developeda new method that modifies the original Mattila approach. Their argument consistsprimarily of two steps. First, one prunes the natural Frostman measure µ on E byremoving bad wave packets at different scales (see Section 3.1 below for the exact pro-cess), and shows that the error introduced in the pruning process can be controlled.Second, one applies a refined decoupling inequality to estimate some L quantity in-volving the pruned good measure. One of the main reasons why the argument doesn’treadily extend to higher dimensions is because of the first step. More precisely, in[9], to make sure that the pruned measure is close enough to the original Frostmanmeasure, one applies a radial projection theorem of Orponen [19] (see Theorem 3.2below) that assumes the measure has dimension α > d −
1. However, when d ≥ α is close enough to d . We overcome this difficulty by ALCONER’S DISTANCE SET PROBLEM IN EVEN DIMENSIONS 3 introducing another ingredient into the process: orthogonal projections of the originalmeasure, which is the main contribution of the present article.
Notation.
Throughout the article, we write A . B if A ≤ CB for some absoluteconstant C ; A ∼ B if A . B and B . A ; A . ǫ B if A ≤ C ǫ B for all ǫ > A / B if A ≤ C ǫ R ǫ B for any ǫ > R > R , RapDec( R ) denotes those quantities that are boundedby a huge (absolute) negative power of R , i.e. RapDec( R ) ≤ C N R − N for arbitrarilylarge N >
0. Such quantities are negligible in our argument. We say a functionis essentially supported in a region if (the appropriate norm of) the tail outside theregion is RapDec( R ) for the underlying parameter R . Acknowledgements.
XD is supported by NSF DMS-1856475. AI was partially sup-ported by NSF HDR TRIPODS 1934985. YO is supported by NSF DMS-1854148.HW is funded by the S.S. Chern Foundation and NSF. RZ is supported by NSF DMS-1856541. Setup and main estimates
In this section, we set up the problem and outline two main estimates, from whichTheorem 1.2 follows.Let E ⊂ R d be a compact set with positive α -dimensional Hausdorff measure, d <α < d + 1. Without loss of generality, assume that E is contained in the unit ball.Then there exists a probability measure µ supported on E such that(2.1) µ ( B ( x, r )) . r α , ∀ x ∈ R d , ∀ r > . According to results on Hausdorff dimension of projections proved in [15] (also pre-sented in [18, Section 5.3]), there exists a ( d + 1)-dimensional subspace P ⊂ R d suchthat ( π P ) ∗ ( µ ), the pushforward measure of µ under the orthogonal projection from R d onto P , is still α -dimensional, in the sense that( π P ) ∗ ( µ )( B ( x, r )) . r α , ∀ x ∈ P, ∀ r > . Since P will be fixed throughout the proof, in the following we will drop it from thenotation and write π = π P for short.Similarly as in [9], it would be helpful to consider two disjoint subsets E , E of E andfocus on showing that the distance set between E , E already has positive Lebesguemeasure.The two subsets will be chosen as follows. First, it is elementary that one can findtwo subsets ˜ E , ˜ E ⊂ π ( E ) with positive α -dimensional Hausdorff measure satisfying d ( ˜ E , ˜ E ) &
1. For i = 1 ,
2, let˜ µ i := π ∗ ( µ )( ˜ E i ) − π ∗ ( µ ) χ ˜ E i . It is easy to see that ˜ µ i is an α -dimensional probability measure supported on ˜ E i .Next, define E i := π − ( ˜ E i ) ∩ E , i = 1 ,
2. Then one has E , E ⊂ E and d ( E , E ) & µ i := µ ( E i ) − µχ E i , i = 1 ,
2, one obtains a pair of α -dimensional prob-ability measures µ , µ that are supported on E , E respectively. It is straightforward X. DU, A. IOSEVICH, Y. OU, H. WANG, AND R. ZHANG to check that π ∗ ( µ i ) = ˜ µ i . Our goal in the following is to show that when α > d + ,there exits x ∈ E such that | ∆ x ( E ) | >
0. Note that in the above, for the orthogonalprojection π to be well defined, we have already used the assumption that d is an eveninteger.To relate the measures discussed above to the distance set, it is useful to considertheir pushforward measures under the distance map. More precisely, let x ∈ E be anyfixed point and let d x ( y ) := | x − y | be its induced distance map. Then, the pushforwardmeasure d x ∗ ( µ ), defined as Z R ψ ( t ) d x ∗ ( µ ) = Z E ψ ( | x − y | ) dµ ( y ) , is a natural measure that is supported on ∆ x ( E ).In the following, we will construct another complex valued measure µ ,g that is the good part of µ with respect to µ , and study its pushforward under the map d x . Themain estimates are the following. Proposition 2.1.
Let d ≥ be an even integer and α > d . If we choose R largeenough in the construction of µ ,g in Section 3.1 below, then there is a subset E ′ ⊂ E so that µ ( E ′ ) ≥ − and for each x ∈ E ′ , k d x ∗ ( µ ) − d x ∗ ( µ ,g ) k L < . Proposition 2.2.
Let d ≥ be an even integer and α > d + , then for sufficientlysmall δ in terms of α in the construction of µ ,g in Section 3.1 below, Z E k d x ∗ ( µ ,g ) k L dµ ( x ) < + ∞ . It is a routine exercise to check that Theorem 1.2 is immediately implied by the twopropositions above.
Proof of Theorem 1.2 using Proposition 2.1 and Proposition 2.2.
The two propositionstell us that there is a point x ∈ E so that(2.2) k d x ∗ ( µ ) − d x ∗ ( µ ,g ) k L < / , and(2.3) k d x ∗ ( µ ,g ) k L < + ∞ . Since d x ∗ ( µ ) is a probability measure, (2.2) guarantees that k d x ∗ ( µ ,g ) k L ≥ − / . Note that the support of d x ∗ ( µ ) is contained in ∆ x ( E ). Therefore Z ∆ x ( E ) | d x ∗ µ ,g | = Z | d x ∗ ( µ ,g ) | − Z ∆ x ( E ) c | d x ∗ ( µ ,g ) |≥ − − Z | d x ∗ ( µ ) − d x ∗ ( µ ,g ) | ≥ − . ALCONER’S DISTANCE SET PROBLEM IN EVEN DIMENSIONS 5
But on the other hand,(2.4) Z ∆ x ( E ) | d x ∗ µ ,g | ≤ | ∆ x ( E ) | / (cid:18)Z | d x ∗ µ ,g | (cid:19) / . Since (2.3) tells us that R | d x ∗ µ ,g | is finite, it follows that | ∆ x ( E ) | is positive. (cid:3) Note that the two propositions in the above are parallel to [9, Proposition 2.1, 2.2].The main novelty of our proof is the construction of the good measure µ ,g and thejustification of Proposition 2.1. Once that step is completed, the proof of Proposition2.2 proceeds very similarly to its corresponding version in [9].3. Construction of good measure and Proposition 2.1
Construction of good measure.
Our plan is to define µ ,g by eliminating cer-tain bad wave packets from µ . We will show that this can be done at a single scaleat each time and the error between µ ,g and µ has sufficient decay. This procedureproceeds very similarly to [9]. Heuristically, we would like to define a wave packet tobe bad if its projection onto the ( d + 1)-dimensional subspace P has ˜ µ -mass that issignificantly higher than average.Here are the details. Let R be a large number that will be determined later, andlet R j = 2 j R . Cover the annulus R j − ≤ | ω | ≤ R j by rectangular blocks τ withdimensions approximately R / j × · · · × R / j × R j , with the long direction of each block τ being the radial direction. Choose a partition of unity subordinate to this cover suchthat 1 = ψ + X j ≥ ,τ ψ j,τ . Let δ > j, τ ), cover the unitdisk with tubes T of dimensions approximately R − / δj × · · · × R − / δj × τ . The covering has uniformly bounded overlap,each T intersects at most C ( d ) other tubes. We denote the collection of all these tubesas T j,τ . Let η T be a partition of unity subordinate to this covering, so that for eachchoice of j and τ , P T ∈ T j,τ η T is equal to 1 on the disk of radius 2 and each η T is smooth.For each T ∈ T j,τ , define an operator M T f := η T ( ψ j,τ ˆ f ) ∨ , which, morally speaking, maps f to the part of it that has Fourier support in τ andphysical support in T . Define also M f := ( ψ ˆ f ) ∨ . We denote T j = ∪ τ T j,τ and T = ∪ j ≥ T j . Hence, for any L function f supported on the unit disk, one has thedecomposition f = M f + X T ∈ T M T f + RapDec( R ) k f k L . See [9, Lemma 3.4] for a justification of the above decomposition. (Even though [9,Lemma 3.4] is stated in two dimensions, the argument obviously extends to higherdimensions.)
X. DU, A. IOSEVICH, Y. OU, H. WANG, AND R. ZHANG
Let c ( α ) > T denote theconcentric tube of four times the radius. We say a tube T ∈ T j,τ is bad if µ (4 T ) ≥ R − d/ c ( α ) δj . Note that the above definition is completely parallel to the one used in [9], withthe only difference being the choice of the mass threshold R − d/ c ( α ) δj for a tube T tobe bad. This threshold is carefully chosen so that the error estimate, i.e. proof ofProposition 2.1, below can work through.A tube T is good if it is not bad, and we define µ ,g := M µ + X T ∈ T ,T good M T µ . We point out that µ ,g is only a complex valued measure, and is essentially supported inthe R − / δ -neighborhood of E with a rapidly decaying tail away from it (see Lemma[9, Lemma 5.2] for a proof, which is presented in the two dimensional case but worksin all dimensions).3.2. Proof of Proposition 2.1.
We would like to relate k d x ∗ ( µ ) − d x ∗ ( µ ,g ) k L to thegeometry of bad tubes. To start with, recall the following lemma: Lemma 3.1. [9, Lemma 3.5]
For any point x ∈ E , defineBad j ( x ) := [ T ∈ T j : x ∈ T and T is bad T, ∀ j ≥ . Then there holds k d x ∗ ( µ ,g ) − d x ∗ ( µ ) k L . X j ≥ R δj µ ( Bad j ( x )) + RapDec ( R ) . Note that the proof of this lemma has nothing to do with the actual definition ofbad tubes and the ambient dimension of the space, so it applies directly to our setting.To estimate the measure of Bad j ( x ), defineBad j := { ( x , x ) ∈ E × E : there is a bad T ∈ T j so that 2 T contains x and x } . We claim that Proposition 2.1 would follow if one can show for sufficiently large con-stant c ( α ) > µ × µ (Bad j ) . R − δj , ∀ j ≥ . Indeed, since µ × µ (Bad j ) = Z µ (Bad j ( x )) dµ ( x ) , the estimate (3.1) ensures that there exists B j ⊂ E so that µ ( B j ) ≤ R − (1 / δj and forall x ∈ E \ B j , µ (Bad j ( x )) . R − (3 / δj . ALCONER’S DISTANCE SET PROBLEM IN EVEN DIMENSIONS 7
Let E ′ = E \ S j ≥ B j and choose R sufficiently large (depending on δ and α ). Oneobviously has µ ( E ′ ) ≥ − , and for each x ∈ E ′ the bound k d x ∗ ( µ ,g ) − d x ∗ ( µ ) k L . R − (1 / δ ≤ , according to Lemma 3.1.In order to prove estimate (3.1), we apply the following radial projection theorem ofOrponen [19]. The choice of the threshold in the definition of bad tubes in the abovewill play an important role in this step. In order to state the theorem, we first definea radial projection map P y : R n \ { y } → S n − by P y ( x ) = x − y | x − y | . Theorem 3.2. [19, Orponen]
For every β > n − there exists p ( β ) > so that thefollowing holds. Suppose that ν and ν are measures on the unit ball in R n with disjointsupports and that for every ball B ( x, r ) , ν i ( B ( x, r )) . r β . Then Z k P y ν k pL p dν ( y ) < + ∞ . Note that we cannot apply the above theorem directly to our problem in R d , becausethe measures µ , µ we are dealing with have dimension α that is barely larger than d (hence fails to satisfy α > d − µ i = π ∗ ( µ i ) instead.Recall from the definition that for i = 1 ,
2, ˜ µ i is a measure on the ( d + 1)-dimensionalsubspace P ⊂ R d and satisfies ˜ µ i ( B ( x, r )) . r α for every ball B ( x, r ). Whenever α > d ,one has α > ( d + 1) −
1. Therefore, Theorem 3.2 does apply to ˜ µ , ˜ µ , and one has(3.2) Z k P y ˜ µ k pL p d ˜ µ ( y ) < + ∞ . To prove estimate (3.1), we first define a set g Bad j in P .We have chosen the sets E , E at the beginning such that d ( E , E ) & d ( π ( E ) , π ( E )) &
1. By definition of Bad j , it suffices to consider tubes T ∈ T j that intersect both E and E . Hence, the projected tube π ( T ) ⊂ P also looks like atube, with side length ∼ ∼ R − + δj in the rest of the direc-tions. Therefore, T j gives rise to a collection ˜ T j that contains tubes in P of dimensionsroughly 1 × R − + δj × · · · × R − + δj .One can similarly define a tube ˜ T ∈ ˜ T j to be bad if ˜ µ (4 ˜ T ) ≥ R − d + c ( α ) δj . It is easyto see that the badness of a tube is preserved under the projection. Indeed, if T ∈ T j is bad, then ˜ µ (4 π ( T )) ≥ µ (4 T ) ≥ R − d + c ( α ) δj . Define g Bad j := { ( x , x ) ∈ P : there is a bad ˜ T ∈ ˜ T j so that 2 ˜ T contains x and x } . X. DU, A. IOSEVICH, Y. OU, H. WANG, AND R. ZHANG
Then one has µ × µ (Bad j ) ≤ ˜ µ × ˜ µ ( g Bad j ) = Z ˜ µ ( g Bad j ( y )) d ˜ µ ( y ) , where g Bad j ( y ) := [ ˜ T ∈ ˜ T j : y ∈ T and ˜ T is bad T .
With bound (3.2), the desired estimate (3.1) follows by an argument identical to [9,Proof of Lemma 3.6]. We sketch the argument here for the sake of completeness.Let ˜ T ∈ ˜ T j be a bad tube and y ∈ T ∩ π ( E ). Let A ( ˜ T ) be the cap of the sphere S d whose center corresponds to the direction of the long axis of ˜ T and with radius ∼ R − / δj . Since d ( π ( E ) , π ( E )) &
1, one has P y (4 ˜ T ∩ π ( E )) ⊂ A ( ˜ T ), hence(3.3) P y ˜ µ ( A ( ˜ T )) ≥ ˜ µ (4 ˜ T ) ≥ R − d + c ( α ) δj . Therefore, P y ( g Bad j ( y )) can be covered by caps A ( ˜ T ) of radius ∼ R − / δj which eachsatisfies (3.3). By the Vitali covering lemma, there exists a disjoint subset of A ( ˜ T )so that 5 A ( ˜ T ) covers P y ( g Bad j ( y )). Hence, the total number of disjoint A ( ˜ T ) in thecovering is bounded by R d − c ( α ) δj , which implies | P y ( g Bad j ( y )) | . R d − c ( α ) δj · R d ( − / δ ) j = R − ( c ( α ) − d ) δj , where | · | denotes the surface measure on S d . Therefore, by H¨older’s inequality andby choosing c ( α ) sufficiently large, one has µ × µ (Bad j ) ≤ Z ˜ µ ( g Bad j ( y )) d ˜ µ ( y ) ≤ Z Z P y ( g Bad j ( y )) P y ˜ µ ! d ˜ µ ( y ) ≤ sup y | P y ( g Bad j ( y )) | − p Z k P y ˜ µ k L p d ˜ µ . R − δj . This completes the justification of (3.1) thus the proof of Proposition 2.1.4.
Refined decoupling and Proposition 2.2
In this section, we prove Proposition 2.2, which will complete the proof of Theorem1.2. This part of the argument proceeds very similarly as [9, Proof of Proposition 2.2],with the only difference being the change of the definition of good tubes.Let σ r be the normalized surface measure on the sphere of radius r . The mainestimate in the proof of Proposition 2.2 is the following: Lemma 4.1.
For any α > , r > , and δ sufficiently small depending on α, ǫ : Z E | µ ,g ∗ ˆ σ r ( x ) | dµ ( x ) ≤ C ( R ) r − d d +1) − ( d − αd +1 + ǫ r − ( d − Z | ˆ µ | ψ r dξ + RapDec( r ) , ALCONER’S DISTANCE SET PROBLEM IN EVEN DIMENSIONS 9 where ψ r is a weight function which is ∼ on the annulus r − ≤ | ξ | ≤ r + 1 anddecays off of it. To be precise, we could take ψ r ( ξ ) = (1 + | r − | ξ || ) − . To see how this lemma implies the desired estimate in Proposition 2.2, one firstobserves that d x ∗ ( µ ,g )( t ) = t d − µ ,g ∗ σ t ( x ) . Since µ ,g is essentially supported in the R − / δ -neighborhood of E , for x ∈ E , weonly need to consider t ∼
1. Hence, Z E k d x ∗ ( µ ,g ) k L dµ ( x ) . Z ∞ Z E | µ ,g ∗ σ t ( x ) | dµ ( x ) t d − dt ∼ Z ∞ Z E | µ ,g ∗ ˆ σ r ( x ) | dµ ( x ) r d − dr, where in the second step, we have used an L -identity proved by Liu [16].Applying Lemma 4.1 for each r > . R Z ∞ Z R d r − d d +1) − ( d − αd +1 + ǫ ψ r ( ξ ) | ˆ µ ( ξ ) | dξdr . Z R d | ξ | − d d +1) − ( d − αd +1 + ǫ | ˆ µ ( ξ ) | dξ ∼ I β ( µ ) , where β = d − d d +1) − ( d − αd +1 + ǫ and the energy of measure is defined as I β ( µ ) := Z | x − y | − β µ ( x ) µ ( y ) = c d,β Z R d | ξ | − ( d − β ) | ˆ µ ( ξ ) | dξ. One thus has I β ( µ ) < ∞ if β < α , which is equivalent to α > d + . The proof ofProposition 2.2 is thus complete upon verification of Lemma 4.1.4.1. Refined decoupling estimates.
The key ingredient in the proof of Lemma 4.1 isthe following refined decoupling theorem, which is derived by applying the l decouplingtheorem of Bourgain and Demeter [1] at many different scales.Here is the setup. Suppose that S ⊂ R d is a compact and strictly convex C hypersurface with Gaussian curvature ∼
1. For any ǫ >
0, suppose there exists 0 <δ ≪ ǫ satisfying the following. Suppose that the 1-neighborhood of RS is partitionedinto R / × ... × R / × θ . For each θ , let T θ be a set of tubes of dimensions R − / δ × θ , and let T = ∪ θ T θ . Each T ∈ T belongsto T θ for a single θ , and we let θ ( T ) denote this θ . We say that f is microlocalized to( T, θ ( T )) if f is essentially supported in 2 T and ˆ f is essentially supported in 2 θ ( T ). Theorem 4.2. [9, Corollary 4.3]
Let p be in the range ≤ p ≤ d +1) d − . For any ǫ > ,suppose there exists < δ ≪ ǫ satisfying the following. Let W ⊂ T and suppose thateach T ∈ W lies in the unit ball. Let W = | W | . Suppose that f = P T ∈ W f T , where f T is microlocalized to ( T, θ ( T )) . Suppose that k f T k L p is ∼ constant for each T ∈ W . Let Y be a union of R − / -cubes in the unit ball each of which intersects at most M tubes T ∈ W . Then k f k L p ( Y ) . ǫ R ǫ (cid:18) MW (cid:19) − p X T ∈ W k f T k L p ! / . Proof of Lemma 4.1.
Assume r > R (we omit the r < R case, which ismuch easier and can be dealt with by the same argument at the end of Section 5 in[9]). By definition, µ ,g ∗ ˆ σ r = X R j ∼ r X τ X T ∈ T j,τ : T good M T µ ∗ ˆ σ r + RapDec( r ) . The contribution of RapDec( r ) is already taken into account in the statement ofLemma 4.1. Hence without loss of generality we may ignore the tail RapDec( r ) in theargument below.Let η be a bump function adapted to the unit ball and define f T = η ( M T µ ∗ ˆ σ r ) . One can easily verify that f T is microlocalized to ( T, θ ( T )).After dyadic pigeonholing, there exists λ > Z | µ ,g ∗ ˆ σ r ( x ) | dµ ( x ) . log r Z | f λ ( x ) | dµ ( x ) , where f λ = X T ∈ W λ f T , W λ := [ R j ∼ r [ τ n T ∈ T j,τ : T good , k f T k L p ∼ λ o . To simplify the argument, we do another pigeonholing: divide the unit ball into r − / -cubes q and sort them. This then reduces the integration domain of | f λ | in the aboveto Y M = S q ∈Q M q for some M , where Q M := { r − / -cubes q : q intersects ∼ M tubes T ∈ W λ } . Since f λ only involves good wave packets, by considering the quantity X q ∈Q M X T ∈ W λ : T ∩ q = ∅ µ ( q ) , we get(4.1) M µ ( N r − / ( Y M )) . | W λ | r − d + c ( α ) δ . The rest of the proof of Lemma 4.1 will follow from Theorem 4.2 and estimate (4.1).Let p = d +1) d − . By H¨older’s inequality and the observation that f λ has Fouriersupport in the 1-neighborhood of the sphere of radius r , one has Z Y M | f λ ( x ) | dµ ( x ) . (cid:18)Z Y M | f λ | p (cid:19) /p (cid:18)Z Y M | µ ∗ η /r | p/ ( p − (cid:19) − /p , where η /r is a bump function with integral 1 that is essentially supported on the ballof radius 1 /r . ALCONER’S DISTANCE SET PROBLEM IN EVEN DIMENSIONS 11
To bound the second factor, we note that η /r ∼ r d on the ball of radius 1 /r andrapidly decaying off it. Using the fact that µ is α -dimensional, we have k µ ∗ η /r k ∞ . r d − α . Therefore, Z Y M | µ ∗ η /r | p/ ( p − . k µ ∗ η /r k / ( p − ∞ Z Y M dµ ∗ η /r . r d − α ) / ( p − µ ( N r − / ( Y M )) . By Theorem 4.2, the first factor can be bounded as follows: (cid:18)Z Y M | f λ | p (cid:19) /p / (cid:18) M W λ (cid:19) − /p X T ∈ W λ k f T k L p . r − d + c ( α ) δ µ ( N r − / ( Y M )) ! − /p X T ∈ W λ k f T k L p , where the second step follows from (4.1).Combining the two estimates together, one obtains Z Y M | f λ ( x ) | dµ ( x ) . r O α ( δ )+( p − ) d − αp X T ∈ W λ k f T k L p . Observe that k f T k L p has the following simple bound: k f T k L p . k f T k L ∞ | T | /p . σ r ( θ ( T )) / | T | /p k \ M T µ k L ( dσ r ) = r − ( p + )( d − O α ( δ ) k \ M T µ k L ( dσ r ) . Plugging this back into the above formula, one obtains Z Y M | f λ ( x ) | dµ ( x ) . r O α ( δ )+( p − ) d − αp + p + X T ∈ W λ k \ M T µ k L ( dσ r ) . r − d d +1) − ( d − αd +1 + ǫ r − ( d − Z | ˆ µ | ψ r dξ, where p = 2( d +1) / ( d −
1) and we have used orthogonality and chosen δ sufficiently smalldepending on α , ǫ . The proof of Lemma 4.1 and hence Proposition 2.2 is complete.5. Further comments
Generalization to other norms.
Similarly as the two-dimensional case in [9],Theorem 1.1 and 1.2 still hold if ∆( E ) and ∆ x ( E ) are replaced by∆ K ( E ) = {|| x − y || K : x, y ∈ E } and ∆ Kx ( E ) = {|| x − y || K : y ∈ E } respectively, where K is a symmetric convex body whose boundary ∂K is C ∞ smoothand has everywhere positive Gaussian curvature bounded from above and below, and || · || K is the distance induced by the norm determined by K .The argument is identical to the one given in Section 7 of [9], where the mainadditional ingredient is the celebrated stationary phase formula due to Herz [11], whichsays b σ K ( ξ ) = C (cid:18) ξ | ξ | (cid:19) | ξ | − d − (cid:18) cos (cid:18) π (cid:18) k ξ k K ∗ − d − (cid:19)(cid:19)(cid:19) , where σ K is the normalized surface measure on S = { x ∈ R d : k x k K = 1 } and k · k K ∗ is the dual norm defined by k ξ k K ∗ = sup x ∈ K x · ξ. We omit the details.5.2.
Why our method fails in odd dimensions.
In odd dimension d , in order tomake use of the Orponen’s radial projection theorem to control the bad part, we project α -dimensional measure µ onto a d +12 -dimensional plane P , since the condition α > d only guarantees that α > d +12 −
1. To make the proof for bad part work through, weneed to choose the mass threshold for bad tubes as follows: T ∈ T j,τ is bad if µ (4 T ) & R − ( d − / c ( α ) δj . Then the numerology for good part gives us the following dimensional threshold forFalconer’s distance set problem: d d , which is not as good as the previously best known result d + + d − from [4].In fact, when d is odd, there exists counterexample that prevents one from removinga larger bad part from the measure. More precisely, consider a set E ⊂ R d thatis contained in some d +12 dimensional subspace of R d with positive d +12 dimensionalLebesgue measure. For instance, let E be the unit ball B d +12 . Then for every T ∈ T j,τ , µ ( T ) ∼ R − d +12 − + δj = R − d − + δj . Hence it is impossible to further lower the bad threshold.6.
Connections with the Erd˝os distance problem
The following definition is due to the second listed author, Rudnev and Uriarte-Tuero([14]).
ALCONER’S DISTANCE SET PROBLEM IN EVEN DIMENSIONS 13
Definition 6.1.
Let P be a set of N points contained in [0 , d . Define the measure (6.2) dµ sP ( x ) = N − · N ds · X p ∈ P χ B ( N s ( x − p )) dx, where χ B is the indicator function of the ball of radius centered at the origin. Wesay that P is s -adaptable if there exists C independent of N such that (6.3) I s ( µ P ) = Z Z | x − y | − s dµ sP ( x ) dµ sP ( y ) ≤ C. It is not difficult to check that if the points in set P are separated by distance cN − /s ,then (6.3) is equivalent to the condition(6.4) 1 N X p = p ′ | p − p ′ | − s ≤ C, where the exact value of C may be different from line to line. In dimension d , it is alsoeasy to check that if the distance between any two points of P is & N − /d , then (6.4)holds for any s ∈ [0 , d ), and hence P is s -adaptable.Let K be a symmetric convex body as in Section 5.1. We will prove that if P is s -adaptable and d is even, then for some x ∈ P , | ∆ Kx ( P ) | ' N d . Moreover, the proof below shows that we get this many distinct N − s -separateddistances, where s = d + . The best currently known bounds for distance sets inhigher dimensions with respect to the Euclidean metric are due to Solymosi and Vu[20]. While their exponent is larger than ours and applies to general point sets, theirmethod does not yield separated distances and does not apply to general metrics. Forthe best previously known bounds in higher dimensions for general metrics, see, forexample, [12] and [13].Fix s ∈ ( d + , d ) and define dµ sP as above. Note that the support of dµ sP is P N − s ,the N − s -neighborhood of P . Since I s ( µ sP ) is uniformly bounded, the proof of (thegeneral norm case of) Theorem 1.2 implies that there exists x ∈ P N − /s so that L (∆ Kx ( P N − s )) ≥ c > , where the constant c only depends on the value of C in (6.4).Let x be a point of P with | x − x | ≤ N − /s . It follows that for any y , k x − y k K = k x − y k K + O ( N − /s ). Let E N − /s (cid:0) ∆ Kx ( P ) (cid:1) be the smallest number of N − /s -intervalsneeded to cover ∆ Kx ( P ). We know that ∆ Kx ( P N − /s ) is contained in the O ( N − /s )neighborhood of ∆ Kx ( P ), and so L (∆ Kx ( P N − s )) . N − s E N − /s (cid:0) ∆ Kx ( P ) (cid:1) . Then our lower bound on L (∆ Kx ( P N − s )) gives E N − /s (cid:0) ∆ Kx ( P ) (cid:1) & N /s . In other words, ∆ Kx ( P ) contains & N /s different distances that are pairwise sepa-rated by & N − /s . In particular, | ∆ Kx ( P ) | & N /s . Since this holds for every s > d + ,we get | ∆ Kx ( P ) | ' N d as desired. References [1] J. Bourgain and C. Demeter,
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On the dimension and smoothness of radial projections , Anal. PDE (2019), no.5, 1273–1294[20] J. Solymosi and V. Vu, Near optimal bounds for the Erd˝os distinct distances problem in highdimensions , Combinatorica (2008), no. 1, 113–125.[21] T. Wolff, Decay of circular means of Fourier transforms of measures , Int. Math. Res. Not. (1999), 547–567. ALCONER’S DISTANCE SET PROBLEM IN EVEN DIMENSIONS 15 (X. Du)
Department of Mathematics, University of Maryland, College Park, MD20742 (A. Iosevich)
Department of Mathematics, University of Rochester, Rochester, NY14627 (Y. Ou)
Department of Mathematics, Baruch College, City University of New York,New York, NY 10010 (H. Wang)
School of Mathematics, Institute for Advanced Study, Princeton, NJ08540 (R. Zhang)(R. Zhang)