Pinned distance problem, slicing measures and local smoothing estimates
Abstract.
We improve the PeresSchlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with
we prove that for any , there exists a probability measure on such that for a.e. ,

if ;

has positive Lebesgue measure if ;

has nonempty interior if .
We also show that in the case when , for a.e. ,
has positive Lebesgue measure. This describes dimensions of slicing subsets of , sliced by spheres centered at .
In our proof, local smoothing estimates of Fourier integral operators (FIO) plays a crucial role. In turn, we obtain results on sharpness of local smoothing estimates by constructing geometric counterexamples.
Key words and phrases:
1. Introduction
1.1. Distance problem and pinned distance problem
One of the most important open problems in geometric measure theory is the Falconer distance conjecture ([4]). It says that given any set , , its distance set
has positive Lebesgue measure whenever . The best currently known results are due to Wolff in ([20]) and Erdogan in higher dimensions ([2]). They proved that has positive Lebesgue measure whenever . There are also results where is assumed to have special structures ([6], [14], [16]).
More generally, we may ask how large the Hausdorff dimensions of need to be to ensure the set
has positive Lebesgue for suitable functions . In [3], Eswarathasan, Iosevich and Taylor prove that if has nonzero MongeAmpere determinant (also called PhongStein rotational curvature condition), i.e.
(1.1) 
then has positive Lebesgue measure whenever .
A corollary of this result is that if is a closed subset of a closed compact Riemannian manifold of dimension of Hausdorff dimension , then has positive Lebesgue measure, where denotes the Riemannian metric on .
Another interesting version of the Falconer distance problem is the pinned distance problem. Given , we ask whether for ”many” points , the pinned distance set
has positive Lebesgue measure. This problem was first studied by Peres and Schlag ([15]). They considered a very large class of functions that generalize orthogonal projections , . For Euclidean distances, one can take and their result implies the following.
Theorem 1.1 (Peres, Schlag, (2000) ([15])).
(1.2) 
(1.3) 
(1.4) 
In 2016, Iosevich, Taylor and UriarteTuero ([7]) gave a straightforward proof of (1.3) for functions satisfying the PhongStein rotational curvature condition (1.1) in the range . Shmerkin ([17]) recently proved that
for any such that , where denotes the packing dimension of .
It is interesting to note that the PeresSchlag bound is in general sharp. One can simply take and where has Hausdorff dimension . Then while
whose Hausdorff dimension is . Therefore, if we want to improve PeresSchlag’s bound for, say Euclidean distances, we need a stronger assumption that rules out the case .
Notice that the key difference between and is that if is fixed, is a sphere with nonzero Gaussian curvature, while is just a hyperplane. With this in mind, we obtain the following improvement of the aforementioned PeresSchlag’s result when .
Definition 1.2.
We say that satisfies Sogge’s cinematic curvature condition ([18]) if
(1.5)  for any , , has nonzero Gaussian curvature. 
Theorem 1.3.
Improvement on pinned distance problem then follows easily.
Corollary 1.4.
One can check that (1.9) improves (1.2) when ; (1.10) improves (1.3) when and (1.11) improves (1.4) when .
In particular, when , we have the following sharp corollary.
Corollary 1.5.
Suppose , . With notations above,
1.2. Dimensions of slicing sets
Given a family of functions and such that whenever , then slices into pieces and we can study not only the size of images , but also the size of those slices . The most classical result is due to Marstrand.
Theorem 1.6 (Marstrand, 1954).
Suppose is a Borel set and let . Then for almost all ,
(1.12)  
Moreover, when , the typical lines with direction which intersect intersect it in dimension , i.e.,
(1.13) 
A variety of generalizations of this result have been obtained over the years. See, for example, [9] and the references contained therein.
What Peres and Schlag proved in [15] generalized (1.12) from orthogonal projections to a large class of functions socalled generalized projections, including, in particular, Euclidean distances , . In [13], Orponen generalized (1.13) from orthogonal projections to generalized projections.
Since Theorem 1.3 improves PeresSchlag’s result on dimensions of images of projections, it is natural to ask whether an improvement on dimensions of slicing sets can also be obtained. Its geometric meaning is that under the assumptions of Theorem 1.3, a typical distance occurs statistically often.
Theorem 1.7.
Under the assumptions of Theorem 1.3, with and
there exists a probability measure on such that for a.e. ,
and
1.3. Local smoothing of Fourier integral operators
The study of Fourier integral operators arises in the study of the wave equation,
whose solution is given by the real part of
More generally, we may consider the Fourier integral operators (FIO), introduced by Hörmander in 1970s ([5]). In this paper, we only present a simple form of FIO for convenience. For general information, see e.g. [19] and the references therein.
Definition 1.8.
Suppose , where are open domains. For any , one can define
Throughout this paper, we use the notation
It is known that if satisfies the PhongStein rotational curvature condition (1.1), then for any fixed ,
for any . This result is sharp if is fixed. But if satisfies the cinematic curvature condition (1.5), there is a gain of regularity for by taking average in . This phenomenon is called local smoothing.
Theorem 1.9 ( local smoothing estimate, Seeger, Sogge, Stein, 1991).
It is known that (1.14) does not hold with for any . Also breaks down for any .
It is more interesting to consider local smoothing estimate because it is related to the BochnerRiesz conjecture. Sogge’s local smoothing conjecture ([18]) takes the following form in our setup.
Conjecture 1.10 ( local smoothing estimate, Sogge, 1991).
The best currently known result, accumulating efforts by Sogge ([18]), Mockenhaupt, Seeger, Sogge ([12]) and Wolff ([21]), is due to Bourgain and Demeter ([1]). They proved that (1.16) holds with
(1.18) 
It is known that is the best possible for due to the sharpness of BochnerRiesz conjecture. There are also examples showing fails for all . In [11], Minicozzi and Sogge construct a metric on odd dimensional spaces such that works only if . Whether Sogge’s conjecture is optimal in the range is an open question.
We will see that the proof of Theorem 1.3 relies on local smoothing estimates. Therefore, in turn, geometric counterexamples of Theorem 1.3 imply sharpness of local smoothing estimates.
Theorem 1.11 (Sharpness of Theorem 1.9).
Notation

denotes the Hausdorff dimension of .

means there exists a constant such that . means the implicit constant only depends on .


.
Acknowledgements. The second listed author would like to thank Professor KaSing Lau for the financial support of research assistantship in Chinese University of Hong Kong.
2. Proof of Theorem 1.3
For any , there exists Frostman measures supported on it that reflects its Hausdorff dimension.
Lemma 2.1 (Frostman Lemma, see e.g. [8]).
Denote as the dimensional Hausdorff measure and . Then if and only if there exists a probability measure on , such that
The definition of Hausdorff dimension states that , denoted by . So by Lemma 2.1, for any and , there exists on such that
(2.1) 
To study the size of the support of a measure , we need the following lemma.
Lemma 2.2 (see, e.g. Theorem 5.4 in [10]).
Suppose is a probability measure on and .

If , the support of has Hausdorff dimension at least .

If , the support of has positive Lebesgue measure.

If , the support of has nonempty interior.
Fix , define a measure on
by
(2.2)  
Therefore, in the sense of distribution,
which will be denoted as as in Definition 1.8.
By Lemma 2.2, to prove Theorem 1.3, it suffices to show that for any of Hausdorff dimension and for any Frostman measure on in (2.1),
(2.3) 
We first decompose into LittlewoodPaley pieces. There exists , supported on such that
Denote , , , , then
and
(2.4)  
To complete the proof, we need the following two lemmas, whose proofs are given at the end of this section.
Lemma 2.3.
There exists such that whenever ,
Lemma 2.4.
In particular,
By Hölder’s inequality, Theorem 1.9 and Lemma 2.4, for any ,
(2.5)  
Together with Lemma 2.3, (2.4) becomes
(2.6)  
Take and to be small enough, then and the integral above is finite whenever
as desired.
proof of Lemma 2.3.
By interpolation we may assume is an integer, then
and by Plancherel in ,
(2.7)  
By the definition of LittlewoodPaley decomposition,
So it suffices to consider in (2.7) in a bounded domain. So it approximately equals
Denote the phase function as , i.e.,
Then
where , .
Since the PhongStein rotational curvature condition (1.1) holds, . If , the lemma follows by integration by parts. If not, must be . Therefore if , it follows that and the Lemma follows by integration by parts. ∎
3. Proof of Theorem 1.7
In the last section we proved that on