# A Sharp Mizohata–Takeuchi Type Estimate for the Cone in $\mathbb{R}^{3}$

Alex Ortiz

(Date: July 20, 2023)

###### Abstract.

We prove an analog of the Mizohata–Takeuchi conjecture for the cone in
$\mathbb{R}^{3}$ and the 1-dimensional weights.

###### Contents

  1. 1 Introduction
    1. 1.1 Weighted Fourier extension estimates
    2. 1.2 Decay of Fourier means
    3. 1.3 Maximal estimates and point-circle duality
  2. 2 List of notation
  3. 3 Wolff’s maximal estimate
  4. 4 Point-circle duality and geometric considerations
    1. 4.1 Rectangles and lightplanks
    2. 4.2 Geometry of comparability
    3. 4.3 Packing rectangles
  5. 5 Application of the maximal function estimate
    1. 5.1 Nearly lightlike pairs
  6. 6 Proof of Theorem 4 and sharpness
  7. 7 Discussion and related questions
  8. A
    1. A.1 Duality arguments and the proof of Theorem 1
    2. A.2 Proof of Fourier transform estimate $|\widecheck{\sigma}(x)|$

##  1\. Introduction

###  1.1. Weighted Fourier extension estimates

In $\mathbb{R}^{3}$, the Fourier extension operator with respect to a smooth
measure $\sigma$ for the cone segment
$\mathbb{C}one^{2}=\\{(\xi,\tau):1<|\xi|<2,\tau=|\xi|\\}$ is the linear
operator taking functions $f$ defined on
$\\{\xi\in\mathbb{R}^{2}:1<|\xi|<2\\}$ to the function $Ef$ on
$\mathbb{R}^{3}$ defined by

$Ef(x,t)=\int_{\\{\xi\in\mathbb{R}^{2}:1<|\xi|<2\\}}f(\xi)e^{2\pi
i(x\cdot\xi+t|\xi|)}\sigma(\xi)\,d\xi,\qquad(x,t)\in\mathbb{R}^{2}\times\mathbb{R}.$

By naturally lifting both $f$ and the measure $\sigma$ to the cone segment
$\mathbb{C}one^{2}$, we can equivalently regard $Ef$ as the Fourier transform
of the complex measure (on $\mathbb{C}one^{2}$) $f\sigma$:

$Ef(x)=\widecheck{f\sigma}(x)=\int_{\mathbb{C}one^{2}}f(\omega)e^{2\pi
ix\cdot\omega}\,d\sigma(\omega),\qquad x\in\mathbb{R}^{3}.$

For a measure $\nu$ in $\mathbb{R}^{3}$, an important problem in Fourier
analysis is proving $L^{2}(\nu)$ estimates for $Ef$ that reflect the geometry
of $\nu$ in some way.

An important open problem of this kind is the Mizohata–Takeuchi conjecture for
the parabola in the plane. By lattice unit squares, we mean squares of the
form $v+[0,1]^{2}\subset\mathbb{R}^{2}$ with $v\in\mathbb{Z}^{2}$.

######  Conjecture 1 (Mizohata–Takeuchi for the parabola).

Let

$Ef(x,t)=\int_{[0,1]}f(\xi)e^{2\pi
i(x\xi+t\xi^{2})}\,d\xi,\qquad(x,t)\in\mathbb{R}\times\mathbb{R},$

be the Fourier extension of the unit parabola in the plane. Suppose $\nu$ is a
positive measure supported in $B_{R}:=[0,R]^{2}$ that agrees with the Lebesgue
measure on a union of lattice unit squares, and let

$\mathbf{T}(\nu)=\sup\\{\nu(T):T\subset\mathbb{R}^{2}\ \text{is a $1\times R$
rectangle (with any orientation)}\\}.$

Then for each $\epsilon>0$, there is a constant $C_{\epsilon}$ so the _a
priori_ estimate

$\int_{[0,R]^{2}}|Ef|^{2}\,d\nu\leq
C_{\epsilon}R^{\epsilon}\,\mathbf{T}(\nu)\,\|f\|_{L^{2}([0,1])}^{2}$

holds for all $R>1$.

Besides its intrinsic interest as a problem in Fourier analysis, propositions
analogous to Conjecture 1 have been proved and have applications to
convergence to initial data for dispersive partial differential equations. See
for example the important work of Du–Zhang on the Schrödinger equation [8]
which established that $Ef(x,t)\to f(x)$ almost everywhere for $f$ in a
critical $L^{2}$-based Sobolev space as a corollary of a fractal restriction
theorem for the paraboloid.

There are a few important examples of measures $\nu$ where Mizohata–Takeuchi
(MT) for the parabola is known to hold—see for example the article of Carbery,
Iliopoulou, and Wang [7]—but the full conjecture lies out of reach at time of
writing. An important obstacle to proving MT reflects one of the objectives of
studying MT to begin with: namely we would like to improve our understanding
of the _shape_ of the level sets $\\{x:|Ef(x)|\approx\lambda\\}$. For some
background on this theme in Fourier analysis, Larry Guth’s article [11] based
on his 2022 ICM talk is a good resource.

The goal of this paper is to prove a sharp $L^{2}(\nu)$ estimate of the
Fourier extension of the cone segment for 1-dimensional $\nu$ that is
analogous to MT for the parabola. Our theorem for the cone only applies to
$1$-dimensional measures, whereas Conjecture 1 considers all measures. We
provide some discussion of other problems and variations in Section 7. To
state our main theorem, we need a geometric definition.

######  Definition 1.

A _lightplank_ in $\mathbb{R}^{3}$ is a rectangular parallelepiped $P$ of
dimensions $C_{0}\times AC_{0}\times A^{2}C_{0}$ such that for some unit
vector $v\in\mathbb{R}^{2}$, the longest edge of $P$ is in the direction
$(v,1)$, and whose shortest edge is in the direction $(v,-1)$.

By lattice unit cubes, we mean cubes of the form $v+[0,1]^{3}$ with
$v\in\mathbb{Z}^{3}$. Our main result is the following $L^{2}(\nu)$ cone
restriction theorem for 1-dimensional $\nu$:

######  Theorem 1.

For each $\epsilon>0$, there is a constant $C_{\epsilon}$ so the following
holds for each $R>1$. Suppose $\nu$ is a measure supported in
$B_{R}:=[0,R]^{2}\times[R,2R]$ that agrees with the Lebesgue measure on a
union of lattice unit cubes $X\subset B_{R}$ and satisfies the 1-dimensional
Frostman condition

$\nu(B^{3}(x_{0},r))\lesssim r,\qquad x_{0}\in\mathbb{R}^{3},r>1.$

Let $\mathbf{P}(\nu)$ be the quantity

$\mathbf{P}(\nu)=\max\\{\nu(P):P\ \text{is a lightplank of dimensions $1\times
R^{1/2}\times R$}\\}.$

Then the estimate

$\int_{[0,R]^{2}\times[R,2R]}|Ef|^{2}\,d\nu\leq
C_{\epsilon}R^{\epsilon}\,\mathbf{P}(\nu)^{1/2}\|f\|_{L^{2}(d\sigma)}^{2}$

holds.

The estimate of Theorem 1 is sharp in the sense that for each $R>1$, and each
$\gamma\in[1,R]$, there is a measure $\nu$ of the stated form with
$\mathbf{P}(\nu)\sim\gamma$, and a function $f$ with
$\|f\|_{L^{2}(d\sigma)}=1$ such that $\int|Ef|^{2}\,d\nu\sim\gamma^{1/2}$.

Our approach is based on a duality argument that connects weighted extension
estimates with another well studied problem in Fourier analysis, namely the
decay of Fourier means, where we make a new contribution that we describe
presently.

###  1.2. Decay of Fourier means

If $\Gamma$ is a compact submanifold of $\mathbb{R}^{d}$ and $\sigma$ is a
smooth surface measure on $\Gamma$, one can ask the following question about
the Fourier transform of $\sigma$: If $\nu$ is a measure in $\mathbb{R}^{d}$,
how fast does the Fourier average
$\int_{\Gamma}|\widehat{\nu}(R\xi)|^{2}d\sigma(\xi)$ with respect to $\sigma$
decay as $R\to\infty$? A particular line of investigation that has received
much attention is to study $\alpha$-dimensional measures $\nu$. One way to
make a precise question is to introduce the $\alpha$-dimensional energy of a
measure. For $\alpha\in(0,d)$, the $\alpha$-dimensional energy of $\nu$ is the
quantity

$I_{\alpha}(\nu)=\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{-\alpha}\,d\nu(x)d\nu(y).$

The energy is a quadratic function of $\nu$, and a precise question is for
fixed $\alpha\in(0,d)$, what is the supremum $\beta_{d}(\alpha)$ of numbers
$\beta>0$ for which we have the estimate

$\int_{\Gamma}|\widehat{\nu}(R\xi)|^{2}\,d\sigma(\xi)\leq
C_{\beta}R^{-\beta}I_{\alpha}(\nu)$

for all $R>1$ and all $\nu$ with $\operatorname{supp}\nu\subset B(0,1)$? In
[5], using ideas of wave packets from restriction theory, Wolff established a
lower bound on $\beta_{2}(\alpha)$ for all $\alpha\in(0,2)$ for the unit
circle. In the range $\alpha\in[1,2)$, Wolff’s lower bound was new at the
time, and it matches examples presented in the same paper, closing the
question (as far as $\beta_{2}(\alpha)$) on the unit circle in
$\mathbb{R}^{2}$:

######  Theorem 2 (Wolff, 1999 [5]).

Fix $\alpha\in(0,2)$. For any $\epsilon>0$, there is a constant $C_{\epsilon}$
such that the following is true. Let $\nu$ be a positive measure in
$\mathbb{R}^{2}$ supported in the unit disc and with $\alpha$-dimensional
energy $I_{\alpha}(\nu)=1$. Then for any $R>1$,

$\int_{0}^{2\pi}|\widehat{\nu}(Re^{i\theta})|^{2}\,d\theta\leq
C_{\epsilon}R^{\epsilon}R^{-\alpha/2}$

holds. This bound is sharp in the sense that $\beta_{2}(\alpha)=\alpha/2$ for
$\alpha\in[1,2]$.

As for the cone in $\mathbb{R}^{3}$, before Erdoğan’s work in [1], the sharp
exponent $\beta_{3}(\alpha)$ for $\mathbb{C}one^{2}$ was known in the ranges
$\alpha\in(0,1]\cup[2,3)$. Combining ideas from Wolff’s investigation of
Fourier decay on dilations of the circle with techniques from bilinear
restriction theory (in particular, a Whitney decomposition of the cone),
Erdoğan established the values of $\beta_{3}(\alpha)$ for $\alpha\in(1,2)$ for
$\mathbb{C}one^{2}$:

######  Theorem 3 (Erdoğan, 2004 [1]).

Fix $\alpha\in[1,2]$. If $\nu$ is a compactly supported measure in
$\mathbb{R}^{3}$ with $I_{\alpha}(\nu)=1$, and $\sigma$ is a smooth surface
measure on $\mathbb{C}one^{2}$, then for each $\epsilon>0$ there is a constant
$C_{\epsilon}$ so the estimate

$\int_{\mathbb{C}one^{2}}|\widehat{\nu}(R\xi)|^{2}\,d\sigma(\xi)\leq
C_{\epsilon}R^{\epsilon}R^{-\alpha/2}$

holds for all $R>1$. Moreover, this bound is sharp in the sense that
$\beta_{3}(\alpha)=\alpha/2$ for $\alpha\in[1,2]$.

Note in particular how the rate of decay in Theorem 3 matches that of the
circle in Theorem 2 for $\alpha\in[1,2)$.

The present paper makes a contribution to this line of investigation by
proving a geometric sharpening of Erdoğan’s estimate at $\alpha=1$. Our main
theorem regarding the decay of Fourier means is the following:

######  Theorem 4.

For each $\epsilon>0$, there is a constant $C_{\epsilon}$ so the following
holds for each $R>1$. Suppose $\nu$ is a measure supported in
$B_{R}:=[0,R]^{2}\times[R,2R]$ that agrees with the Lebesgue measure on a
union of lattice unit cubes $X\subset B_{R}$ and satisfies the 1-dimensional
Frostman condition

$\nu(B^{3}(x_{0},r))\lesssim r,\qquad x_{0}\in\mathbb{R}^{3},r>1.$

Let $\mathbf{P}(\nu)$ be the quantity

$\mathbf{P}(\nu)=\max\\{\nu(P):P\ \text{is a lightplank of dimensions $1\times
R^{1/2}\times R$}\\}.$

Then the estimate

$\int|\widehat{\nu}|^{2}\,d\sigma\leq
C_{\epsilon}R^{\epsilon}\,\mathbf{P}(\nu)^{1/2}\|\nu\|$

holds, where $\|\nu\|=\nu(\mathbb{R}^{3})=|X|$ is the total mass of $\nu$.

Our hypotheses are slightly different from Erdoğan’s; we start from a measure
$\nu$ that agrees with the Lebesgue measure on a 1-dimensional configuration
of lattice unit cubes in $[0,R]^{2}\times[R,2R]$, rather than a measure
supported in the unit ball with $I_{1}(\nu)=1$. To illustrate the connection
between Theorem 3 and our Theorem 4, we show how Theorem 3 implies a weaker
estimate than Theorem 4:

######  Corollary 1.

For each $\epsilon>0$, there is a constant $C_{\epsilon}$ so the following
holds for each $R>1$. Suppose $\nu$ is a measure supported in
$B_{R}:=[0,R]^{2}\times[R,2R]$ that agrees with the Lebesgue measure on a
union of lattice unit cubes $X\subset B_{R}$ and satisfies the 1-dimensional
Frostman condition

$\nu(B^{3}(x_{0},r))\lesssim r,\qquad x_{0}\in\mathbb{R}^{3},r>1.$

Then the estimate

$\int|\widehat{\nu}|^{2}\,d\sigma\leq C_{\epsilon}R^{\epsilon}R^{3/2}$

holds.

###### Proof using Theorem 3.

To make use of Theorem 3, we have to push the measure $\nu$ forward to
$B_{1}:=[0,1]^{2}\times[1,2]$ under the map

$Tx=R^{-1}x,$

as well as normalize the 1-energy of our measure. Let $\mu=T\nu$ be the
pushforward of $\nu$ under $T$. By the definition of $\mu$ and our assumption
on $\nu$,

$\displaystyle
I_{1}(\mu)=\iint\frac{d\nu(x)d\nu(y)}{|R^{-1}x-R^{-1}y|}\lesssim
R\int\sum_{j=1}^{O(\log R)}\frac{\nu(B(x,2^{j}))}{2^{j}}\,d\nu(x)\lesssim(\log
R)R^{2}.$

Hence $\mu_{0}=R^{-1}\mu$ satisfies $I_{1}(\mu_{0})\approx 1$, so by Theorem
3,

$\int|\widehat{\mu_{0}}(R\xi)|^{2}\,d\sigma(\xi)\lessapprox R^{-1/2}.$

On the other hand, $\widehat{\mu_{0}}(R\xi)=R^{-1}\widehat{\nu}(\xi)$, so
substituting and rearranging, we obtain

$\int|\widehat{\nu}|^{2}\,d\sigma\lessapprox R^{3/2}.$

∎

If $\nu$ is a measure satisfying the assumptions of Theorem 4, then
$\mathbf{P}(\nu)\leq R$ and $\|\nu\|\leq R$, so Theorem 4 also immediately
implies Corollary 1. However, for measures $\nu$ with $\mathbf{P}(\nu)\ll R$,
Theorem 4 gives a better estimate than Corollary 1.

As we will show in Section A.1 of the appendix, following a closely related
argument due to Barceló–Bennett–Carbery–Rogers [9], the estimate of Theorem 4
is essentially equivalent to Theorem 1 apart from $R^{\epsilon}$ factors.
Theorem 4 is therefore sharp in the sense that for each $R$ and each
$\gamma\in[1,R]$, there is a measure $\nu$ on $B_{R}$ satisfying the Frostman
condition of exponent 1 with $\mathbf{P}(\nu)\sim\gamma$ such that

$\int|\widehat{\nu}|^{2}\,d\sigma\gtrapprox\gamma^{1/2}\|\nu\|.$

We describe examples illustrating both the sharpness of Theorem 1 and Theorem
4 following the proof of Theorem 4.

The keys to the proof of Theorem 4 are a useful pointwise estimate for the
Fourier transform $|\widecheck{\sigma}(x)|$ of a smooth surface measure
$\sigma$ on $\mathbb{C}one^{2}$, and a maximal function estimate due to Wolff
which was generalized in 2022 by Pramanik–Yang–Zahl in their work on
restricted families of projections [2].

###  1.3. Maximal estimates and point-circle duality

As we mentioned, one of the keys to the proof of Theorem 4 is a useful decay
estimate for the Fourier transform of $\sigma$, a smooth surface measure on
$\mathbb{C}one^{2}$. We do not believe this estimate is new, but we could not
find this precise statement in the literature, so we provide a proof in the
appendix to keep this paper self-contained.

######  Proposition 1.

Let $\sigma$ be a smooth compactly supported surface measure in
$\mathbb{C}one^{2}$. For any $\epsilon>0$ and any $N>1$, there is a constant
$C(\epsilon,N)$ so that

$|\widecheck{\sigma}(x)|\leq
C(\epsilon,N)\frac{1}{(1+|x|)^{\frac{1}{2}-\epsilon}}\frac{1}{(1+d(x,\Gamma_{0}))^{N}}$

holds for all $x\in\mathbb{R}^{3}$.

To a first approximation, this proposition says that up to a rapidly decaying
tail, a majorant for the Fourier transform of $\sigma$ is the “step function”

(1) $S(x)=\frac{1}{(1+|x|)^{1/2}}1_{N_{1}(\Gamma_{0})}(x)$

where $\Gamma_{0}=\\{(x^{\prime},x_{3}):||x^{\prime}|-|x_{3}||=0\\}$ is the
lightcone with vertex $0$ in $\mathbb{R}^{3}$ and $N_{1}(\Gamma_{0})$ denotes
the 1-neighborhood of $\Gamma_{0}$. By Fourier transform properties and this
heuristic,

$\int|\widehat{\nu}|^{2}\,d\sigma=\iint\widecheck{\sigma}(x-y)\,d\nu(x)d\nu(y)\lessapprox\iint
S(x-y)\,d\nu(x)d\nu(y)+R^{-500}.$

By the equation (1) for $S(x-y)$, we see to estimate $\iint
S(x-y)\,d\nu(x)d\nu(y)$, the main contribution will come from pairs
$(x,y)\in\operatorname{supp}\nu\times\operatorname{supp}\nu$ for which $x-y$
is close to the lightcone $\Gamma_{0}$. Equivalently, we can regard points
$x\in\operatorname{supp}\nu\subset[0,R]^{2}\times[R,2R]$ as circles $C_{x}$ in
the plane with centers in $[0,R]^{2}$ and radii in $[R,2R]$ via

$C_{(x^{\prime},x_{3})}=\\{a\in\mathbb{R}^{2}:||a-x^{\prime}|-x_{3}|=0\\}.$

In terms of this point-circle duality, if $x-y$ is nearly lightlike, the
circles $C_{x},C_{y}$ must be nearly internally tangent. The maximal estimates
of Wolff or Pramanik–Yang–Zahl provide the necessary geometric input that
allows us to count such pairs of nearly internally tangent circles.

### Acknowledgments

I would like to acknowledge my advisor Larry Guth for his support and
invaluable discussions. In particular, I would like to thank him for
introducing me to Wolff’s maximal function estimate.

##  2\. List of notation

  * •

$R(=\\!\delta^{-1})$ will denote the large spatial scale.

  * •

$C$ will denote absolute constants that may vary within the same line.

  * •

$B(x,r)$ denotes the Euclidean ball with center $x$ and radius $r$.

  * •

$\Gamma_{0}=\\{(a,r)\in\mathbb{R}^{3}:||a|-|r||=0\\}$ is the lightcone with
vertex $0$.

  * •

$\Gamma_{y}=\Gamma_{0}+y$ is the lightcone with vertex $y$.

  * •

$|X|$ may denote the Lebesgue measure, or the cardinality of $X$ as
appropriate.

For $\epsilon>0$ fixed:

  * •

$A\ll B$: there is a constant $C>0$ so that $A\leq\delta^{C\epsilon}B$.

  * •

$A\lessapprox B$: there is a constant $C>0$ so that
$A\lesssim\delta^{-C\epsilon}B$.

  * •

$A\approx B$: $A\lessapprox B$ and $B\lessapprox A$ (with possibly different
implied constants).

##  3\. Wolff’s maximal estimate

Fix $\delta>0$ and, for $a\in\mathbb{R}^{2}$ and $r\in[\frac{1}{2},2]$, let
$C_{\delta,a,r}=\\{x\in\mathbb{R}^{2}:r-\delta<|x-a|<r+\delta\\}$. If
$f\colon\mathbb{R}^{2}\to\mathbb{R}$, then we define
$M_{\delta}f\colon[\frac{1}{2},2]\to\mathbb{R}$ via

$M_{\delta}f(r)=\sup_{a\in\mathbb{R}^{2}}\frac{1}{|C_{\delta,a,r}|}\int_{C_{\delta,a,r}}|f(x)|\,dx.$

In [3], Wolff proved the following estimate for the maximal function
$M_{\delta}f$.

######  Theorem 5.

If $\epsilon>0$ then there is a constant $A_{\epsilon}$ such that for all
$\delta>0$ and $f$,

(2) $\|M_{\delta}f\|_{L^{3}([\frac{1}{2},2],dr)}\leq
A_{\epsilon}\delta^{-\epsilon}\|f\|_{L^{3}(\mathbb{R}^{2},dx)}.$

The estimate (2) has a dual form. Suppose that $a(r)$ is a choice of center
for a circle in the plane of radius $r$, and $w(r)$ is a nonnegative weight
function. Define a multiplicity function

$g(x)=\int_{1/2}^{2}w(r)\frac{C_{\delta,a(r),r}}{|C_{\delta,a(r),r}|}(x)\,dr.$

######  Proposition 2 (Dual formulation).

If $\epsilon>0$ then there is a constant $A_{\epsilon}$ such that for all
$\delta>0$, $a(r)$ and $w(r)$,

(3) $\|g\|_{L^{3/2}(\mathbb{R}^{2},dx)}\leq
A_{\epsilon}\delta^{-\epsilon}\|w\|_{L^{3/2}([\frac{1}{2},2],dr)}.$

######  Proposition 3.

Wolff’s maximal estimate is equivalent to its dual formulation.

###### Proof.

Suppose that (2) holds. By duality, for an appropriate $f\in
L^{3}(\mathbb{R}^{2},dx)$ with $\|f\|_{3}=1$,

$\displaystyle\|g\|_{L^{3/2}(\mathbb{R}^{2},dx)}$
$\displaystyle=\int_{\mathbb{R}^{2}}g(x)f(x)\,dx$
$\displaystyle=\int_{\mathbb{R}^{2}}\bigg{(}\int_{1}^{2}w(r)\frac{C_{\delta,a(r),r}}{|C_{\delta,a(r),r}|}(x)\,dr\bigg{)}f(x)\,dx$
$\displaystyle=\int_{1}^{2}w(r)\bigg{(}\frac{1}{|C_{\delta,a(r),r}|}\int_{C_{\delta,a(r),r}}f(x)\,dx\bigg{)}\,dr$
$\displaystyle\leq\int_{1}^{2}w(r)M_{\delta}f(r)\,dr$
$\displaystyle\leq\|w\|_{L^{3/2}([\frac{1}{2},2],dr)}\|M_{\delta}f\|_{L^{3/2}([\frac{1}{2},2],dr)}$
$\displaystyle\leq
A_{\epsilon}\delta^{-\epsilon}\|w\|_{L^{3/2}([\frac{1}{2},2],dr)}.$

Likewise, if (3) holds, then by linearizing the maximal function, given $f\in
L^{3}(\mathbb{R}^{2},dx)$, for an appropriate $a(r)$ we have

$M_{\delta}f(r)=\frac{1}{|C_{\delta,a(r),r}|}\int_{C_{\delta,a(r),r}}|f(x)|\,dx.$

By duality, for an appropriate $w\in L^{3/2}([\frac{1}{2},2],dr)$ with
$\|w\|_{3/2}=1$,

$\displaystyle\|M_{\delta}f\|_{L^{3}([1,2],dr)}$
$\displaystyle=\int_{1}^{2}M_{\delta}f(r)w(r)\,dr$
$\displaystyle=\int_{1}^{2}\bigg{(}\frac{1}{|C_{\delta,a(r),r}|}\int_{C_{\delta,a(r),r}}|f(x)|\,dx\bigg{)}w(r)\,dr$
$\displaystyle=\int_{\mathbb{R}^{2}}|f(x)|\bigg{(}\int_{1}^{2}w(r)\frac{C_{\delta,a(r),r}}{|C_{\delta,a(r),r}|}(x)\,dr\bigg{)}\,dx$
$\displaystyle\leq\|f\|_{L^{3}(\mathbb{R}^{2},dx)}\|g\|_{L^{3/2}(\mathbb{R}^{2},dx)}$
$\displaystyle\leq
A_{\epsilon}\delta^{-\epsilon}\|f\|_{L^{3}(\mathbb{R}^{2},dx)}.$

∎

We will refer to either the original maximal function estimate or its dual
formulation as “Wolff’s maximal estimate.” In the forthcoming arguments, we
will assume that for some small, but universal $\alpha_{0}>0$
($\alpha_{0}=\frac{1}{100}$ works), the centers and radii of the circles
belong to
$Q=[0,2\alpha_{0}]^{2}\times[1-\alpha_{0},1+\alpha_{0}]\subset\mathbb{R}^{3}$.
This only affects the constants in Wolff’s maximal estimate.

######  Example 1 (Wolff).

Suppose $X=\\{x_{i}=(a_{i},r_{i})\\}_{i=1}^{|X|}$ is a set of $|X|\leq R$
circles in $Q$ with at most one radius in each interval of length $\sim\delta$
in $[1-\alpha_{0},1+\alpha_{0}]$, and set

$g_{\delta}(y)=\sum_{x\in X}C_{\delta,x}(y).$

Let $\\{I_{i}\\}_{i=1}^{|X|}$ be the intervals of length $\sim\delta$ that
intersect the set of radii from $X$; then we can also express $g(y)$ as a
weighted integral of the form in Wolff’s maximal estimate:

$g_{\delta}(y)\sim\sum_{i=1}^{|X|}\int_{I_{i}}\frac{C_{\delta,a_{i},r}(y)}{\delta}\,dr\sim\int_{\bigcup_{i=1}^{|X|}I_{i}}\frac{C_{\delta,a_{i},r}(y)}{|C_{\delta,a_{i},r}|}\,dr.$

Thus by Wolff’s maximal estimate with the weight
$w(r)=1_{\bigcup_{i=1}^{|X|}I_{i}}(r)$,

$\|g_{\delta}\|_{L^{3/2}(\mathbb{R}^{2},dy)}\lesssim_{\epsilon}\delta^{-\epsilon}(\delta|X|)^{2/3}.$

In Example 1, the family of circles is very regular in the $r$-parameter. In
2022, Pramanik, Yang, and Zahl generalized Wolff’s maximal estimate in their
work on restricted families of projections [2]. In particular, as a
consequence of the general maximal function estimate they proved, we have the
following generalization of the last example. Now we allow for configurations
of circles satisfying a Frostman condition of exponent at most $1$ jointly in
the centers and radii $(a,r)$. The following theorem is essentially Remark 2
following Theorem 1.7 of [2].

######  Theorem 6 (Pramanik–Yang–Zahl [2], 2022).

Suppose $X\subset Q=[0,2\alpha_{0}]\times[1-\alpha_{0},1+\alpha_{0}]$ is a set
of circles satisfying the 1-dimensional Frostman condition

$|X\cap B(x_{0},r)|\lesssim_{\epsilon}\delta^{-\epsilon}(r/\delta),\qquad
x_{0}\in Q,r\geq\delta.$

and let

$g_{\delta}(y)=\sum_{x\in X}C_{\delta,x}(y).$

Then the estimate

$\|g_{\delta}\|_{L^{3/2}(\mathbb{R}^{2},dy)}\leq\delta^{-C\epsilon}(\delta|X|)^{2/3}$

holds.

In Section 5, we will show how to use the maximal estimate of Example 1 or
Theorem 6 to bound the number of pairs of nearly internally tangent circles.
It will be clear from the argument how any available maximal estimate of the
form

$\|g_{\delta}\|_{L^{p}(\mathbb{R}^{2},dy)}\lesssim\delta^{-\epsilon}(\delta|X|)^{1/p}$

for a configuration of circles $X$ leads to an analogous theorem to Theorem 4.

##  4\. Point-circle duality and geometric considerations

To estimate the integral

$\|\widehat{\nu}\|_{L^{2}(d\sigma)}^{2}=\iint_{B_{R}\times
B_{R}}\widecheck{\sigma}(x-y)\,d\nu(x)\,d\nu(y),$

we take into account the distance $d(x,y)=|x-y|$, as well as the distance from
$y=(y^{\prime},y_{3})$ to the lightcone $\Gamma_{x}$ with vertex
$x=(x^{\prime},x_{3})$:

$\Delta(x,y):=||x^{\prime}-y^{\prime}|-|x_{3}-y_{3}||\sim d(x-y,\Gamma_{0}).$

By the approximation $|\widecheck{\sigma}(x)|\leq S(x)+R^{-500}$, to estimate
the integral appearing above, heuristically, we could pigeonhole a value
$\rho\in[1,R]$ such that most pairs of points in $X\times X$ contributing to
the integral have $d(x,y)\sim\rho$ and $\Delta(x,y)\leq 1$. Each such pair
lies in a $1\times\rho^{1/2}\times\rho$-lightplank, as we will show (see
Proposition 22 for a precise statement). In order to apply the maximal
estimate of Example 1 or Theorem 6 to the estimate of this integral in the
proof of Theorem 4, we have to convert information about pairs of points in
$B_{R}$ into information about pairs of circles in the plane with centers in
$[0,R]^{2}$ and radii in $[R,2R]$. Once we are considering circles and their
thin neighborhoods in the plane, we are in a situation where we can directly
apply a maximal estimate.

The goal of this section is to prove a number of geometric propositions
regarding the overlap patterns of thin annuli, as well as lightplanks. By
point-circle duality, we can convert between incidences of thin annuli and
incidences of lightplanks. Besides being natural, an attractive feature of
working with lightplanks is that lightplanks are flat shapes, and certain
propositions are simpler to prove when phrased in terms of lightplanks. Our
main result in this direction is Proposition 15. We believe the results of
this section may have an independent interest.

###  4.1. Rectangles and lightplanks

Given a point $(x^{\prime},x_{3})\in\mathbb{R}^{2}\times(0,\infty)$, we can
associate a circle $C_{(x^{\prime},x_{3})}$ in the plane defined by

$C_{(x^{\prime},x_{3})}=\\{a\in\mathbb{R}^{2}:||a-x^{\prime}|-x_{3}|=0\\},$

and conversely a circle in the plane naturally determines a point in the upper
half-space with coordinates its center-radius pair. In this first subsection,
we will extend this fundamental duality between points and circles to shapes
in $\mathbb{R}^{2}\times(0,\infty)$ and subregions of thin annuli in the
plane.

From now until the end of Section 5, let $\epsilon>0$ be fixed. We will assume
that $\delta<\delta_{0}(\epsilon)$ is small enough so that
$\delta_{0}^{\epsilon}<10^{-3}$ to ensure that approximations such as
$\cos\theta\sim 1-\theta^{2}/2$ hold up to constant factors if
$|\theta|\leq\delta^{\epsilon}$. All the circles we consider will be assumed
to lie in $Q=[0,2\alpha_{0}]^{2}\times[1-\alpha_{0},1+\alpha_{0}]$ unless
mentioned otherwise.

######  Definition 2 ($\delta,\tau$-rectangle).

For $\delta^{1/2}\leq\tau\ll 1$, a _$\delta,\tau$ -rectangle_ is the
$\delta$-neighborhood of an arc of length $\tau$ on some circle of radius
$r\in[1-\alpha_{0},1+\alpha_{0}]$. We will sometimes refer to the implicit
circle in this definition as the _core circle_ of $\Omega$, and we may write
$\Omega=\Omega^{(v)}$ if $v$ is the core circle of $\Omega$. The midpoint of
the core arc of $\Omega$ will be referred to as the _center_ of $\Omega$.

######  Definition 3 (Comparable).

For $1<A<\delta^{-C\epsilon}$, we say two $\delta,\tau$-rectangles
$\Omega_{1},\Omega_{2}$ are _$A$ -comparable_ if there is an
$A^{2}\delta,A\tau$-rectangle $\Omega^{\prime}$ such that
$\Omega_{1}\cup\Omega_{2}\subset\Omega^{\prime}$. If $\Omega_{1},\Omega_{2}$
are not $A$-comparable, we say they are _$A$ -incomparable._ A collection
$\mathcal{R}$ of $\delta,\tau$-rectangles is _pairwise $A$-incomparable_ if no
two members of $\mathcal{R}$ are $A$-comparable.

With these definitions in hand, we can state the main goal of Section 4 is to
prove Proposition 10 and Proposition 16. The first says that being
$A$-comparable is almost a transitive relation on $\delta,\tau$-rectangles.
The second proposition says that we cannot fit too many $A$-incomparable
$\delta,\tau$-rectangles in a slightly larger rectangle. We need both of these
propositions for the application of the maximal function estimate.

######  Remark 1.

The definitions of $\delta,\tau$-rectangle and $A$-comparable make sense for
any numbers $\delta,\tau,A$, but in our application we only need to work with
$\delta^{1/2}\leq\tau\leq\delta^{C\epsilon}$ and $1\leq
A\leq\delta^{-C\epsilon}$. The choice of $A^{2}$ in the definition of
$A$-comparable makes the numerology in the forthcoming rectangle-lightplank
duality nicer, but it is not an important point since we always work with
$A\approx 1$.

If $\tau\lesssim\delta^{1/2}$, then a $\delta,\tau$-rectangle is a rectangle
in the usual sense, while if $\tau$ is much larger than $\delta^{1/2}$, a
$\delta,\tau$-rectangle will be a “curved” subset of a $\delta$-thick annulus.

######  Definition 4 (Tangency).

We say a $\delta,\tau$-rectangle $\Omega$ is _$\lambda$ -tangent_ to the
circle $x$ if $\Omega\subset C_{\lambda\delta,x}$. We let
$\mathcal{C}_{\lambda\delta}(\Omega)=\\{x\in Q:\Omega\subset
C_{\lambda\delta,x}\\}$ be the collection of $\lambda$-tangent circles to
$\Omega$ in $Q$.

If $\Omega=\Omega^{(v)}$ is a $\delta,\tau$-rectangle, then the core circle
$v$ is $1$-tangent to $\Omega$. Besides $v$, there are many other nearby
circles $w\in Q$ which may well serve as an _approximate_ core circle of
$\Omega$. The set of all such $w$ has the simple and important shape of a
$\approx\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$-lightplank centered
on $v$ when regarded as a subset of $\mathbb{R}^{3}$. We will describe this
correspondence more and it will be an important ingredient in some of the
proofs.

######  Proposition 4.

If $\Omega$ is a $\delta,\delta^{1/2}$-rectangle, then
$\mathcal{C}_{\delta}(\Omega)$ is the union of two lightplanks of dimensions
$\sim\delta\times\delta^{1/2}\times 1$ contained in
$\mathbb{R}^{2}\times[1-\alpha_{0},1+\alpha_{0}]$.

###### Proof.

By translating and rotating our coordinate system, we may assume that
$\Omega=[-\delta,\delta]\times[-\delta^{1/2},\delta^{1/2}]$. The circles
$(a_{1},a_{2},r)$ that have $\Omega\subset C_{\delta,(a_{1},a_{2},r)}$ satisfy
by definition

$(O(\delta)-a_{1})^{2}+(O(\delta^{1/2})-a_{2})^{2}=(r+O(\delta))^{2}.$

Simplifying and neglecting terms of $O(\delta^{2})$, we find

$O(\delta)a_{1}+O(\delta^{1/2})a_{2}+||a|^{2}-r^{2}|=O(\delta).$

In order for this equation to be satisfied, we must have $a_{1}=\pm 1+O(1)$,
$a_{2}=O(\delta^{1/2})$, and $r=|a|+O(\delta)$, which are the equations for
the union of two lightplanks of dimensions $\sim\delta\times\delta^{1/2}\times
1$ contained in $\mathbb{R}^{2}\times[1-\alpha_{0},1+\alpha_{0}]$. ∎

As a minor variation on the last proposition, we can also describe the shape
of $\mathcal{C}_{A\delta}(\Omega)$ if $\Omega$ is a
$\delta,\delta^{1/2}$-rectangle. We will need this result later when we prove
Proposition 9. The proof is just a reiteration of the proof of Proposition 4.

######  Proposition 5.

If $\Omega$ is a $\delta,\delta^{1/2}$-rectangle, then
$\mathcal{C}_{A\delta}(\Omega)$ is the the union of two lightplanks with
dimensions $\sim A\delta\times A\delta^{1/2}\times 1$ contained in
$\mathbb{R}^{2}\times[1-\alpha_{0},1+\alpha_{0}]$.

To describe the shape of $\mathcal{C}_{\delta}(\Omega)$ when $\Omega$ is a
$\delta,\tau$-rectangle and $\tau\gg\delta^{1/2}$, we make a definition.

######  Definition 5.

A _lightlike basis_ for $\mathbb{R}^{3}$ is an orthonormal basis $\mathcal{E}$
of $\mathbb{R}^{3}$, such that for some $e\in\mathbb{R}^{2}$, $|e|=1$,
$\mathcal{E}=\\{\frac{1}{\sqrt{2}}(e,1),\frac{1}{\sqrt{2}}(e,-1),(e^{\perp},0)\\}$.
A _lightlike coordinate system_ $(x_{1},x_{2},x_{3}),o$ is the usual
rectangular coordinate system with respect to a lightlike basis with the point
$o\in\mathbb{R}^{3}$ as the designated origin.

######  Proposition 6.

If $\delta^{1/2}\ll\tau\ll 1$ and $\Omega^{(o)}$ is a $\delta,\tau$-rectangle
with core circle $o$, then $\mathcal{C}_{\delta}(\Omega)$ contains and is
contained in $O(1)$-dilations of a
$\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$ lightplank contained in
$\mathbb{R}^{2}\times[1-\alpha_{0},1+\alpha_{0}]$ with center $o$.

###### Proof.

We will use complex notation, so a point in the plane can be represented as
$re^{i\theta}$ for some $r,\theta$. By rotating, translating, and scaling by a
factor $\sim 1$, we may assume that in our chosen coordinate system, the core
circle $o$ of $\Omega$ is $(0,0,1)$ and $\Omega$ is the region

$\Omega=\\{a\in\mathbb{R}^{2}:||a|-1|<\delta,|a-1|\leq\tau\\}.$

Let $N\sim\tau\delta^{-1/2}$ be a dyadic integer, so we may write $\Omega$ as
a union of $N$-many $\delta,\delta^{1/2}$-rectangles

$\Omega_{k}=\\{a\in\mathbb{R}^{2}:||a|-1|<\delta,|a-e^{ik\sqrt{\delta}}|<\sqrt{\delta}\\},\quad|k|\lesssim
N.$

By definition, $\mathcal{C}_{\delta}(\Omega)=\bigcap_{|k|\lesssim
N}\mathcal{C}_{\delta}(\Omega_{k})$, so we analyze the shape of this latter
intersection. By Proposition 4, each $\mathcal{C}_{\delta}(\Omega_{k})$ is a
union of two lightplanks $P_{1}(k),P_{2}(k)$, only one of which contains
$o$—say it is $P_{1}(k)$ for each $k$. Thus, $\mathcal{C}_{\delta}(\Omega)$ is
the intersection of the “bush” of lightplanks $\bigcap_{|k|\lesssim
N}P_{1}(k)$. Our claim is that the intersection of this bush is essentially a
smaller lightplank of the prescribed dimensions and same orientation as
$P_{1}(0)$.

Let

$e_{m}=\begin{pmatrix}0\\\ 1\\\
0\end{pmatrix},e_{l}=\frac{1}{\sqrt{2}}\begin{pmatrix}-1\\\ 0\\\
1\end{pmatrix},e_{s}=\frac{1}{\sqrt{2}}\begin{pmatrix}-1\\\ 0\\\
-1\end{pmatrix}$

be the direction vectors for the intermediate, long, and short edges of the
lightplank $P_{1}(0)$, respectively, and consider the associated lightlike
coordinate system $(x_{m},x_{l},x_{s}),o$ with $o=(0,0,1)$ as the designated
origin. In this coordinate system, we claim that
$\mathcal{C}_{\delta}(\Omega)$ is contained in and contains $O(1)$-dilations
of the set

$P:=\\{(x_{m},x_{l},x_{s}):|x_{m}|\lesssim\delta\tau^{-1},|x_{l}|\lesssim\delta\tau^{-2},|x_{s}|\leq\delta\\}.$

Indeed, for each $1\leq|k|\lesssim N$, consider the intersection
$R_{k}:=P_{1}(k)\cap P_{1}(-k)\cap P_{1}(0)$. In the lightlike coordinate
system $(x_{m},x_{l},x_{s}),o$, this region is contained in and contains
$O(1)$-dilations of the set

$\\{(x_{m},x_{l},x_{s}):|x_{m}|\lesssim k^{-1}\delta^{1/2},|x_{l}|\lesssim
k^{-2},|x_{s}|\leq\delta\\}$

as can be seen by considering the intersection of the core planes of
$P_{1}(k),P_{1}(-k)$ with the lightplank $P_{1}(0)$. Taking the intersection
of $R_{k}$ over $|k|\lesssim N$ and using the definition of $N$ gives that
$\mathcal{C}_{\delta}(\Omega)$ contains and is contained in $O(1)$-dilations
of the set

$\\{(x_{m},x_{l},x_{s}):|x_{m}|\lesssim\delta\tau^{-1},|x_{l}|\lesssim\delta\tau^{-2},|x_{s}|\leq\delta\\},$

which is a lightplank of dimensions of
$\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$, as claimed. ∎

Like the sets $\mathcal{C}_{\delta}(\Omega)$ for
$\Omega\subset\mathbb{R}^{2}$, there is an appropriate “dual” for subsets
$E\subset\mathbb{R}^{3}$.

######  Definition 6.

If $E\subset\mathbb{R}^{3}$, and $\delta>0$, let

$\mathcal{V}_{\delta}(E)=\\{a\in\mathbb{R}^{2}:E\subset\Gamma_{\delta,a}\\}$

where $\Gamma_{\delta,a}$ is the $\delta$-neighborhood of $\Gamma_{a}$.

The fundamental relationship we need between $\mathcal{C}$ and $\mathcal{V}$
is the one between
$\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$-lightplanks and
$\delta,\tau$-rectangles.

######  Proposition 7.

If $P$ is a $\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$ lightplank
contained in $Q$, then $\mathcal{V}_{\delta}(P)$ is a $\delta,\tau$-rectangle
in the plane.

###### Proof.

Let $o=(o^{\prime},o_{3})$ be the center of $P$, and let $\ell_{o}$ be the
lightray parallel to the long edge of $P$ passing through $o$. Let $a_{0}$ be
the point of intersection of $\ell_{o}$ and $\mathbb{R}^{2}\times\\{0\\}$, and
let $\Omega$ be the $\delta,\tau$ rectangle with core circle $o$ centered on
$a_{0}$. We claim that $\mathcal{V}_{\delta}(P)=\Omega$. By rotating,
translating, and scaling by a factor $\sim 1$, we may assume that in our
chosen coordinates, $o=(0,0,1)$ and $a_{0}=(1,0,0)$. If
$a\in\mathcal{V}_{\delta}(P)$, then in particular, $o\in\Gamma_{\delta,a}$, so
$||a|-1|<\delta$. We claim that $|a-a_{0}|\lesssim\tau$, which together with
$||a|-1|<\delta$ implies that $a$ is contained in a constant dilation of
$\Omega$, as desired.

Borrowing notation from Proposition 6, for any $k$, let

$\Omega_{k}=\\{a\in\mathbb{R}^{2}:||a|-1|<\delta,|a-e^{ik\sqrt{\delta}}|<\sqrt{\delta}\\},$

and let $P_{1}(k)$ be the component of $\mathcal{C}_{\delta}(\Omega_{k})$
containing $o$. Let $k_{0}\geq 1$ be the largest $k$ such that
$P\subset\bigcap_{k\leq k_{0}}P_{1}(k)$. In the lightlike coordinate system
$(x_{m},x_{l},x_{s}),o$ associated to the lightplank $P_{1}(0)$, we see that

$P\subset\bigcap_{k\leq
k_{0}}P_{1}(k)\subset\\{(x_{m},x_{l},x_{s}):|x_{m}|\lesssim
k_{0}^{-1}\delta^{1/2},|x_{l}|\lesssim k_{0}^{-2},|x_{s}|\leq\delta\\}.$

As $P$ is a $\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$ lightplank, it
must be the case that $k_{0}^{-1}\delta^{1/2}\gtrsim\delta\tau^{-1}$, or in
other words that $k_{0}\lesssim\delta^{-1/2}\tau$. Hence, if
$P\subset\Gamma_{a,\delta}$,
$|a-a_{0}|\lesssim\delta^{1/2}\cdot(\delta^{-1/2}\tau)=\tau$. This finishes
the proof. ∎

Taken together, Proposition 6 and Proposition 7 give us the following result
concerning the duality between rectangles and lightplanks.

######  Corollary 2 (Rectangle-lightplank duality).

$\delta,\tau$-rectangles in $\mathbb{R}^{2}\times\\{0\\}$ and
$\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$-lightplanks contained in
$\mathbb{R}^{2}\times[1-\alpha_{0},1+\alpha_{0}]$ are duals of one another.

###  4.2. Geometry of comparability

In addition to the duality laid out in Corollary 2, we would like to transform
statements about comparable rectangles into statements about comparable
lightplanks, and vice versa.

To begin, we need a lemma which describes the measure of the intersection of
thin annuli, and which appears in various forms throughout the literature. The
form we use here is part (a) of Lemma 3.1 in Wolff’s survey on then-recent
work on the Kakeya problem [4]. To introduce it, we set up some more notation.
Given a pair of circles $x_{i}=(a_{i},r_{i})\in Q$, $i=1,2$, we define the
numbers

$\displaystyle d(x_{1},x_{2})$ $\displaystyle=|a_{1}-a_{2}|+|r_{1}-r_{2}|,$
$\displaystyle\Delta(x_{1},x_{2})$
$\displaystyle=||a_{1}-a_{2}|-|r_{1}-r_{2}||.$

The number $d$ is the usual distance between circles thought of as points in
$\mathbb{R}^{3}$, and the number $\Delta$ is a measure of how nearly the
circles $x_{1},x_{2}$ are to being internally tangent. For instance,
$\Delta(x_{1},x_{2})=0$ if and only if the circles $x_{1},x_{2}$ are
internally tangent, or equivalently if and only if the vector $x_{1}-x_{2}$ is
lightlike. The number $\Delta(x_{1},x_{2})$ is also (up to a multiplicative
constant) the distance from $x_{2}$ to the lightcone $\Gamma_{x_{1}}$ with
vertex $x_{1}$, and vice-versa.

######  Lemma 1 (Lemma 3.1 (a), [4]).

Assume that $v,w$ are two circles in $Q$. Let $d=d(v,w)$, and
$\Delta=\Delta(v,w)$. Then

$|C_{\delta,v}\cap
C_{\delta,w}|\lesssim\delta\cdot\frac{\delta}{\sqrt{(d+\delta)(\Delta+\delta)}}.$

######  Corollary 3.

If $\Omega\subset C_{\delta,v}\cap C_{\delta,w}$ is a $\delta,\tau$-rectangle,
then

$\tau\lesssim\frac{\delta}{\sqrt{(d(v,w)+\delta)(\Delta(v,w)+\delta)}}$

###### Proof.

Let $d=d(v,w)$, $\Delta=\Delta(v,w)$. By Lemma 1,

$\delta\tau\sim|\Omega|\leq|C_{\delta,v}\cap
C_{\delta,w}|\lesssim\delta\cdot\frac{\delta}{\sqrt{(d+\delta)(\Delta+\delta)}}.$

Canceling $\delta$ from both sides of the inequality gives the desired result.
∎

######  Proposition 8.

Suppose $C_{1},C_{2}$ are two circles in $Q$ which intersect at a point
$a\in\mathbb{R}^{2}$. Let $u_{1},u_{2}$ be the unit tangent vectors to
$C_{1},C_{2}$, respectively at the point $a$. Then $\angle
u_{1},u_{2}\sim\sqrt{d(C_{1},C_{2})\Delta(C_{1},C_{2})}$

###### Proof.

Without loss of generality, suppose $C_{1}=(0,0,r)$ and $C_{2}=(b,0,s)$ with
$1-\alpha_{0}\leq s\leq r\leq 1+\alpha_{0}$ and $b>0$. With these choices, we
have $d(C_{1},C_{2})=b+(r-s)$ and $\Delta(C_{1},C_{2})=|b-(r-s)|$. Consider
the triangle $T$ in the plane whose vertices are $a$, $(0,0)$ and $(b,0)$. By
elementary geometry, the angle $\phi$ at the vertex $a$ of $T$ is the same as
$\angle u_{1},u_{2}$. By the law of cosines,

$b^{2}=r^{2}+s^{2}-2rs\cos\phi.$

Adding and subtracting $2rs$ and completing the square yields

$b^{2}=(r-s)^{2}+2rs(1-\cos\phi).$

Note that by definition,
$d(C_{1},C_{2})\Delta(C_{1},C_{2})=|b^{2}-(r-s)^{2}|$. Suppose $b>r-s$ (with a
similar conclusion in case $b\leq r-s$), so rearranging, we have

$\frac{d(C_{1},C_{2})\Delta(C_{1},C_{2})}{rs}=2(1-\cos\phi).$

Using the approximations $r,s\sim 1$ and $\cos\phi\sim 1-\frac{\phi^{2}}{2}$,
we obtain

$d(C_{1},C_{2})\Delta(C_{1},C_{2})\sim\phi^{2}.$

Taking square roots yields the claim. ∎

The following proposition, and its proof, is very similar to Lemma 1.2 of [3].
In that context, the assumption that $\Omega$ is contained in the intersection
of thin annuli is replaced with the assumption that the intersection is
nonempty, and the conclusion gives an estimate of the size of $\tau$ such that
$C_{\delta,v}\cap C_{A\delta,w}$ contains a $\delta,\tau$-rectangle in terms
of $d(v,w)$ and $\Delta(v,w)$.

######  Proposition 9 (Engulfing).

Let $1<A,B<\delta^{-C\epsilon}$ and $\Omega^{(v)}$ be a
$\delta,\tau$-rectangle contained in $C_{\delta,v}\cap C_{A\delta,w}$ for
$v,w\in Q$. Suppose $\overline{\Omega}^{(w)}$ is an $A\delta,B\tau$-rectangle
contained in $C_{A\delta,w}$ which contains $\Omega$. Then there exists a
universal constant $A_{1}>1$ such that $\overline{\Omega}\subset
C_{A_{1}AB^{2}\delta,v}\cap C_{A\delta,w}$.

###### Proof.

Let $d=d(v,w)$, $\Delta=\Delta(v,w)$, and let $\gamma,\overline{\gamma}$
denote the core arcs of $\Omega,\overline{\Omega}$. We make the simplifying
technical assumption that there exists a point
$x\in\gamma\cap\overline{\gamma}\subset\Omega$. To remove this assumption, we
note by replacing $v$ with a concentric circle of a slightly smaller or larger
radius, we can arrange for $\gamma\cap\overline{\gamma}\neq\emptyset$, while
keeping $\Omega^{(v)}\subset\overline{\Omega}^{(w)}$.

By translating, scaling by a factor $\sim 1$, and rotating our coordinate
system if necessary, assume that $w=(0,0,1)$, $v=(a_{1},a_{2},s)=:(a,s)$, and
that $x=1$ is on the positive real axis (see Figure 1). With this choice of
coordinate system, it suffices to show (using complex notation) that for
$|r-1|<A\delta$ and $|\theta|\lesssim B\tau$, we have

$|re^{i\theta}-a|=s+O(AB^{2})\delta,$

since in our chosen coordinate system,
$\overline{\Omega}\subset\\{re^{i\theta}:|r-1|<A\delta,|\theta|\lesssim
B\tau\\}$.

So assume $|r-1|<A\delta$ and $|\theta|\lesssim B\tau$. It follows by the
triangle inequality we can replace $re^{i\theta}$ with $e^{i\theta}$ at the
cost of $A\delta$. Next, because we assume $1\in\gamma\cap\overline{\gamma}$,
we have $s=|1-a|$, so we can substitute $|1-a|$ for $s$, and we are left with
estimating

$||e^{i\theta}-a|-|1-a||.$

Because our circles lie in
$Q=[0,\alpha_{0}]^{2}\times[1-\alpha_{0},1+\alpha_{0}]$, we have the estimate
$|e^{i\theta}-a|+|1-a|\sim 1$. Therefore multiplying by this expression, we
only have to show

$|e^{i\theta}-a|^{2}-|1-a|^{2}=O(AB^{2})\delta.$

The upshot is we can use the trigonometric identity

$\displaystyle|e^{i\theta}-a|^{2}-|1-a|^{2}$
$\displaystyle=2\,\mathrm{Re}(a)(1-\cos\theta)-2\,\mathrm{Im}(a)\sin\theta$
$\displaystyle=O(\mathrm{Re}(a))\theta^{2}+O(\mathrm{Im}(a))|\theta|$
$\displaystyle\lesssim O(\mathrm{Re}(a))B^{2}\tau^{2}+O(\mathrm{Im}(a))B\tau,$

and it suffices to estimate the components $a_{1}=\mathrm{Re}(a)$ and
$a_{2}=\mathrm{Im}(a)$. We can use rectangle-lightplank duality to estimate
both components simultaneously. We note $\Omega\subset C_{A\delta,w}$, so we
have $w\in\mathcal{C}_{A\delta}(\Omega)$, which is contained in an
$O(A)\delta\times O(A)\delta\tau^{-1}\times\delta\tau^{-2}$-lightplank, by a
straightforward variation on Proposition 6 analogous to Proposition 5. By
projecting this lightplank down to the plane $\mathbb{R}^{2}\times\\{0\\}$,
this shows that $|\mathrm{Re}(a)|\leq d\lesssim\delta\tau^{-2}$, and
$|\mathrm{Im}(a)|\lesssim A\delta\tau^{-1}$.

Collecting the estimates we have made so far, we have shown for arbitrary
$|r-1|<A\delta$ and $|\theta|\lesssim B\tau$,

$||re^{i\theta}-a|-s|\lesssim A\delta+\delta\tau^{-2}\cdot
B^{2}\tau^{2}+A\delta\tau^{-1}\cdot B\tau=O(AB^{2})\delta.$

This finishes the proof.

Figure 1. Illustration of the conclusion of Proposition 9.

∎

######  Corollary 4.

Let $\Omega^{(v)}$ be a $\delta,\tau$-rectangle contained in
$\overline{\Omega}^{(w)}$, an $A^{2}\delta,A\tau$-rectangle. Then
$v\in\mathcal{C}_{A_{1}A^{5}\delta}(\overline{\Omega})$.

###### Proof.

By Proposition 9, and since $\overline{\Omega}$ is an
$A^{2}\delta,A\tau$-rectangle, by definition,
$v\in\mathcal{C}_{A_{1}A^{5}\delta}(\overline{\Omega})$. ∎

Now we are ready to give the proof that being comparable is almost a
transitive relation. For the purpose of stating it succinctly, if
$\Omega,\Omega^{\prime}$ are $A$-comparable, we write
$\Omega\asymp_{A}\Omega^{\prime}$.

######  Proposition 10 (Almost-transitivity).

There is an absolute constant $C>1$ such that if
$\Omega_{1}\asymp_{A}\Omega_{2}$ and $\Omega_{2}\asymp_{A}\Omega_{3}$, then
$\Omega_{1}\asymp_{A^{C}}\Omega_{3}$.

###### Proof.

Suppose the core circles of $\Omega_{i}$ are $v_{i}\in Q$, $i=1,2,3$. Consider
the radial projection $\pi\colon\mathbb{R}^{2}\setminus\\{v_{2}^{\prime}\\}\to
C_{v_{2}}$ onto $C_{v_{2}}$. By assumption, $\pi(\Omega_{1}\cup\Omega_{3})$ is
contained in an arc of length $\sim A\tau$ containing the core arc of
$\Omega_{2}$. Therefore, it suffices to show that $\Omega_{1}$ and
$\Omega_{3}$ are contained in $C_{A^{O(1)}\delta,v_{2}}$, as this will imply
that $\Omega_{1}\cup\Omega_{3}$ is contained in an
$A^{O(1)}\delta,O(A)\tau$-rectangle, which is the desired relation.

Let $\overline{\Omega}_{12}\supset\Omega_{1}\cup\Omega_{2}$ and
$\overline{\Omega}_{23}\supset\Omega_{2}\cup\Omega_{3}$ be the rectangles
coming from the assumption $\Omega_{1}\asymp_{A}\Omega_{2}$,
$\Omega_{2}\asymp_{A}\Omega_{3}$. By Proposition 9,
$\overline{\Omega}_{12}\cup\overline{\Omega}_{23}\subset
C_{A^{O(1)}\delta,v_{2}}$, which finishes the proof. ∎

To turn statements about comparable rectangles into statements about nearly
overlapping lightplanks, we need a lemma that relates different lightlike
coordinate systems.

######  Proposition 11.

Let $\mathcal{E}=e_{s},e_{m},e_{l}$ and
$\overline{\mathcal{E}}=\bar{e}_{s},\bar{e}_{m},\bar{e}_{l}$ be two lightlike
bases of $\mathbb{R}^{3}$. If $\theta=\angle e_{m},\bar{e}_{m}$, then the
following relationship holds between $\mathcal{E}$ and
$\overline{\mathcal{E}}$:

$\begin{matrix}\bar{e}_{s}&=&-\frac{\sin\theta}{\sqrt{2}}e_{m}&+&\frac{\cos\theta-1}{2}e_{l}&+&\frac{\cos\theta+1}{2}e_{s},\\\
\bar{e}_{m}&=&\cos\theta\,e_{m}&+&\frac{\sin\theta}{\sqrt{2}}e_{l}&+&\frac{\sin\theta}{\sqrt{2}}e_{s},\\\
\bar{e}_{l}&=&-\frac{\sin\theta}{\sqrt{2}}e_{m}&+&\frac{\cos\theta+1}{2}e_{l}&+&\frac{\cos\theta-1}{2}e_{s}.\end{matrix}$

###### Proof.

By rotating our coordinate system if necessary, we may assume that in our
chosen coordinates, we have

$e_{m}=\begin{pmatrix}0\\\ 1\\\
0\end{pmatrix},e_{l}=\frac{1}{\sqrt{2}}\begin{pmatrix}-1\\\ 0\\\
1\end{pmatrix},e_{s}=\frac{1}{\sqrt{2}}\begin{pmatrix}-1\\\ 0\\\
-1\end{pmatrix}.$

By our assumption that $\angle e_{m},\bar{e}_{m}=\theta$, we can write

$\bar{e}_{m}=\begin{pmatrix}-\sin\theta\\\ \cos\theta\\\
0\end{pmatrix},\bar{e}_{l}=\frac{1}{\sqrt{2}}\begin{pmatrix}-\cos\theta\\\
-\sin\theta\\\
1\end{pmatrix},\bar{e}_{s}=\frac{1}{\sqrt{2}}\begin{pmatrix}-\cos\theta\\\
-\sin\theta\\\ -1\end{pmatrix}.$

As the bases $\mathcal{E}$, $\overline{\mathcal{E}}$ are orthonormal, the
conclusion follows by computing the 9 inner products
$\langle\bar{e}_{s},e_{m}\rangle$, $\langle\bar{e}_{s},e_{l}\rangle$, etc. ∎

In the next proposition, we do not make a serious attempt to optimize the
exponent of $A$, since the only point is to establish a bound of the form
$A^{C}\tau$ for an absolute $C$.

######  Proposition 12.

Suppose $\Omega^{(v)}\subset\overline{\Omega}^{(w)}$, where $\Omega^{(v)}$ is
a $\delta,\tau$-rectangle, and $\overline{\Omega}^{(w)}$ is an
$A^{2}\delta,A\tau$-rectangle. Let $a,\bar{a}$ be the center points of
$\Omega^{(v)}$ and $\overline{\Omega}^{(w)}$, respectively, and let
$e_{m},\bar{e}_{m}$ be the positively oriented tangent vectors to $v,w$ at the
points $a,\bar{a}$, respectively. Then $\angle e_{m},\bar{e}_{m}\lesssim
A^{4}\tau$.

###### Proof.

Let $d=d(v,w)$, $\Delta=\Delta(v,w)$, and let $\gamma,\overline{\gamma}$
denote the core arcs of $\Omega,\overline{\Omega}$. We make the same
simplifying technical assumption as in Proposition 9, that there exists a
point $x\in\gamma\cap\overline{\gamma}$ to make use of Proposition 8. To
remove this assumption, we note by replacing $v$ with a concentric circle of a
slightly smaller or larger radius, we can arrange for
$\gamma\cap\overline{\gamma}\neq\emptyset$, while keeping $\angle
e_{m},\bar{e}_{m}$ and $\Omega^{(v)}\subset\overline{\Omega}^{(w)}$. By
Proposition 8, the angle between $v$ and $w$ at $x$ is $\sim\sqrt{d\Delta}$.

By assumption $\Omega^{(v)}\subset\overline{\Omega}^{(w)}$,

$|C_{A^{2}\delta,v}\cap C_{A^{2}\delta,w}|\geq|\Omega|\sim\delta\tau.$

On the other hand, by Lemma 1,

$|C_{A^{2}\delta,v}\cap
C_{A^{2}\delta,w}|\lesssim\frac{(A^{2}\delta)^{2}}{\sqrt{(d+A^{2}\delta)(\Delta+A^{2}\delta)}}.$

Rearranging the inequality and using our a priori assumption
$\delta^{1/2}\leq\tau$ gives

$\sqrt{d\Delta}\lesssim A^{4}\delta\tau^{-1}\leq A^{4}\tau.$

Finally, because $\mathrm{dist}(a,x)+\mathrm{dist}(x,\bar{a})\lesssim A\tau$,
by comparing angles at $a$ and $\bar{a}$, we conclude $\angle
e_{m},\bar{e}_{m}=O(A^{4})\tau+O(A)\tau=O(A^{4})\tau$. ∎

######  Proposition 13.

Suppose $\Omega^{(v)}\subset\overline{\Omega}^{(w)}$, where $\Omega^{(v)}$ is
a $\delta,\tau$-rectangle, and $\overline{\Omega}^{(w)}$ is an
$A^{2}\delta,A\tau$-rectangle. Let $P=\mathcal{C}_{\delta}(\Omega)$ and
$\overline{P}=\mathcal{C}_{A^{2}\delta}(\overline{\Omega})$. Then for an
absolute constant $C>1$, $P\subset A^{C}\overline{P}$.

###### Proof.

Let $\mathcal{E}=e_{s},e_{m},e_{l}$ and
$\overline{\mathcal{E}}=\bar{e}_{s},\bar{e}_{m},\bar{e}_{l}$ be the lightlike
bases associated with the lightplanks $P$ and $\overline{P}$, respectively.
Let $\theta=\angle e_{m},\bar{e}_{m}$. Because $\Omega\subset
C_{A^{2}\delta,w}$, $w\in\mathcal{C}_{A^{2}\delta}(\Omega)$, which is
contained in an $A^{2}$-dilation of $P$ by Proposition 6.

Hence, by Proposition 11, in $\mathcal{E}$, we have

$\begin{matrix}|\langle v-w,\bar{e}_{s}\rangle|&\leq&|\langle
v-w,e_{s}\rangle|&+&O(\theta)|\langle
v-w,e_{m}\rangle|&+&O(\theta^{2})|\langle v-w,e_{l}\rangle|\\\
&\leq&A^{2}\delta&+&O(\theta)A^{2}\delta\tau^{-1}&+&O(\theta^{2})A^{2}\delta\tau^{-2}.\end{matrix}$

By Proposition 12, $|\theta|\lesssim A^{4}\tau$, so $|\langle
v-w,\bar{e}_{s}\rangle|\lesssim A^{10}\delta$. Analogous considerations using
Proposition 11 and $|\theta|\lesssim A^{4}\tau$ show $|\langle
v-w,\bar{e}_{m}\rangle|\lesssim A^{6}\delta\tau^{-1}$ and $|\langle
v-w,\bar{e}_{l}\rangle|\lesssim A^{2}\delta\tau^{-2}$. Since $\overline{P}$ is
a $A^{2}\delta\times A\delta\tau^{-1}\times\delta\tau^{-2}$-lightplank, we
find $v\in CA^{8}\overline{P}$. Now it suffices to prove that for any $x\in
P=P^{(v)}$, the inequalities

$\displaystyle|\langle x-v,\bar{e}_{s}\rangle|$ $\displaystyle\lesssim
A^{C}\delta$ $\displaystyle|\langle x-v,\bar{e}_{m}\rangle|$
$\displaystyle\lesssim A^{C}\delta\tau^{-1}$ $\displaystyle|\langle
x-v,\bar{e}_{l}\rangle|$ $\displaystyle\lesssim A^{C}\delta\tau^{-2}$

all hold. We provide the details to estimate $|\langle
x-v,\bar{e}_{s}\rangle|$ as the proofs of the remaining inequalities are
entirely analogous. By Proposition 11 again, we have

(4) $|\langle x-v,\bar{e}_{s}\rangle|\lesssim O(1)|\langle
x-v,e_{s}\rangle|+O(\theta)|\langle x-v,e_{m}\rangle|+O(\theta^{2})|\langle
x-v,e_{l}\rangle|.$

Since $x\in P$, in the lightlike coordinate system $\mathcal{E}$, we have

$\displaystyle|\langle x-v,e_{s}\rangle|$ $\displaystyle\lesssim\delta$
$\displaystyle|\langle x-v,e_{m}\rangle|$
$\displaystyle\lesssim\delta\tau^{-1}$ $\displaystyle|\langle
x-v,e_{l}\rangle|$ $\displaystyle\lesssim\delta\tau^{-2}.$

Substituting these estimates into (4) with $|\theta|\lesssim A^{4}\tau$, we
obtain

$|\langle x-v,\bar{e}_{s}\rangle|\lesssim A^{8}\delta.$

Using the remaining two relations from Proposition 11 provides the required
estimates for $|\langle x-v,\bar{e}_{m}\rangle|$ and $|\langle
x-v,\bar{e}_{l}\rangle|$, and this finishes the proof. ∎

In the other direction, assuming
$P\subset\overline{P}\subset\mathbb{R}^{2}\times[1-\alpha_{0},1+\alpha_{0}]$
for two lightplanks, we can say something about the corresponding dual
rectangles.

######  Proposition 14.

Suppose
$P^{(v)}\subset\overline{P}^{(w)}\subset\mathbb{R}^{2}\times[1-\alpha_{0},1+\alpha_{0}]$,
where $P^{(v)}$ is a
$\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$-rectangle centered on $v$,
and $\overline{P}^{(w)}$ is an $A^{2}\delta\times
A\delta\tau^{-1}\times\delta\tau^{-2}$-lightplank centered on $w$. Let
$\Omega^{(v)}=\mathcal{V}_{\delta}(P)$ and
$\overline{\Omega}^{(w)}=\mathcal{V}_{\delta}(\overline{P})$. Then there is an
$A^{6}\delta,A^{2}\tau$-rectangle containing $\Omega\cup\overline{\Omega}$.

###### Proof.

By Corollary 2, $\Omega^{(v)}\subset\overline{\Omega}^{(w)}$, where $\Omega$
is a $\delta,\tau$-rectangle, and $\overline{\Omega}$ is an
$A^{2}\delta,A\tau$-rectangle. By Corollary 4, $\overline{\Omega}\subset
C_{A^{5}\delta,v}$, so it suffices to prove that the angle $\theta:=\angle
e_{m},\bar{e}_{m}$ between the intermediate edges of the lightplanks
$P,\overline{P}$, respectively is at most $O(A^{2})\tau$. If this is done, it
shows that $\Omega\cup\overline{\Omega}$ is contained in an
$A^{5}\delta,A^{2}\tau$-rectangle. Consider the plane $\Pi$ containing the
lower $A\delta\tau^{-1}\times A^{2}\delta$ face of the lightplank
$\overline{P}$ (see Figure 2). Considering the edges of $P,\overline{P}$ in
the plane $\Pi$, we have

$\theta\sim\sin\theta\leq\frac{A^{2}\delta}{\delta\tau^{-1}}=A^{2}\tau,$

and this finishes the proof.

Figure 2. Bottom faces of the lightplanks $P,\overline{P}$ in the plane $\Pi$.

∎

The next proposition combines the last few propositions to characterize
comparability of $\delta,\tau$-rectangles in terms of an analogous statement
concerning their dual lightplanks. In the statement of the proposition, the
absolute constant $C$ can vary within the same line, but the only important
point is that in each instance the constant is absolute.

######  Proposition 15 (Comparability dictionary).

Suppose $\Omega_{1},\Omega_{2}$ are $\delta,\tau$-rectangles in the plane with
corresponding lightplanks
$P_{1},P_{2}\subset\mathbb{R}^{2}\times[1-\alpha_{0},1+\alpha_{0}]$. If
$\Omega_{1},\Omega_{2}$ are $A$-comparable, then there is a $A^{C}\delta\times
A^{C}\delta\tau^{-1}\times A^{C}\delta\tau^{-2}$-lightplank $\overline{P}$
containing $P_{1}\cup P_{2}$. Conversely, if $P_{1}\cup
P_{2}\subset\overline{P}$ for some $A^{C}\delta\times
A^{C}\delta\tau^{-1}\times A^{C}\delta\tau^{-2}$-lightplank $\overline{P}$,
then $\Omega_{1},\Omega_{2}$ are $A^{C}$-comparable.

###### Proof.

Suppose that $\Omega_{1},\Omega_{2}$ are $A$-comparable, and let
$\overline{\Omega}$ be an $A^{2}\delta,A\tau$-rectangle containing
$\Omega_{1}\cup\Omega_{2}$, and
$\overline{P}=\mathcal{C}_{A^{2}\delta}(\overline{\Omega})$ its dual
lightplank. By Proposition 13, $P_{1}\cup P_{2}$ is contained in an
$A^{C}$-dilation of $\overline{P}$.

Conversely, if $P_{1}\cup P_{2}\subset\overline{P}$ for an $A^{C}\delta\times
A^{C}\delta\tau^{-1}\times A^{C}\delta\tau^{-2}$-lightplank $\overline{P}$,
then by Proposition 14, $\Omega_{1}\cup\Omega_{2}$ is contained in a single
$A^{C}\delta,A^{C}\tau$-rectangle, so $\Omega_{1},\Omega_{2}$ are
$A^{C}$-comparable. ∎

###  4.3. Packing rectangles

The next proposition is a minor refinement of Lemma 1.2 in [6]. The refinement
comes in the form of being more explicit about the shape of the constant, and
the only important point is it is at most $(A_{0}A)^{C}$ for an absolute
constant $C>1$ (rather than an intolerable exponential growth, e.g.
$e^{A_{0}A}$). We remark that much of the work we have done in this section
was for the sake of having a concise proof of this proposition.

######  Proposition 16 (Packing).

For any $A_{0}\geq 1$, the number of pairwise $A$-incomparable
$\delta,\tau$-rectangles contained in an
$A_{0}A^{2}\delta,A_{0}A\tau$-rectangle is $\lesssim(A_{0}A)^{C}$.

###### Proof.

Let $\overline{\Omega}$ be an $A_{0}A^{2}\delta,A_{0}A\tau$-rectangle. By
covering $\overline{\Omega}$ with $O(A_{0})$ finitely overlapping
$A_{0}A^{2}\delta,A\tau$-rectangles, it suffices to check that the number of
pairwise $A$-incomparable $\delta,\tau$-rectangles contained in an
$A_{0}A^{2}\delta,A\tau$-rectangle, that we also denote by
$\overline{\Omega}$, is at most $C(A_{0}A)^{C}$.

Let $\\{\Omega^{(v_{i})}\\}_{i=1}^{M}$ be a maximal pairwise $A$-incomparable
collection of $\delta,\tau$-rectangles contained in $\overline{\Omega}^{(o)}$.

Let $\overline{\mathcal{E}}=\bar{e}_{s},\bar{e}_{m},\bar{e}_{l}$ be the
lightlike basis associated to the lightplank $\overline{P}$ with center $o$.
By Proposition 13, for each $i$, $v_{i}\in(A_{0}A)^{C}\overline{P}$, so each
of the following inequalities holds for every $i,j\in\\{1,\dots,M\\}$:

  * •

$|\langle v_{i}-v_{j},\bar{e}_{s}\rangle|\lesssim(A_{0}A)^{C}\delta$

  * •

$|\langle v_{i}-v_{j},\bar{e}_{m}\rangle|\lesssim(A_{0}A)^{C}\delta\tau^{-1}$

  * •

$|\langle v_{i}-v_{j},\bar{e}_{l}\rangle|\lesssim(A_{0}A)^{C}\delta\tau^{-2}$.

As the circles $v_{1},\dots,v_{M}$ contained in $(A_{0}A)^{C}\overline{P}$ are
$A$-incomparable, for each $i\neq j$, at least one of the following
inequalities must hold by Proposition 15:

  * •

$|\langle v_{i}-v_{j},\bar{e}_{s}\rangle|\gg A^{C}\delta$

  * •

$|\langle v_{i}-v_{j},\bar{e}_{m}\rangle|\gg A^{C}\delta\tau^{-1}$

  * •

$|\langle v_{i}-v_{j},\bar{e}_{l}\rangle|\gg A^{C}\delta\tau^{-2}$.

Therefore, $M\lesssim(A_{0}A)^{C}$, and the claim is proved. ∎

##  5\. Application of the maximal function estimate

In this section, we show how to combine the geometric considerations from
Section 4 with the maximal function estimate to count pairs of nearly
internally tangent circles. Throughout, assume $X\subset Q$ is a set of at
most $R=\delta^{-1}$ circles, let $\epsilon>0$ be fixed, and
$\delta<\delta_{0}(\epsilon)$. We define a family of multiplicity functions by

$g_{\lambda\delta}(y)=\sum_{x\in X}C_{\lambda\delta,x}(y),\qquad
y\in\mathbb{R}^{2},\lambda\geq 1.$

Throughout, we work with $\lambda\lessapprox 1$.

######  Proposition 17.

There is an absolute constant $C\geq 1$ such that the following holds. Let
$\mathcal{R}$ be an $A$-incomparable collection of $\delta,\tau$-rectangles
contained in $\bigcup_{x\in X}C_{\delta,x}$. For each $x\in X$, and each
$\lambda\geq A$,

$\sum_{\begin{subarray}{c}\Omega\in\mathcal{R}\\\
x\in\mathcal{C}_{\lambda\delta}(\Omega)\end{subarray}}\Omega(y)\lesssim\lambda^{C}C_{\lambda\delta,x}(y).$

###### Proof.

For each fixed $y\in\mathbb{R}^{2}$, the $\Omega\in\mathcal{R}$ which contain
$y$ and which are contained in $C_{\lambda\delta,x}$ are contained in a
$\lambda\delta,C\tau$-rectangle. As the $\Omega\in\mathcal{R}$ are pairwise
$A$-incomparable, by Proposition 16, the number of such $\Omega$ is at most
$\lambda^{C}$ for some absolute constant $C$. ∎

Given $\lambda\geq 1$, and a $\delta,\tau$-rectangle $\Omega$, we let

$\nu_{\lambda\delta}(\Omega)=|X\cap\mathcal{C}_{\lambda\delta}(\Omega)|.$

For $\lambda\approx 1$, the number $\nu_{\lambda\delta}(\Omega)$ counts the
number of circles in $X$ which contain an arc of length $\approx\tau$ that is
$\approx\delta$-close to the true core arc of $\Omega$. As we saw throughout
Section 4, in addition to $\delta,\tau$-rectangles, we have to consider
slightly larger rectangles. For this reason we have to define and work with
the numbers $\nu_{\lambda\delta}(\Omega)$ for $\lambda\approx 1$.

######  Proposition 18.

If $\mathcal{R}$ is a pairwise $A$-incomparable collection of
$\delta,\tau$-rectangles contained in $\bigcup_{x\in X}C_{\delta,x}$ and
$\lambda\geq A$, then

$\sum_{\Omega\in\mathcal{R}}\nu_{\lambda\delta}(\Omega)\Omega(y)\lesssim\lambda^{C}g_{\lambda\delta}(y).$

###### Proof.

For each $\Omega\in\mathcal{R}$ and each $\lambda\geq 1$,

$\nu_{\lambda\delta}(\Omega)\Omega(y)=\sum_{x\in\mathcal{C}_{\lambda\delta}(\Omega)}\Omega(y)C_{\lambda\delta,x}(y),$

by the definition of $\mathcal{C}_{\lambda\delta}(\Omega)$. Summing over
$\Omega\in\mathcal{R}$ and changing the order of summation,

$\displaystyle\sum_{\Omega\in\mathcal{R}}\nu_{\lambda\delta}(\Omega)\Omega(y)$
$\displaystyle=\sum_{\Omega\in\mathcal{R}}\sum_{x\in\mathcal{C}_{\lambda\delta}(\Omega)}\Omega(y)C_{\lambda\delta,x}(y)$
$\displaystyle=\sum_{x\in
X}C_{\lambda\delta,x}(y)\sum_{\begin{subarray}{c}\Omega\in\mathcal{R}\\\
x\in\mathcal{C}_{\lambda\delta}(\Omega)\end{subarray}}\Omega(y).$

By Proposition 17, for each $\lambda\geq A$, the inner sum is bounded by
$\lambda^{C}C_{\lambda\delta,x}(y)$. Combining this with the definition of
$g_{\lambda\delta}(y)$ finishes the proof. ∎

For $\delta^{1/2}<\tau<\delta^{\epsilon}$, let

$\gamma_{\tau}=\max\\{|X\cap\mathcal{C}_{\delta}(\Omega)|:\Omega\subset\mathbb{R}^{2}\
\text{is a $\delta,\tau$-rectangle}\\},$

or equivalently by Corollary 2,

$\gamma_{\tau}=\max\\{|X\cap P|:P\subset Q\ \text{is a lightplank of
dimensions\ }\delta\times\delta\tau^{-1}\times\delta\tau^{-2}\\}.$

######  Proposition 19.

If $\Omega$ is a $\delta,\tau$-rectangle, then for each $\lambda\approx 1$,

$\nu_{\lambda\delta}(\Omega)\lessapprox\gamma_{\tau}.$

###### Proof.

Notation aside, the proposition says that the number of points in $X$
contained in a given lightplank with dimensions slightly larger than
$\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$, as quantified by
$\lambda$, is not much more than the maximal number of points of $X$ contained
in a $\delta\times\delta\tau^{-1}\times\delta\tau^{-2}$-lightplank. The proof
is a routine covering and pigeonholing argument, so we omit the details. ∎

For a dyadic number $M\in[1,\gamma_{\tau}]$ a number $\lambda\approx 1$, and a
collection $\mathcal{R}$ of $\delta,\tau$-rectangles, let

$\mathcal{R}_{M,\lambda}=\\{\Omega\in\mathcal{R}:\nu_{\lambda\delta}(\Omega)\sim
M\\}.$

The next proposition is the culmination of this subsection which will
ultimately allow us to estimate pairs of nearly internally tangent circles.
Recall the notation
$Q=[0,2\alpha_{0}]^{2}\times[1-\alpha_{0},1+\alpha_{0}]\subset\mathbb{R}^{3}$.

######  Proposition 20.

Suppose that $X\subset Q$ is a set of at most $R$ circles in $Q$ either having
one radius per interval of length $\sim\delta$, or else satisfying the
1-dimensional Frostman condition

$|X\cap B(x_{0},r)|\lesssim_{\epsilon}\delta^{-\epsilon}(r/\delta),\ \qquad
x_{0}\in Q,r\geq\delta.$

If $\mathcal{R}$ is any pairwise $A$-incomparable collection of
$\delta,\tau$-rectangles contained in $\bigcup_{x\in X}C_{\delta,x}$, then for
each $M\in[1,\gamma_{\tau}]$ and $A\leq\lambda\lessapprox 1$,

$M^{3/2}|\mathcal{R}_{M,\lambda}|\lesssim\delta^{-C\epsilon}\tau^{-1}|X|.$

###### Proof.

By Proposition 18, if $\lambda\geq A$, we have

$\lambda^{C}g_{\lambda\delta}(y)\gtrsim\sum_{\Omega\in\mathcal{R}}\nu_{\lambda\delta}(\Omega)\Omega(y).$

We organize the sum on the right-hand side by the dyadic level sets of
$\nu_{\lambda\delta}(\Omega)$, noting that
$\nu_{\lambda\delta}(\Omega)\lessapprox\gamma_{\tau}$:

$\lambda^{C}g_{\lambda\delta}(y)\gtrsim\sum_{\begin{subarray}{c}1<M<\gamma_{\tau}\\\
M\
\text{dyadic}\end{subarray}}M\sum_{\Omega\in\mathcal{R}_{M,\lambda}}\Omega(y).$

By Example 1, or the estimate of Theorem 6, and the embedding
$\ell^{1}\hookrightarrow\ell^{3/2}$,

$\delta^{-C\epsilon}\cdot\delta|X|\gtrsim\lambda^{C}\int
g_{\lambda\delta}^{3/2}\gtrsim M^{3/2}|\mathcal{R}_{M,\lambda}|\cdot|\Omega|.$

Dividing by $|\Omega|\sim\delta\tau$ finishes the proof. ∎

In the next subsection we will specialize the value of $\tau$ for our
application.

###  5.1. Nearly lightlike pairs

Fix a set $X\subset Q$ of circles satisfying the Frostman condition

$|X\cap
B^{3}(x_{0},r)|\lesssim_{\epsilon}\delta^{-\epsilon}(r/\delta)\quad\text{for
all $x_{0}\in Q$, $r\geq\delta$}.$

In particular, $|X|\lessapprox R$, though $|X|$ can be much smaller. For
dyadic numbers $\delta<\Delta\leq D<1$, define a collection

$\mathcal{L}_{D,\Delta}=\\{(x,y)\in X\times X:d(x,y)\sim
D,\Delta(x,y)\sim\Delta\\}.$

We will be interested in the cardinality of the collection
$\mathcal{L}_{D,\Delta}$ when $D>\delta^{1-C\epsilon}$ and
$\Delta<\delta^{1-\epsilon}$ as this is the only range of the parameters where
we require a nontrivial estimate of $|\mathcal{L}_{D,\Delta}|$. We will refer
to a pair $(x,y)\in\mathcal{L}_{D,\Delta}$ as _nearly lightlike_ when
$\Delta<\delta^{1-\epsilon}$. In order to estimate the number of nearly
lightlike pairs, we will ultimately use Proposition 20. Let
$\tau_{D}=\delta^{1/2}D^{-1/2}$.

######  Definition 7.

We say two circles $C_{v},C_{w}$ are $\gtrapprox\delta,\tau$-tangent if there
are $\approx 1$-comparable $\delta,\tau$-rectangles $\Omega^{(v)}\subset
C_{\delta,v}$, $\Omega^{(w)}\subset C_{\delta,w}$.

######  Proposition 21.

If $(v,w)\in\mathcal{L}_{D,\Delta}$ for $D\gg\delta$ and
$\Delta\lessapprox\delta$, then $C_{v},C_{w}$ are
$\gtrapprox\delta,\tau_{D}$-tangent.

###### Proof.

Suppose $D\gg\delta$, $\Delta\lessapprox\delta$, and
$(v,w)\in\mathcal{L}_{D,\Delta}$. We will find a lightplank
$P\subset\mathbb{R}^{2}\times[1-\alpha_{0},1+\alpha_{0}]$ of dimensions
$\approx\delta\times\delta\tau_{D}^{-1}\times\delta\tau_{D}^{-2}$ such that
both $v,w\in P$. By duality, for an appropriate $A\approx 1$,
$\Omega^{(v)}:=\mathcal{V}_{A\delta}(P)\cap C_{\delta,v}$ and
$\Omega^{(w)}:=\mathcal{V}_{A\delta}(P)\cap C_{\delta,w}$ are $\approx
1$-comparable $\delta,\tau_{D}$-rectangles contained in $C_{\delta,v}$ and
$C_{\delta,w}$, respectively, so this will finish the proof.

Let $w_{0}\in\Gamma_{v}$ be the nearest point to $w$ in the lightcone with
vertex $v$. By definition, $v-w_{0}$ is a lightlike vector, and since
$\delta\ll D$, we have $|v-w_{0}|\sim|v-w|\sim D$ and
$|w_{0}-w|=\Delta(v,w)\lessapprox\delta$. Let $e_{m}$ be a unit tangent vector
to the $x_{3}$-slice of $\Gamma_{v}$ containing $w_{0}$; let $e_{l}$ be the
unit vector in the direction $v-w_{0}$, and let $e_{s}$ be such that
$\mathcal{E}=e_{l},e_{m},e_{s}$ is an orthonormal basis. To show that $v,w$
both belong to a common lightplank of the required dimensions, it suffices to
show that

  * (i)

$|\langle v-w,e_{l}\rangle|\lessapprox\delta\tau_{D}^{-2}$,

  * (ii)

$|\langle v-w,e_{m}\rangle|\lessapprox\delta\tau_{D}^{-1}$, and

  * (iii)

$|\langle v-w,e_{s}\rangle|\lessapprox\delta$.

Since $\delta\tau_{D}^{-2}=D\sim|\langle v-w,e_{l}\rangle|$, and
$\Delta(v,w)=|w-w_{0}|\sim|\langle v-w,e_{s}\rangle|\lessapprox\delta$, only
point (ii) needs elaboration. But by elementary geometry considerations, this
is a simple consequence of the assumption $d(v,w)\sim D$ and
$\Delta(v,w)\lessapprox\delta$. ∎

######  Proposition 22.

There is an absolute constant $C>1$ so that the following holds. If
$(v,w)\in\mathcal{L}_{D,\Delta}$ and $\mathcal{R}$ is a maximal pairwise
$A$-incomparable collection of $\delta,\tau_{D}$-rectangles contained in
$\bigcup_{x\in X}C_{\delta,x}$, then there exists $\Omega\in\mathcal{R}$ so
that $v,w\in\mathcal{C}_{A^{C}\delta}(\Omega)$.

###### Proof.

By Proposition 21, there are $\approx 1$-comparable $\delta,\tau_{D}$
rectangles $\Omega^{(v)},\Omega^{(w)}$ in $C_{\delta,v},C_{\delta,w}$
respectively. By maximality of $\mathcal{R}$ with respect to
$A$-incomparability, there is some $\Omega\in\mathcal{R}$ such that
$\Omega^{(v)}$ (say) is $A$-comparable to $\Omega$. This shows that
$v\in\mathcal{C}_{\lambda\delta}(\Omega)$ for some $\lambda=O(A^{C})$.

As $\Omega^{(v)}$ and $\Omega^{(w)}$ are $A$-comparable, by almost-
transitivity (Proposition 10), $\Omega$ and $\Omega^{(w)}$ are
$\lambda^{C}=A^{O(1)}$-comparable. Hence
$w\in\mathcal{C}_{\lambda^{C}\delta}(\Omega)$ for a large enough absolute
constant $C$, and the claim is proved as
$v\in\mathcal{C}_{\lambda\delta}(\Omega)\subset\mathcal{C}_{\lambda^{C}\delta}(\Omega)$.
∎

######  Proposition 23.

If $\Delta\lesssim\delta^{1-\epsilon}$, and $D>\delta^{1-C\epsilon}$ then
$|\mathcal{L}_{D,\Delta}|\lessapprox\gamma_{\tau_{D}}^{1/2}(RD)^{1/2}|X|$.

###### Proof.

Let $A\approx 1$ be a parameter (take $A=\delta^{-\epsilon}$ for
definiteness), and fix an arbitrary maximal pairwise $A$-incomparable
collection $\mathcal{R}$ of $\delta,\tau_{D}$-rectangles contained in
$\bigcup_{x\in X}C_{\delta,x}$.

By Proposition 22, for a given $(v,w)\in\mathcal{L}_{D,\Delta}$, we can find a
rectangle $\Omega\in\mathcal{R}$ such that
$v,w\in\mathcal{C}_{\lambda\delta}(\Omega)$ for some $\lambda=A^{O(1)}$, and
we can write

$\mathcal{L}_{D,\Delta}\subset\bigcup_{\Omega\in\mathcal{R}}\\{(v,w)\in
X\times X:v,w\in\mathcal{C}_{\lambda\delta}(\Omega)\\}.$

By the union bound,

(5)
$|\mathcal{L}_{D,\Delta}|\leq\sum_{\Omega\in\mathcal{R}}|X\cap\mathcal{C}_{\lambda\delta}(\Omega)|^{2}=:\sum_{\Omega\in\mathcal{R}}\nu_{\lambda\delta}(\Omega)^{2}.$

Recall that by Proposition 19,
$\nu_{\lambda\delta}(\Omega)\lessapprox\gamma_{\tau_{D}}$. We organize the
last sum on the right-hand side of (5) by the dyadic value of
$\nu_{\lambda\delta}(\Omega)$, up to $\gamma_{\tau_{D}}$, the same as we did
in Proposition 20. Letting
$\mathcal{R}_{M,\lambda}=\\{\Omega\in\mathcal{R}:\nu_{\lambda\delta}(\Omega)\sim
M\\}$, we estimate (5) by

$\sum_{\begin{subarray}{c}1<M<\gamma_{\tau_{D}}\\\ M\
\text{dyadic}\end{subarray}}M^{2}|\mathcal{R}_{M,\lambda}|.$

By Proposition 20, for each $M$,
$M^{3/2}|\mathcal{R}_{M,\lambda}|\lessapprox\tau_{D}^{-1}|X|=(RD)^{\frac{1}{2}}|X|$.
As $\gamma_{\tau_{D}}\leq\delta^{-1}$ _a priori_ , there are $\approx 1$-many
values of $M$ in the sum, so we have shown
$|\mathcal{L}_{D,\Delta}|\lessapprox\gamma_{\tau_{D}}^{1/2}(RD)^{1/2}|X|$.
This finishes the proof. ∎

##  6\. Proof of Theorem 4 and sharpness

Besides the geometric considerations of Sections 4 and 5, the main ingredient
we need for the proof of Theorem 4 is a stationary phase estimate for the
Fourier transform of $\sigma$, a smooth surface measure on the cone segment.
Recall $\Gamma_{0}=\\{(x^{\prime},x_{3}):||x^{\prime}|-|x_{3}||=0\\}$ is the
lightcone in $\mathbb{R}^{3}$ with vertex 0. We state the version of the
estimate we will use in the proof of Theorem 4 here. The proof of this lemma
is contained in the appendix.

######  Lemma 2.

Let $\sigma$ be a smooth compactly supported surface measure in
$\mathbb{C}one^{2}$. For any $\epsilon>0$, there is a constant $C_{\epsilon}$
so that

$|\widecheck{\sigma}(x)|\leq
C_{\epsilon}\frac{1}{(1+|x|)^{\frac{1}{2}-\epsilon}}\frac{1}{(1+d(x,\Gamma_{0}))^{100\epsilon^{-1}}}.$

Now we are ready to give the proof of Theorem 4, whose statement we recollect
here.

######  Theorem 7.

For each $\epsilon>0$, there is a constant $C_{\epsilon}$ so the following
holds for each $R>1$. Suppose $\nu$ is a measure that agrees with the Lebesgue
measure on a union of lattice unit cubes
$X\subset[0,R]^{2}\times[R,2R]=:B_{R}$ and satisfies the 1-dimensional
Frostman condition

$\nu(B(x_{0},r))\lesssim r,\qquad x_{0}\in\mathbb{R}^{3},r>1.$

Let $\mathbf{P}(\nu)$ be the quantity

$\mathbf{P}(\nu)=\max\\{\nu(P):P\ \text{is a lightplank of dimensions $1\times
R^{1/2}\times R$}\\}.$

Then the estimate

$\int|\widehat{\nu}|^{2}\,d\sigma\leq
C_{\epsilon}R^{\epsilon}\,\mathbf{P}(\nu)^{1/2}\|\nu\|$

holds, where $\|\nu\|:=\nu(B_{R})=|X|$ is the total mass of $\nu$.

###### Proof.

By duality and Fourier transform properties,

$\displaystyle\|\widehat{\nu}\|_{L^{2}(d\sigma)}^{2}=\int\widehat{\nu}\overline{\widehat{\nu}}\,d\sigma=\iint_{B_{R}\times
B_{R}}\widecheck{\sigma}(x-y)\,d\nu(x)d\nu(y).$

We will estimate this integral by dividing the domain of integration into
regions where we have good control on the integrand. For instance,

$\displaystyle\iint\widecheck{\sigma}(x-y)\,d\nu(x)d\nu(y)$
$\displaystyle=\iint_{|x-y|\leq
R^{10\epsilon}}\widecheck{\sigma}(x-y)\,d\nu(x)d\nu(y)+\iint_{|x-y|>R^{10\epsilon}}\widecheck{\sigma}(x-y)\,d\nu(x)d\nu(y)$
$\displaystyle=:I+II.$

Because $|\widecheck{\sigma}|\lesssim 1$ everywhere, we can estimate
$|I|\lesssim R^{10\epsilon}|X|$. To estimate $|II|$, by slight abuse of
notation, let $X=\\{(a_{i},r_{i}):i=1,\dots,|X|\\}$ be the collection of
centers of the cubes in the support of $\nu$. For any $x\in B_{R}$, we have a
corresponding point $\tilde{x}\in B_{1}$ defined by $\tilde{x}=R^{-1}x$. For
each pair $x,y\in X$, we consider the numbers

$d(\tilde{x},\tilde{y})=R^{-1}|x-y|\quad\text{and}\quad\Delta(\tilde{x},\tilde{y})=R^{-1}||x^{\prime}-y^{\prime}|-|x_{3}-y_{3}||.$

These are simply the scaled down values of $d(x,y)$ and $\Delta(x,y)$. We
write things this way to use the results of Section 5 which are phrased at
scales $\leq 1$. For any dyadic number $\delta^{1-10\epsilon}<D<1$, we let

$\displaystyle\mathcal{L}_{D,\lessapprox\delta}$ $\displaystyle=\\{(x,y)\in
X\times X:d(\tilde{x},\tilde{y})\sim
D,\Delta(\tilde{x},\tilde{y})\leq\delta^{1-\epsilon}\\}$
$\displaystyle\mathcal{L}_{D,\gg\delta}$ $\displaystyle=\\{(x,y)\in X\times
X:d(\tilde{x},\tilde{y})\sim
D,\Delta(\tilde{x},\tilde{y})>\delta^{1-\epsilon}\\}.$

We let $\mathcal{L}_{D,\lessapprox\delta}^{1},\mathcal{L}_{D,\gg\delta}^{1}$
denote the $1$-neighborhoods of
$\mathcal{L}_{D,\lessapprox\delta},\mathcal{L}_{D,\gg\delta}$ in
$\mathbb{R}^{3}\times\mathbb{R}^{3}$, respectively. We organize the integral
$II$ by writing

$\displaystyle II$
$\displaystyle\leq\sum_{\delta^{1-10\epsilon}<D<1}\iint_{\mathcal{L}_{D,\lessapprox\delta}^{1}}\widecheck{\sigma}(x-y)\,d\nu(x)d\nu(y)$
$\displaystyle\qquad+\sum_{\delta^{1-10\epsilon}<D<1}\iint_{\mathcal{L}_{D,\gg\delta}^{1}}\widecheck{\sigma}(x-y)\,d\nu(x)d\nu(y).$

We claim that the second sum in this decomposition of $II$ is $O(R^{-95})$, so
it is negligible. Postponing the proof of this for a moment, we only have to
show that the first sum is bounded by the quantity in the statement of Theorem
4.

For $D>\delta^{1-10\epsilon},\Delta\leq\delta^{1-\epsilon}$, and
$(x,y)\in\mathcal{L}_{D,\Delta}^{1}$ we use the Fourier transform estimate of
Lemma 2,

$|\widecheck{\sigma}(x-y)|\lessapprox\frac{1}{(RD)^{1/2}},$

together with the estimate of Proposition 23 for $|\mathcal{L}_{D,\Delta}|$,
$\Delta\lessapprox\delta$:

$|\mathcal{L}_{D,\lessapprox\delta}|\lessapprox\mathbf{P}(\nu)^{1/2}(RD)^{1/2}|X|.$

Putting these estimates together gives

$\displaystyle\sum_{\delta^{1-10\epsilon}<D<1}\iint_{\mathcal{L}_{D,\lessapprox\delta}^{1}}|\widecheck{\sigma}(x-y)|\,d\nu(x)d\nu(y)$
$\displaystyle\lesssim\sum_{\delta^{1-10\epsilon}<D<1}|\mathcal{L}_{D,\lessapprox\delta}|\cdot\frac{1}{(RD)^{1/2}}$
$\displaystyle\lessapprox\mathbf{P}(\nu)^{1/2}|X|\sum_{\delta^{1-10\epsilon}<D<1}1\approx\mathbf{P}(\nu)^{1/2}|X|.$

Now we estimate the contribution from the second sum in the decomposition of
$II$. We write the contribution as

$\displaystyle\sum_{\delta^{1-10\epsilon}<D<1}\sum_{\delta^{1-\epsilon}<\Delta<D}\iint_{\begin{subarray}{c}d(\tilde{x},\tilde{y})\sim
D\\\
\Delta(\tilde{x},\tilde{y})\sim\Delta\end{subarray}}|\widecheck{\sigma}(x-y)|\,d\nu(x)d\nu(y).$

By the estimate of Lemma 2, for $(x,y)\in\mathcal{L}_{D,\Delta}^{1}$ with
$\Delta>\delta^{1-\epsilon}$, we have

$|\widecheck{\sigma}(x-y)|\lesssim_{\epsilon}\frac{1}{(RD)^{1/2}}\cdot\frac{1}{(R\Delta)^{100\epsilon^{-1}}}\leq\frac{1}{R^{100}}.$

Since $|\mathcal{L}_{D,\Delta}|\leq R^{2}$ _a priori_ (for any $D,\Delta$),
and since there are a logarithmic number of summands, we have a total
contribution of no more than (say) $C_{\epsilon}R^{-100+2+\epsilon}$. This
finishes the estimate of $|II|$, and the proof. ∎

Lastly, we describe examples that establish the sharpness of Theorem 4.

######  Proposition 24.

For each $R>1$, and each $\gamma\in[1,R]$, there is a measure $\nu$ with
$\mathbf{P}(\nu)\sim\gamma$, that agrees with the Lebesgue measure on a union
$X$ of lattice unit cubes in $B_{R}$ satisfying the 1-dimensional Frostman
condition

$|X\cap B(x_{0},r)|\lesssim r,\qquad x_{0}\in\mathbb{R}^{3},r>1,$

such that

$\int|\widehat{\nu}|^{2}\,d\sigma\gtrapprox\gamma^{1/2}\|\nu\|.$

###### Proof.

By Corollary 5, and the results in the appendix, given $R>1$, and
$\gamma\in[1,R]$, to illustrate the sharpness of the theorem, it suffices to
produce a measure $\nu$ of the required form satisfying the Frostman condition
of exponent 1, and an $f\in L^{2}(d\sigma)$ such that

$\int|Ef|^{2}\,d\nu\gtrsim\gamma^{1/2}\|f\|_{L^{2}(d\sigma)}^{2}.$

  1. (1)

Let $f_{0}(\xi)=1_{\theta}(\xi)$, where
$\theta=[1,2]\times[0,\gamma^{-1/2}]\subset\\{1<|\xi|<2\\}$, be the Knapp
example of the given dimensions. Let $f$ be an appropriate modulation of
$f_{0}$ so that $|Ef|\gtrsim|\theta|1_{P}=\gamma^{-1/2}1_{P}$, where $P$ is a
lightplank in $B_{R}$ of dimensions $1\times\gamma^{1/2}\times\gamma$.

Let $X$ be any $1\times 1\times\gamma$ tube contained in the lightplank $P$,
and let the measure $\nu$ agree with the Lebesgue measure on $X$. By
construction, $\nu$ obeys the Frostman condition and is of the desired form,
with $\mathbf{P}(\nu)\sim\gamma$. Then,
$\|f\|_{L^{2}(d\sigma)}^{2}=|\theta|=\gamma^{-1/2}$, and

$\int|Ef|^{2}\,d\nu\gtrsim\gamma^{-1}\nu(P)\sim
1\sim\gamma^{1/2}\|f\|_{L^{2}}^{2},$

as desired.

  2. (2)

As a small variation on the last example, we can also normalize $\|\nu\|=R$,
with $\mathbf{P}(\nu)\sim\gamma$. Let $f$, and $P$ be the same as in the first
example, and let $X_{P}$ be any $1\times 1\times\gamma$ tube contained in the
lightplank $P$, and $X_{V}$ be a $1\times 1\times(R-\gamma)$-tube whose long
direction is parallel to $e_{3}=(0,0,1)$. Let $X=X_{P}\cup X_{V}$, and let the
measure $\nu$ agree with the Lebesgue measure on $X$. By construction,
$\|\nu\|=R$, $\nu$ obeys the Frostman condition of exponent $1$, and
$\mathbf{P}(\nu)\sim\gamma$.

By the same computation of the first example,
$\int|Ef|^{2}\,d\nu\gtrsim\gamma^{1/2}\|f\|_{L^{2}(d\sigma)}^{2}$.

∎

##  7\. Discussion and related questions

Theorem 1 is a sharp Mizohata–Takeuchi type estimate for the cone segment in
$\mathbb{R}^{3}$ for the 1-dimensional measures $\nu$. However, it would be
interesting to go beyond Theorem 1 and prove even more refined estimates which
capture the wave patterns within lightplanks of dimensions $1\times
R^{1/2}\times R$. As the proof of Theorem 4 shows, we do not take advantage of
potential cancellations of $\widecheck{\sigma}(x-y)$ within lightplanks.

We describe three related further problems below.

Given a measure $\nu$ that agrees with the Lebesgue measure on a union $X$ of
lattice unit cubes in $B_{R}$, let $U(\nu)$ be the smallest constant such that

$\int_{B_{R}}|Ef|^{2}\,d\nu\leq U(\nu)\|f\|_{L^{2}(d\sigma)}^{2}$

holds for all $f$. By Theorem 1, $U(\nu)\lessapprox\mathbf{P}(\nu)^{1/2}$
holds for the 1-dimensional measures.

  1. (i)

Give examples of 1-dimensional measures $\nu$ as in Theorem 4, such that
$\int|\widehat{\nu}|^{2}\,d\sigma$ is much smaller than
$\mathbf{P}(\nu)^{1/2}\|\nu\|$. Equivalently, describe a 1-dimensional measure
$\nu$ such that $U(\nu)\ll\mathbf{P}(\nu)^{1/2}$.

  2. (ii)

Assume Conjecture 1 for the parabola is true; let $E$ be the Fourier extension
for $\mathbb{C}one^{2}$. Recognizing that within an angular strip of
dimensions $1\times.01$, the cone segment is nearly a parallel stack of
parabolas in the lightlike basis associated with the strip, can we prove a
further refined estimate for $\int|Ef|^{2}\,d\nu$ along the lines of

(6) $\int_{B_{R}}|Ef|^{2}\,d\nu\leq
C_{\epsilon}R^{\epsilon}\max_{T}\nu(T)^{1/2}\|f\|_{L^{2}(d\sigma)}^{2},$

where the maximum is taken over $1\times 1\times R$-tubes $T$ pointing in
lightlike directions? As a small step in this direction, we note that any
$1\times R^{1/2}\times R$ lightplank $P$ may be covered by $R^{1/2}$-many
$1\times 1\times R$ lightlike tubes contained in $P$, so by Theorem 1 and the
pigeonhole principle, if $\nu$ is a 1-dimensional measure,

$\int_{B_{R}}|Ef|^{2}\,d\nu\lessapprox
R^{1/4}\max_{T}\nu(T)^{1/2}\|f\|_{L^{2}(d\sigma)}^{2},$

which is (6) for the 1-dimensional measures with an $R^{1/4}$-loss.

  3. (iii)

The estimate of Theorem 4 applies to the $1$-dimensional family of measures
$\nu$ because of the available maximal function estimates. What can we say
about measures satisfying a Frostman condition of exponent $s\neq 1$? It seems
natural to conjecture that bounds of the shape

$\int|\widehat{\nu}|^{2}\,d\sigma\lessapprox
R^{a(s)}\mathbf{P}(\nu)^{b(s)}\|\nu\|^{c(s)}$

for some $a,b,c$ continue to hold for other values of $s$.

## Appendix A

###  A.1. Duality arguments and the proof of Theorem 1

In this section we prove a general theorem relating $L^{1}(\nu)$ and
$L^{2}(\nu)$ estimates of $Ef$ that will be one of the last elements in the
proof of Theorem 1. The theorem here is essentially contained in the proof of
Lemma C.1 in Appendix C of [9].

######  Theorem 8 (Barceló–Bennett–Carbery–Rogers, [9]).

Suppose $(\Gamma,d\sigma)$ is a compact submanifold of $\mathbb{R}^{d}$ with a
smooth surface measure $\sigma$, and let

$Ef(x)=\widecheck{f\sigma}(x).$

For a measure $\nu$, and a family of $\mathfrak{P}$ of measurable sets, let

$\gamma(\nu)=\gamma_{\mathfrak{P}}(\nu)=\sup\\{\nu(P):P\in\mathfrak{P}\\}.$

Then for each $\epsilon>0$, the following are equivalent (possibly with
different implied absolute constants):

  * (L1)

For all $f$, and all measures $\nu$ supported in $B_{R}$,

$\|Ef\|_{L^{1}(\nu)}\leq
C_{\epsilon}R^{\epsilon}\gamma(\nu)^{1/4}\|\nu\|^{1/2}\|f\|_{L^{2}(d\sigma)},$

  * (L2)

For all $f$, and all measures $\nu$ supported in $B_{R}$,

$\|Ef\|_{L^{2}(\nu)}\leq
C_{\epsilon}R^{\epsilon}\gamma(\nu)^{1/4}\|f\|_{L^{2}(d\sigma)}.$

###### Proof.

For all $f$, Hölder’s inequality immediately gives
$\|Ef\|_{L^{1}(\nu)}\leq\|Ef\|_{L^{2}(\nu)}\|\nu\|^{1/2}$, so if (L2) holds,
so does (L1).

Conversely, suppose (L1) holds, and let $\nu$ be a measure supported in
$B_{R}$. For a measurable set $U$, let $d\mu=1_{U}\,d\nu$. Then by (L1)
applied to the measure $\mu$,

(7) $\int_{U}|Ef|\,d\nu=\int|Ef|\,d\mu\leq
C_{\epsilon}R^{\epsilon}\gamma(\mu)^{1/4}\|\mu\|^{1/2}\|f\|_{L^{2}(d\sigma)}.$

Note that by the definition of $\mu$ and $\gamma$,

$\displaystyle\gamma(\mu)$ $\displaystyle=\sup\\{\nu(P\cap
U):P\in\mathfrak{P}\\}\leq\min(\gamma(\nu),\nu(U)).$

For each $\lambda>0$, plug $U=\\{|Ef|>\lambda\\}$ into (7), together with the
upper bound $\gamma(\mu)\leq\gamma(\nu)$ to find

(8) $\lambda\,\nu(\\{|Ef|>\lambda\\})^{1/2}\leq
C_{\epsilon}R^{\epsilon}\gamma(\nu)^{1/4}\|f\|_{L^{2}(d\sigma)}.$

By dyadic pigeonholing, we can produce a particular $\lambda>0$ such that

$(\int|Ef|^{2}\,d\nu)^{1/2}\approx\lambda\,\nu(\\{|Ef|>\lambda\\})^{1/2}.$

Together with the estimate (8), this proves (L2) holds. ∎

The next proposition is essentially Proposition 15.11 from Mattila’s book
[10]. It shows how estimates of $\|\widehat{\nu}\|_{L^{2}(d\sigma)}^{2}$ are
essentially equivalent to $L^{1}(\nu)$ estimates for $Ef$.

######  Proposition 25.

Suppose $c_{0}(\nu)$ is a monotone quantity in the sense that if $\mu\ll\nu$
($\mu$ is absolutely continuous with respect to $\nu$) are positive measures
and $0\leq\frac{d\mu}{d\nu}\leq 1$, then $c_{0}(\mu)\leq c_{0}(\nu)$. If
$\|\widehat{\nu}\|_{L^{2}(d\sigma)}^{2}\lesssim c_{0}(\nu)$ holds for all
positive measures $\nu$, then for all measures $\nu$, and all $f\in
L^{2}(d\sigma)$,

$\int|Ef|\,d\nu\lesssim c_{0}(\nu)^{1/2}\|f\|_{L^{2}(d\sigma)}$

also holds. Conversely, if $\int|Ef|\,d\nu\lesssim c_{0}(\nu)^{1/2}$ for all
$f\in L^{2}(d\sigma)$, then $\|\widehat{\nu}\|_{L^{2}(d\sigma)}^{2}\lesssim
c_{0}(\nu)$.

###### Proof.

Suppose $\|\widehat{\nu}\|_{L^{2}(d\sigma)}^{2}\lesssim c_{0}(\nu)$ holds for
all positive measures $\nu$, and let $\|f\|_{L^{2}(d\sigma)}=1$. By
definition,

$\int|Ef|\,d\nu=\sup_{\|h\|_{L^{\infty}(d\nu)}=1}\int Ef\,h\,d\nu.$

Writing $h=h_{1}-h_{2}+i(h_{3}-h_{4})$ as a linear combination of 4 positive
functions in the canonical way, for each $j=1,\dots,4$, we have

$\displaystyle|\int Ef\,h_{j}\,d\nu|$ $\displaystyle=|(Ef,h_{j}\nu)|$
$\displaystyle\leq\sup_{\|f\|_{L^{2}(d\sigma)}=1}|(Ef,h_{j}\nu)|$
$\displaystyle=\sup_{\|f\|_{L^{2}(d\sigma)}=1}|(f,\widehat{h_{j}\nu})|$
$\displaystyle=\|\widehat{h_{j}\nu}\|_{L^{2}(d\sigma)},$

where $(f,g)$ denotes the distributional pairing of $f$ and $g$. Since
$h_{j}\nu$ is a positive measure, we have
$\|\widehat{h_{j}\nu}\|_{L^{2}(d\sigma)}\lesssim c_{0}(h_{j}\nu)^{1/2}\leq
c_{0}(\nu)^{1/2}$ by assumption, and monotonicity of $c_{0}$, respectively.
Therefore,

$\int|Ef|\,d\nu\lesssim c_{0}(\nu)^{1/2},$

as desired. The converse follows immediately by a similar duality argument. ∎

######  Corollary 5.

Theorem 1 holds.

###### Proof.

By Theorem 4, we have

$\|\widehat{\nu}\|_{L^{2}(d\sigma)}^{2}\lessapprox\mathbf{P}(\nu)^{1/2}\|\nu\|.$

Since $c_{0}(\nu):=\mathbf{P}(\nu)^{1/2}\|\nu\|$ is monotone in the sense of
Proposition 25, we have the $L^{1}(\nu)$ estimate

$\int|Ef|\,d\nu\lessapprox\mathbf{P}(\nu)^{1/4}\|\nu\|^{1/2}\|f\|_{L^{2}(d\sigma)}.$

By Theorem 8, we therefore also have the $L^{2}(\nu)$ estimate

$(\int|Ef|^{2}\,d\nu)^{1/2}\lessapprox\mathbf{P}(\nu)^{1/4}\|f\|_{L^{2}(d\sigma)},$

which was to be shown. ∎

###  A.2. Proof of Fourier transform estimate $|\widecheck{\sigma}(x)|$

In this subsection we recollect and prove the Fourier transform estimate of
Proposition 1 that was the second key to the proof of Theorem 4.

######  Proposition 26.

Let $\sigma$ be a smooth compactly supported surface measure in
$\mathbb{C}one^{2}$. For any $\epsilon>0$ and any $N>1$, there is a constant
$C(\epsilon,N)$ so that

$|\widecheck{\sigma}(x)|\leq
C(\epsilon,N)\frac{1}{(1+|x|)^{\frac{1}{2}-\epsilon}}\frac{1}{(1+d(x,\Gamma_{0}))^{N}}$

holds for all $x\in\mathbb{R}^{3}$.

###### Proof.

We will prove this by combining two estimates for $|\widecheck{\sigma}(x)|$:

  * (i)

$|\widecheck{\sigma}(x)|\lesssim(1+|x|)^{-\frac{1}{2}+\epsilon}$

  * (ii)

For every $N$, $|\widecheck{\sigma}(x)|\lesssim_{N}(1+d(x,\Gamma_{0}))^{-N}$.

The conclusion follows by taking an appropriate geometric average of these two
estimates. We may assume that $|x|\geq C$ for an appropriately large constant
since $|\widecheck{\sigma}(x)|\lesssim 1$ for $|x|\lesssim 1$.

We will start with (i). Suppose $|x|\sim r\gg 1$; our aim is to show
$|\widecheck{\sigma}(x)|\lesssim r^{-\frac{1}{2}+\epsilon}$. We divide
$\mathbb{C}one^{2}$ into $\sim r^{\frac{1}{2}-\epsilon}$-many strips $\theta$
of angular width $r^{-\frac{1}{2}+\epsilon}$ and let $\\{\eta_{\theta}\\}$ be
a smooth partition of unity subordinate to $\\{\theta\\}$. Then with
$\sigma_{\theta}=\sigma\eta_{\theta}$,

$\widecheck{\sigma}(x)=\sum_{\theta}\widecheck{\sigma}_{\theta}(x).$

For each $\theta$, we let $\theta^{*}$ be the lightplank containing the origin
of dimensions $1\times r^{\frac{1}{2}-\epsilon}\times r^{1-2\epsilon}$ dual to
the $r^{-1+2\epsilon}$-neighborhood of $\theta$. By the Schwartz decay of
$\widecheck{\sigma}_{\theta}(x)$, we have

$|\widecheck{\sigma}_{\theta}(x)|\lesssim_{N}|\theta|\sum_{j=0}^{\infty}2^{-jN}1_{2^{j}\theta^{*}}(x).$

Since we assume $|x|\sim r\gg r^{1-2\epsilon}$, and the directions of
$\theta^{*}$ are $r^{-\frac{1}{2}+\epsilon}$-separated, $x$ lies in at most
$\lessapprox 1$ of the $\theta^{*}$. Therefore,

$|\widecheck{\sigma}_{\theta}(x)|\lessapprox|\theta|\sim
r^{-\frac{1}{2}+\epsilon}.$

Now we prove (ii), but instead of using wave packets, we give a proof based on
stationary phase considerations. Let $x=(x^{\prime},x_{3})$ with $|x|\gg 1$.
Suppose that $x$ is spacelike and lies in the upper half-space, so
$|x^{\prime}|>x_{3}>0$. The case of $|x^{\prime}|<x_{3}$ is similar. For an
appropriate smooth and compactly supported function $a(\xi)$ in
$\\{1<|\xi|<2\\}$, we can write

$\widecheck{\sigma}(x)=\int_{1<|\xi|<2}a(\xi)e^{i(x^{\prime}\cdot\xi+x_{3}|\xi|)}\,d\xi=:Ea(x).$

Here $E$ is the extension operator for the cone.

Let $w$ be the nearest point on the cone $\Gamma_{0}$ to $x$. By elementary
geometry, $x-w$ is orthogonal to the lightcone at $w$, and from this, we can
compute the coordinates of $w$ in terms of $x$:

$w=w(x)=\left(\frac{|x^{\prime}|+x_{3}}{2}\frac{x^{\prime}}{|x^{\prime}|},\frac{|x^{\prime}|+x_{3}}{2}\right).$

Note from this formula for $w$ that

$d(x,\Gamma_{0})=|x-w|\leq|x^{\prime}|-x_{3}\lesssim|x-w|=d(x,\Gamma_{0}),$

so $d(x,\Gamma_{0})\sim||x^{\prime}|-x_{3}|$. Write

$\displaystyle Ea(x)=Ea(w+(x-w))$ $\displaystyle=\int
a(\xi)e^{i(w^{\prime}\cdot\xi+w_{3}|\xi|)}e^{i[(x^{\prime}-w^{\prime})\cdot\xi+(x_{3}-w_{3})|\xi|]}\,d\xi$
$\displaystyle=\int
a(\xi)e^{i(w^{\prime}\cdot\xi+w_{3}|\xi|)}e^{i\frac{|x^{\prime}|-x_{3}}{2}(\frac{x^{\prime}}{|x^{\prime}|}\cdot\xi-|\xi|)}$
$\displaystyle=\int a(\xi)e^{i\phi_{1}(\xi)}e^{i\lambda\phi_{2}(\xi)}\,d\xi,$

where

$\displaystyle\phi_{1}(\xi)$ $\displaystyle=w^{\prime}\cdot\xi+w_{3}|\xi|$
$\displaystyle\phi_{2}(\xi)$
$\displaystyle=\frac{x^{\prime}}{|x^{\prime}|}\cdot\xi-|\xi|$
$\displaystyle\lambda=\lambda(x)$
$\displaystyle=\frac{|x^{\prime}|-x_{3}}{2}.$

Let $\Sigma_{1}=\\{1<|\xi|<2:\nabla\phi_{1}(\xi)=0\\}$, and similarly denote
$\Sigma_{2}$ as the set of critical points of $\phi_{2}$. Since $w$ is the
nearest point to $x$ lying in $\Gamma_{0}$, the critical points of
$\phi_{1}(\xi)$ in $\\{1<|\xi|<2\\}$ are precisely the line segment

$\Sigma_{1}=\\{1<|\xi|<2:\frac{\xi}{|\xi|}=-\frac{x^{\prime}}{|x^{\prime}|}\\}.$

Likewise, $\Sigma_{2}$ is the line segment

$\Sigma_{2}=\\{1<|\xi|<2:\frac{\xi}{|\xi|}=\frac{x^{\prime}}{|x^{\prime}|}\\}.$

Consider the open sets

$U_{1}=\\{1<|\xi|<2:\angle(\xi,-x^{\prime})>0.1\\},\qquad
U_{2}=\\{1<|\xi|<2:\angle(\xi,-x^{\prime})<0.2\\},$

and a smooth partition of unity $\eta_{1},\eta_{2}$ subordinate to
$U_{1},U_{2}$. Then $Ea=E(a\eta_{1})+E(a\eta_{2})$. Since the phase
$x^{\prime}\cdot\xi+x_{3}|\xi|$ has no critical points in $U_{1}$,
$|E(a\eta_{1})(x)|\lesssim_{N}(1+|x|)^{-N}$ via integration by parts. So we
only have to show that $|E(a\eta_{2})(x)|\lesssim_{N}d(x,\Gamma_{0})^{-N}$.

Since we only work with $a\eta_{2}$ from now on, to reduce clutter, we let $a$
denote $a\eta_{2}$, so $\mathrm{dist}(\operatorname{supp}a,\Sigma_{2})\gtrsim
1$. Lastly, we note that the phase $\phi_{2}$ satisfies

$\|\phi_{2}\|_{C^{N}(1<|\xi|<2)}\lesssim_{N}1.$

Consider the following vector field and its transpose

$L=\frac{1}{i\lambda}\mathbf{v}\cdot\nabla,\quad
L^{t}f=-\frac{1}{i\lambda}\nabla\cdot(f\mathbf{v})$

where $\mathbf{v}=\nabla\phi_{2}/|\nabla\phi_{2}|$. By definition,
$Le^{i\lambda\phi_{2}}=e^{i\lambda\phi_{2}}$, and consequently integrating by
parts one time,

$\displaystyle Ea(x)$ $\displaystyle=\int
L^{t}(ae^{i\phi_{1}})e^{i\lambda\phi_{2}}$
$\displaystyle=-\frac{1}{i\lambda}\int\nabla\cdot(ae^{i\phi_{1}}\mathbf{v})e^{i\lambda\phi_{2}}.$

Using the vector calculus identity

$\nabla\cdot(fg\mathbf{v})=f\nabla g\cdot\mathbf{v}+g\nabla
f\cdot\mathbf{v}+fg\nabla\cdot\mathbf{v},$

we get

$\displaystyle Ea(x)=-\frac{1}{i\lambda}\bigg{(}\int
a(ie^{i\phi_{1}}\nabla\phi_{1}\cdot\mathbf{v})e^{i\lambda\phi_{2}}+\int
e^{i\phi_{1}}\underbrace{(\nabla
a\cdot\nabla\mathbf{v}+a\nabla\cdot\mathbf{v})}_{\equiv\
a^{\prime}}e^{i\lambda\phi_{2}}\bigg{)}.$

Note that

$\nabla\phi_{1}=w^{\prime}+w_{3}\frac{\xi}{|\xi|}=(\frac{|x^{\prime}|+x_{3}}{2})(\frac{x^{\prime}}{|x^{\prime}|}+\frac{\xi}{|\xi|})$

and

$\mathbf{v}=\frac{1}{|\nabla\phi_{2}|}(\frac{x^{\prime}}{|x^{\prime}|}-\frac{\xi}{|\xi|}).$

Therefore, $\nabla\phi_{1}\cdot\mathbf{v}=0$, so $Ea(x)$ simplifies to

$Ea(x)=-\frac{1}{i\lambda}\int a^{\prime}e^{i\phi_{1}}e^{i\lambda\phi_{2}}.$

Since $a^{\prime}$ is a smooth phase with all the same essential properties as
those of $a(=a\eta_{2})$, we are ready to run the same integration by parts
argument $N$ times to get

$|Ea(x)|\lesssim_{N}\frac{1}{\lambda^{N}}=\frac{1}{(|x^{\prime}|-x_{3})^{N}}.$

Since $||x^{\prime}|-x_{3}|\sim d(x,\Gamma_{0})$, we have proved

$|Ea(x)|\lesssim_{N}\frac{1}{d(x,\Gamma_{0})^{N}}.$

Together with the proof of (i), this finishes the proof. ∎

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