An -identity and pinned distance problem
Let be a Frostman measure on . The spherical average decay
was originally used to attack Falconer distance conjecture, via Mattila’s integral. In this paper we consider the pinned distance problem, a stronger version of Falconer distance problem, and show that spherical average decay implies the same dimensional threshold on both of them. In particular, with the best known spherical average estimates, we improve Peres-Schlag’s result on pinned distance problem significantly.
The idea is to reduce the pinned distance problem to an integral where spherical averages apply. The key ingredient is the following identity. Using a group action argument, we show that for any Schwartz function on and any ,
where is the normalized surface measure on . An interesting remark is that the right hand side can be easily seen equal to
An alternative derivation of Mattila’s integral via group actions is also given in the Appendix.
Key words and phrases:
1.1. Falconer distance conjecture
Given , , one can define its distance set as
The famous Falconer distance conjecture () states that has full Hausdorff dimension, or even positive Lebesgue measure, whenever the Hausdorff dimension of , denoted by , is greater than . It is known () that is necessary and the dimensional threshold is, up to the end point, optimal. This conjecture can be seen as a continuous version of the Erdős distance problem, which has already been solved by Guth and Katz in the plane ().
Theorem 1.1 (Guth, Katz, 2015).
Suppose is a finite set of points. Then for any there exists a constant such that
Unlike the Erdős distance problem, the Falconer distance conjecture is, however, far from being solved. Accumulating effort of different great mathematicians (see e.g., ,,,,,), the best currently known results are due to Wolff () in the plane and Erdogan () in higher dimensions. They proved that has positive Lebesgue measure whenever . On the other hand, assuming in the plane, Bourgain  showed that there exists an absolute such that . Very recently Keleti and Shmerkin () improve Bourgain’s result by showing that whenever in the plane.
To obtain Wolff-Erdogan’s dimensional exponent (i.e., ), the following tool invented by Mattila  plays an important role. That is, to show that has positive Lebesgue measure, it suffices to prove that there exists a measure on such that
The following lemma provides a family of measures on .
Frostman Lemma (see, e.g. , Theorem 2.7).
Suppose and denote as the -dimensional Hausdorff measure. Then if and only if there exists a probability measure on such that
for any , .
Since by definition , Frostman Lemma implies that for any there exists a probability measure on such that
In fact, the main results in Wolff’s and Erdogan’s papers are, for any satisfying (1.2),
which is known to be finite whenever and small enough (see, e.g. , Section 2.5). Solve it for to obtain .
1.2. Pinned distance problem
A stronger version of the Falconer distance problem is the pinned distance problem, which states the following.
Pinned distance problem: How large the Hausdorff dimension of , needs to be to ensure that there exists such that the pinned distance set has full Hausdorff dimension, or even positive Lebesgue measure?
This problem was first studied by Peres and Schlag (). For , denote as its -dimensional Lebesgue measure.
Theorem 1.3 (Peres, Schlag, 2000).
Given , , then
In particular, if , there exists such that .
Later this problem was studied by different authors (, , , , , ) and the estimate (1.5) was recently improved by Iosevich and the author () when . However, for the pinned distance problem, the best known dimensional exponent is still Peres-Schlag’s . There are also some results on special classes of sets. For example, Keleti and Shmerkin () proved that for planar sets , has full Hausdorff dimension for some if and , where denotes the packing dimension. One can also see , , . As we can see, there is a gap on the known dimensional threshold between Falconer distance problem and pinned distance problem. So it is very natural to ask if the exponent for Falconer distance problem is also sufficient for the pinned distance problem. This is the main result of this paper.
Suppose , . Assuming (1.3), then
In particular, if , there exists such that has positive Lebesgue measure.
Wolff-Erdogan’s estimate on (1.3) implies the following.
Suppose , . Then
In particular, if , there exists such that has positive Lebesgue measure.
The proof relies on an -identity (see Section 1.4) and Wolff-Erdogan’s estimate (see Lemma 3.2). Although Wolff-Erdogan’s estimate was originally used on the Falconer distance problem, this paper shows that it also works on the pinned distance problem, where Mattila’s integral (1.1) is replaced by a new integral (see (1.8)).
1.3. Associated spherical means
Let be a Frostman measure on and be the normalized surface measure on . One can define a measure on by
where , . Therefore as a distribution,
To prove Theorem 1.4, a natural idea is to show that for any ,
there must exist such that the support of has positive Lebesgue measure. Thus it suffices to prove that there exists a measure on such that
If it holds, the Radon-Nikodym derivatives for -a.e. , which implies the support of has positive Lebesgue measure. We shall prove a more general result. Define
which implies the following geometric result.
Suppose , , . Then for any , the -chain set,
has positive -dimensional Lebesgue measure.
1.4. An -identity and weighted Strichartz estimates
The key new ingredient in this paper is the following -identity. Denote as the normalized surface measure on . Also denote .
For any Schwartz function on , and any ,
This identity links the spherical mean value operator (on ) and the extension operator (on ), where restriction estimates apply. Moreover, the right hand side equals, by Plancherel,
where denotes the inverse Fourier transform, and is the standard Laplacian. Similarly, with , it follows that
Therefore, the norm in Theorem 1.7 is, in fact,
In other words, we reduce the pinned distance problem to weighted -estimates for the wave (or Schrödinger) operator. This kind of estimates was first studied by Ruiz and Vega in , where they investigate perturbations of the free equation by time-dependent potentials. More precisely they consider
where , the Morrey-Campanato classes, defined by
One can also see , ,  for related work. An explicit weight, , is discussed in  (see (2.6) there). Although, unfortunately, none of their results helps in the distance problem, it is interesting to see this connection between geometric measure theory and PDE. Notations. means for some constant . means for some constant , depending on .
For any set , denotes its -dimensional Lebesgue measure.
Denote as the normalized surface measure on . Also denote .
is the Fourier transform and is the inverse Fourier transform.
Denote as the standard Laplacian and .
The proof relies on a group action argument. The idea of using group action argument to attack distance problem dates back to the solution to the Erdős distance problem (, ). On Falconer distance problem, authors in  observed that Mattila’s integral (1.1) can be interpreted in terms of Haar measures on , that is,
In this paper our proof of Theorem 1.9 is inspired by , where an alternative derivation of Mattila’s integral (1.1) is given (see Appendix). Similar reduction can also be found in . We may assume is real. Denote as the Haar measure on , the orthogonal group. By the invariance of the Haar measure, we can write
where is arbitrary but fixed. Then
By the invariance of the Haar measure, we may replace by . By polar coordinates , we have . It follows that
3. Some lemmas on Frostman measures
We give the proof below for the sake of completeness.
Take whose Fourier transform is positive on the unit ball. Then
Since has bounded support and satisfies (1.2),
Then the lemma follows by Shur’s test. ∎
Suppose is a compactly supported measure satisfying (1.2). Then
Denote as the -neighborhood of . In Wolff’s and Erdogan’s proof of (1.3), what was proved is, for any supported on ,
One can see, e.g. , Chapter 16 for the reduction.
In our case, since has compact support, one can find such that on the support of . Then is smooth on . Therefore by (3.1),
Since has compact support and is the normalized surface measure on ,
as desired. ∎
By Theorem 1.9,
Then by Lemma 3.2, it is bounded above by
To see this, by Lemma 3.1,
which is if , as desired.
APPENDIX: A derivation of Mattila’s integral (1.1) via group actions
We only sketch the proof. One can see  for details. Denote as the group of rigid motions and as the normalized surface measure on . Roughly speaking there are two ways to define a measure on the distance set ,
where , arbitrary but fixed. Therefore
By the invariance of the Haar measure, we can take , . Choose such that . Now the integral equals
By Plancherel in , it equals
By Plancherel in , it equals
- J. A. Barceló, J. M. Bennett, A. Carbery, A. Ruiz, and M. C. Vilela. A note on weighted estimates for the Schrödinger operator. Rev. Mat. Complut., 21(2):481–488, 2008.
- J. A. Barceló, J. M. Bennett, A. Carbery, A. Ruiz, and M. C. Vilela. Strichartz inequalities with weights in Morrey-Campanato classes. Collect. Math., 61(1):49–56, 2010.
- M. Bennett, A. Iosevich, and K. Taylor. Finite chains inside thin subsets of . Anal. PDE, 9(3):597–614, 2016.
- J. Bourgain. Hausdorff dimension and distance sets. Israel J. Math., 87(1-3):193–201, 1994.
- J. Bourgain. On the Erdős-Volkmann and Katz-Tao ring conjectures. Geom. Funct. Anal., 13(2):334–365, 2003.
- X. D. Du, L. Guth, Y. Ou, H. Wang, B. Wilson, and R. Zhang. Weighted restriction estimates and application to falconer distance set problem. https://arxiv.org/abs/1802.10186, 2018.
- G. Elekes and M. Sharir. Incidences in three dimensions and distinct distances in the plane. Combinatorics, Probability and Computing, 20(4):571–608, 2011.
- M. B. Erdogan. A bilinear Fourier extension theorem and applications to the distance set problem. Int. Math. Res. Not., (23):1411–1425, 2005.
- K. J. Falconer. On the Hausdorff dimensions of distance sets. Mathematika, 32(2):206–212, 1985.
- A. Greenleaf, A. Iosevich, B. Liu, and E. Palsson. A group-theoretic viewpoint on Erdős-Falconer problems and the Mattila integral. Rev. Mat. Iberoam., 31(3):799–810, 2015.
- L. Guth and N. H. Katz. On the Erdős distinct distances problem in the plane. Ann. of Math. (2), 181(1):155–190, 2015.
- K. Hidano, J. Metcalfe, H. F. Smith, C. D. Sogge, and Y. Zhou. On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles. Trans. Amer. Math. Soc., 362(5):2789–2809, 2010.
- A. Iosevich and B. Liu. Pinned distance problem, slicing measures and local smoothing estimates. arXiv preprint arXiv:1706.09851, 2017.
- A. Iosevich, K. Taylor, and I. Uriarte-Tuero. Pinned geometric configurations in euclidean space and riemannian manifolds. https://arxiv.org/pdf/1610.00349v1.pdf, 2016.
- T. Keleti and P. Shmerkin. New bounds on the dimensions of planar distance sets. https://arxiv.org/abs/1801.08745, 2018.
- Y. Koh and I. Seo. On weighted estimates for solutions of the wave equation. Proc. Amer. Math. Soc., 144(7):3047–3061, 2016.
- B. Liu. Group actions, the mattila integral and applications. arXiv preprint arXiv:1705.00560, 2017.
- B. Liu. Improvement on -chains inside thin subsets of euclidean spaces. https://arxiv.org/abs/1709.06814, 2017.
- R. Lucà and K. Rogers. Avergae decay of the fourier transform of measures with appli- cations. to appear in J. Eur. Math. Soc., 2015.
- P. Mattila. Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets. Mathematika, 34(2):207–228, 1987.
- P. Mattila. Fourier analysis and Hausdorff dimension, volume 150. Cambridge University Press, 2015.
- D. Oberlin and R. Oberlin. Spherical means and pinned distance sets. Commun. Korean Math. Soc., 30(1):23–34, 2015.
- T. Orponen. On the distance sets of Ahlfors-David regular sets. Adv. Math., 307:1029–1045, 2017.
- Y. Peres and W. Schlag. Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J., 102(2):193–251, 2000.
- A. Ruiz and L. Vega. Local regularity of solutions to wave equations with time-dependent potentials. Duke Math. J., 76(3):913–940, 1994.
- P. Shmerkin. On distance sets, box-counting and ahlfors-regular sets. arXiv:1604.00308, 2016.
- P. Shmerkin. On the hausdorff dimension of pinned distance sets. arXiv preprint arXiv:1706.00131, 2017.
- P. Sjölin. Estimates of spherical averages of Fourier transforms and dimensions of sets. Mathematika, 40(2):322–330, 1993.
- T. Wolff. Decay of circular means of Fourier transforms of measures. Internat. Math. Res. Notices, (10):547–567, 1999.