1 Introduction

# An L2-identity and pinned distance problem

## Abstract.

Let be a Frostman measure on . The spherical average decay

 ∫Sd−1|ˆμ(rω)|2dω≲r−β

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 ,

 ∫∞0|ωt∗f(x)|2td−1dt=∫∞0|ˆωr∗f(x)|2rd−1dr,

where is the normalized surface measure on . An interesting remark is that the right hand side can be easily seen equal to

 Missing or unrecognized delimiter for \right

An alternative derivation of Mattila’s integral via group actions is also given in the Appendix.

###### Key words and phrases:
The work is supported by Professor Nir Lev’s ERC Starting Grant No. 713927

## 1. Introduction

### 1.1. Falconer distance conjecture

Given , , one can define its distance set as

 Δ(E)={|x−y|:x,y∈E}.

The famous Falconer distance conjecture ([9]) states that has full Hausdorff dimension, or even positive Lebesgue measure, whenever the Hausdorff dimension of , denoted by , is greater than . It is known ([9]) 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 ([11]).

###### Theorem 1.1 (Guth, Katz, 2015).

Suppose is a finite set of points. Then for any there exists a constant such that

 #(Δ(P))≥CϵN1−ϵ.

Unlike the Erdős distance problem, the Falconer distance conjecture is, however, far from being solved. Accumulating effort of different great mathematicians (see e.g., [9],[20],[28],[4],[29],[8]), the best currently known results are due to Wolff ([29]) in the plane and Erdogan ([8]) in higher dimensions. They proved that has positive Lebesgue measure whenever . On the other hand, assuming in the plane, Bourgain [5] showed that there exists an absolute such that . Very recently Keleti and Shmerkin ([15]) 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 [20] 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

 (1.1) M(μ):=∫(∫Sd−1|ˆμ(rω)|2dω)2rd−1dr<∞.

The following lemma provides a family of measures on .

###### Frostman Lemma (see, e.g. [21], Theorem 2.7).

Suppose and denote as the -dimensional Hausdorff measure. Then if and only if there exists a probability measure on such that

 μ(B(x,r))≲rs

for any , .

Since by definition , Frostman Lemma implies that for any there exists a probability measure on such that

 (1.2) μ(B(x,r))≲rsμ, ∀ x∈Rd, r>0.

In fact, the main results in Wolff’s and Erdogan’s papers are, for any satisfying (1.2),

 (1.3) ∫Sd−1|ˆμ(rω)|2dω≲ϵr−β(sμ)+ϵ,

where

 β(s)=d+2s−24,  s∈[d2,d2+1].

When is small, it is proved by Mattila ([20]) that (1.3) holds with

 β(s)={s,s∈(0,d−12]d−12,s∈[d−12,d2].

When is large, the best known result on (1.3) is due to Lucà and Rogers ([19]). Plugging (1.3) into (1.1), it follows that

 M(μ)≲ϵ∫|ˆμ(ξ)|2|ξ|−β(sμ)+ϵdξ,

which is known to be finite whenever and small enough (see, e.g. [21], Section 2.5). Solve it for to obtain .

###### Remark 1.2.

Shortly after this paper came out, Du-Guth-Ou-Want-Wilson-Zhang ([6]) improves (1.3) when , . Now the best known dimensional exponent for Falconer distance conjecture is

 (1.4) dimH(E)>⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩43,d=2 (Wolff)1.8,d=3 (Du \emph{et al.})d2+14+d+14(2d+1)(d−1),d≥4 (Du \emph{et % al.})\par

### 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 ([24]). For , denote as its -dimensional Lebesgue measure.

###### Theorem 1.3 (Peres, Schlag, 2000).

Given , , then

 (1.5) dimH({x∈Rd:|Δx(E)|=0})≤d+1−dimH(E).

In particular, if , there exists such that .

Later this problem was studied by different authors ([22], [14], [26], [13], [27], [15]) and the estimate (1.5) was recently improved by Iosevich and the author ([13]) 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 ([15]) proved that for planar sets , has full Hausdorff dimension for some if and , where denotes the packing dimension. One can also see [23], [26], [27]. 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.

###### Theorem 1.4.

Suppose , . Assuming (1.3), then

 dimH({x∈Rd:|Δx(E)|=0})≤inf{s:dimH(E)+β(s)>d}.

In particular, if , there exists such that has positive Lebesgue measure.

Wolff-Erdogan’s estimate on (1.3) implies the following.

###### Corollary 1.5.

Suppose , . Then

 Unknown environment 'cases%

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)).

###### Remark 1.6.

As we mentioned above in Remark 1.2, the estimate (1.3) has been improved by Du et al. when very recently. Thus our dimensional exponent for the pinned distance problem has been improved to (1.4) as well.

### 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

 ∫f(t)dνx(t)=∫f(|x−y|)dμ(y)=limϵ→0cd∫f(t)(∫Sd−1μϵ(x−tω)dω)td−1dt,

where , . Therefore as a distribution,

 νx(t)=limϵ→0cdtd−1ωt∗μϵ(x).

To prove Theorem 1.4, a natural idea is to show that for any ,

 dimH(F)>{32d+1−2dimH(E),dimH(E)∈[d2,d+12]d−dimH(E),dimH(E)>d+12,

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

 ∫∫t≈1|νx(t)|2dtdλ(x)<∞.

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

 Ttf(x)=ωt∗f(x), Tμtf(x)=ωt∗(fdμ)(x),
 ||f||2˙Hs=∫|ˆf(ξ)|2|ξ|2sdξ.
###### Theorem 1.7.

Suppose is a compactly supported measure satisfying (1.2). Then

 (1.6) ||Ttf||L2(td−1dt×dλ)≲ϵ||f||˙H−β(sλ)2+ϵ.

In particular, if in addition satisfies (1.2) and , then

 (1.7) ||Tμtf||L2(td−1dt×dλ)≲||f||L2(μ).

As we explained above, Theorem 1.4 follows from Theorem 1.7. In fact the dimensional exponent in Theorem 1.4 comes from solving from . A straightforward consequence of Theorem 1.7 is

 ||Tμtk+1∘⋯∘Tμt1f||L2(td−11dt1×⋯×td−1k+1dtk+1×dμ)≲||f||L2(μ), if sμ+β(sμ)>d,

which implies the following geometric result.

###### Corollary 1.8.

Suppose , , . Then for any , the -chain set,

 {(|x1−x2|,…,|xk−xk+1|):xj∈E}

has positive -dimensional Lebesgue measure.

By the results of Wolff and Du et al., this corollary holds whenever (1.4) holds. This improves results in [3], where is obtained, and results in [18], where only is considered.

### 1.4. An L2-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 .

###### Theorem 1.9.

For any Schwartz function on , and any ,

 ∫∞0|ωt∗f(x)|2td−1dt=∫∞0|ˆωr∗f(x)|2rd−1dr.

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,

 ∫∣∣∣∫e−2πitr(ˆfdωr)∨(x)rd−12dr∣∣∣2dt= ∫∣∣∣∬e−2πitre2πix⋅rωˆf(rω)dωrd−12dr∣∣∣2dt = cd∫∣∣∣∬e−2πit|ξ|e2πix⋅ξˆf(ξ)|ξ|−d−12dξ∣∣∣2dt = cd∫∣∣∣D−d−12xe−2πit√−Δf(x)∣∣∣2dt,

where denotes the inverse Fourier transform, and is the standard Laplacian. Similarly, with , it follows that

 ∫|ˆωr∗f(x)|2rd−1dr=12∫|ˆω√r′∗f(x)|2(r′)d−22dr′=c′d∫∣∣∣D−d−22xe2πitΔf(x)∣∣∣2dt.

Therefore, the norm in Theorem 1.7 is, in fact,

 (1.8) ∬|ωt∗f(x)|2td−1dtdλ(x)= ∬|ˆωr∗f(x)|2rd−1drdλ(x) = cd∬∣∣∣D−d−12xe−2πit√−Δf(x)∣∣∣2dtdλ(x) = c′d∬∣∣∣D−d−22xe2πitΔf(x)∣∣∣2dtdλ(x).

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 [25], where they investigate perturbations of the free equation by time-dependent potentials. More precisely they consider

 ∬∣∣e−2πit√−Δf(x)∣∣2V(t,x)dtdx,  ∬∣∣e2πitΔf(x)∣∣2V(t,x)dtdx,

where , the Morrey-Campanato classes, defined by

 ||w||Lα,p=supr,x0rα(r−d∫B(x0,r)|w(x)|pdx)1p<∞.

One can also see [1], [2], [16] for related work. An explicit weight, , is discussed in [12] (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 .

## 2. Proof of Theorem 1.9

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 ([7], [11]). On Falconer distance problem, authors in [10] observed that Mattila’s integral (1.1) can be interpreted in terms of Haar measures on , that is,

 ∫(∫Sd−1|ˆμ(rω)|2dω)2rd−1dr=∫|ˆμ(ξ)|2(∫O(d)|ˆμ(θξ)|2dθ)dξ.

In this paper our proof of Theorem 1.9 is inspired by [17], where an alternative derivation of Mattila’s integral (1.1) is given (see Appendix). Similar reduction can also be found in [18]. We may assume is real. Denote as the Haar measure on , the orthogonal group. By the invariance of the Haar measure, we can write

 Ttf(x)=ωt∗f(x)=∫Sd−1f(x−tω)dω=∫O(d)f(x−tθω0)dθ,

where is arbitrary but fixed. Then

 ∫|Ttf(x)|2td−1dt= ∫(∫Sd−1f(x−tω)dω)(∫O(d)f(x−tθω0)dθ)td−1dt = ∫∫Sd−1f(x−tω)(∫O(d)f(x−tθω0)dθ)dωtd−1dt.

By the invariance of the Haar measure, we may replace by . By polar coordinates , we have . It follows that

 ∫|Ttf(x)|2td−1dt = 1|Sd−1|∬f(x−y)(∫O(d)f(x−θy)dθ)dy = 1|Sd−1|∫O(d)(∫f(x−y)f(x−θy)dy)dθ = 1|Sd−1|∫O(d)(∫ˆf(−ξ)e−2πix⋅ξ ˆf(θξ)e2πix⋅θξdξ)dθ = ∫(∫Sd−1∫Sd−1ˆf(−rω)e−2πix⋅rω ˆf(rω′)e2πix⋅rω′dωdω′)rd−1dr = ∫|ˆ¯¯¯¯ˆfdωr(x)|2rd−1dr = ∫|f∗ˆωr(x)|2rd−1dr,

as desired.

## 3. Some lemmas on Frostman measures

###### Lemma 3.1 ([3], Lemma 2.5).

Suppose satisfies (1.2). Then

 ∫|ξ|≤R|ˆfdμ(ξ)|2dξ≲Rd−sμ||f||2L2(μ).

We give the proof below for the sake of completeness.

###### Proof.

Take whose Fourier transform is positive on the unit ball. Then

 ∫|ξ|≤R|ˆfdμ(ξ)|2dξ ≲∫|ˆfdμ(ξ)|2ˆψ(ξR)dξ =Rd∬ψ(R(x−y))f(x)f(y)dμ(x)dμ(y).

Since has bounded support and satisfies (1.2),

 ∫|ψ(R(x−y))|dμ(x)≲R−sμ, ∫|ψ(R(x−y))|dμ(y)≲R−sμ.

Then the lemma follows by Shur’s test. ∎

###### Lemma 3.2.

Suppose is a compactly supported measure satisfying (1.2). Then

 ∫|ˆgdωR|2dλ≲ϵR−β(sλ)+ϵ||g||2L2(ωR).
###### Proof.

Denote as the -neighborhood of . In Wolff’s and Erdogan’s proof of (1.3), what was proved is, for any supported on ,

 (3.1) ∫|ˆh|2dλ≲ϵRd−1−β(sλ)+ϵ||h||2L2(AR).

One can see, e.g. [21], 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),

 ∫|ˆgdωR|2dλ≤∫|ˆgdωR|2|ˆϕ|2dλ=∫|ˆ(gdωR)∗ϕ|2dλ≲ϵRd−1−β(sλ)+ϵ∫AR|(gdωR)∗ϕ|2.

Since has compact support and is the normalized surface measure on ,

 ∫AR|(gdωR)∗ϕ|2 ≲ ∫AR(∫RSd−1|ϕ(x−y)||g(y)|2dωR(y))(∫RSd−1|ϕ(x−y)|dωR(y))dx ≲ R−d+1∫AR∫RSd−1|ϕ(x−y)||g(y)|2dωR(y)dx ≲ R−d+1||g||2L2(ωR),

as desired. ∎

## 4. Proof of Theorem 1.7

By Theorem 1.9,

 ∬|Ttf(x)|2td−1dtdλ(x)=∬|ˆ¯¯¯¯¯¯¯ˆfdωr(x)|2dλ(x)rd−1dr.

Then by Lemma 3.2, it is bounded above by

 ∫∫rSd−1|ˆf|2dωrr−β(sλ)+ϵrd−1dr=cd∫|ˆf(ξ)|2|ξ|−β(sλ)+ϵdξ,

which completes the proof of (1.6) in Theorem 1.7. For (1.7), it suffices to show, when ,

 ∫|ˆfdμ(ξ)|2|ξ|−β(sλ)+ϵdξ≲||f||2L2(μ).

To see this, by Lemma 3.1,

 ∫|ˆfdμ(ξ)|2|ξ|−β(sλ)+ϵdξ≲ 2j(−β(sλ)+ϵ)∫|ξ|≈2j|ˆfdμ(ξ)|2dξ ≲ ∑j2j(−β(sλ)+d−sμ+ϵ)||f||2L2(μ),

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 [17] 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 ,

 ν(t)=∫|x−y|=tμ(x)μ(y)dσt(x,y)=∫E(d)μ(gxt)μ(gyt)dg,

where , arbitrary but fixed. Therefore

 ∫|ν(t)|2J(t)dt=∫(∫|x−y|=tμ(x)μ(y)(∫E(d)μ(gxt)μ(gyt)dg)dσt(x,y))J(t)dt

By the invariance of the Haar measure, we can take , . Choose such that . Now the integral equals

 Extra open brace or missing close brace

By Plancherel in , it equals

 ∫O(d)∫Rd∣∣∣∫Rdˆμ(ξ)ˆμ(θξ)e2πiz⋅ξdξ∣∣∣2dzdθ.

By Plancherel in , it equals

 ∫O(d)(∫Rd|ˆμ(ξ)|2|ˆμ(θξ)|2dξ)dθ=cd∫(∫Sd−1|ˆμ(rω)|2dω)2rd−1dr,

as desired.

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