# A Furstenberg-type problem for circles, and a Kaufman-type restricted
projection theorem in $\mathbb{R}^{3}$

Malabika Pramanik, Tongou Yang and Joshua Zahl  Department of Mathematics, The
University of British Columbia, Vancouver, B.C. Canada V6T 1Z2
<EMAIL_ADDRESS>Department of Mathematics, The University of Wisconsin-
Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI United States
53706-1325<EMAIL_ADDRESS>Department of Mathematics, The University of
British Columbia, Vancouver, B.C. Canada V6T 1Z2<EMAIL_ADDRESS>
###### Abstract.

We resolve a conjecture of Fässler and Orponen on the dimension of exceptional
projections to one-dimensional subspaces indexed by a space curve in
$\mathbb{R}^{3}$. We do this by obtaining sharp $L^{p}$ bounds for a variant
of the Wolff circular maximal function over fractal sets for a class of
$C^{2}$ curves related to Sogge’s cinematic curvature condition. A key new
tool is the use of lens cutting techniques from discrete geometry.

##  1\. Introduction

In 1954, Marstrand [18] proved that if $Z\subset\mathbb{R}^{2}$ is Borel, then
$\mathrm{dim}\,(Z\cdot v)=\min(\mathrm{dim}\,Z,1)$ for a.e. $v\in S^{1}$. Here
and throughout, “$\mathrm{dim}\,$” will mean Hausdorff dimension. This was
sharpened by Kaufman [14] in 1968, who bounded the size of the set of
exceptional directions for which the above inequality fails.

######  Theorem 1.1 (Kaufman).

Let $Z\subset\mathbb{R}^{2}$ be Borel, and let $0\leq
s<\min\\{\mathrm{dim}\,Z,1\\}$. Then

$\mathrm{dim}\,\\{\theta\in S^{1}\colon\mathrm{dim}\,(Z\cdot\theta)<s\\}\leq
s.$ (1.1)

Marstrand and Kaufman’s results have been generalized to higher dimensions and
other settings, and they are the foundation of projection theory, which is an
active area in geometric measure theory. See [20] for an introduction to the
topic, or [5] for a recent survey of the area.

In this paper, we will be interested in analogues of the above results for
Borel sets $Z\subset\mathbb{R}^{3}$. As a starting point, one analogue of
Marstrand’s theorem says that $\mathrm{dim}\,(Z\cdot
v)=\min(\mathrm{dim}\,Z,1)$ for a.e. $v\in S^{2}$. There are also analogues of
Kaufman’s theorem; for example, if $0\leq s<\min\\{\mathrm{dim}\,Z,1\\}$ then

$\mathrm{dim}\,\\{v\in S^{2}\colon\mathrm{dim}\,(Z\cdot v)<s\\}\leq 1+s.$
(1.2)

See [19] for details. In particular, (1.2) says that the set of $v\in S^{2}$
for which $\mathrm{dim}\,(Z\cdot v)=0$ has dimension at most 1, and this can
be sharp—for example, if $Z$ is a subset of the $z$-axis then $Z\cdot
v=\\{0\\}$ for each $v$ in the great circle $S^{2}\cap\\{z=0\\}$. It is
possible that stronger bounds hold if we restrict the set of projection
directions to a subset of $S^{2}$ that does not concentrate near great
circles. In this direction, Fässler and Orponen [6] conjectured the following
restricted Marstrand-type estimate.

######  Conjecture 1.2.

Let $I\subset\mathbb{R}$ be a compact interval and let $\gamma:I\to S^{2}$ be
a $C^{2}$ curve that satisfies the “escaping great circle” condition

$\operatorname{span}\big{\\{}\gamma(\theta),\dot{\gamma}(\theta),\ddot{\gamma}(\theta)\big{\\}}=\mathbb{R}^{3}\quad\textrm{for
all}\ \theta\in I.$ (1.3)

Let $Z\subset\mathbb{R}^{3}$ be analytic. Then for almost every $\theta\in I$,
$\mathrm{dim}\,(\gamma(\theta)\cdot Z)=\min(\mathrm{dim}\,Z,1)$.

In [6], Fässler and Orponen proved Conjecture 1.2 in the special case
$\mathrm{dim}\,(Z)\leq 1/2$. In [10], He considered the related problem of
estimating the size of the set where $\mathrm{dim}\,(\gamma(\theta)\cdot Z)$
is close to $(\mathrm{dim}\,Z)/3$. In [11], Käenmäki, Orponen, and Venieri
used ideas from incidence geometry, and specifically a circle tangency bound
due to Wolff [27] to prove Conjecture 1.2 (for all $Z$) in the special case
$\gamma(\theta)=\frac{1}{\sqrt{2}}(\cos\theta,\sin\theta,1)$. Our contribution
is the following Kaufman-type sharpening of Conjecture 1.2.

######  Theorem 1.3.

Let $I\subset\mathbb{R}$ be a compact interval and let $\gamma:I\to S^{2}$ be
a $C^{2}$ curve that satisfies the “escaping great circle” condition (1.3).
Let $Z\subset\mathbb{R}^{3}$ be analytic and let $0\leq
s<\min\\{\mathrm{dim}\,Z,1\\}$. Then

$\mathrm{dim}\,\\{\theta\in
I\colon\mathrm{dim}\,(Z\cdot\gamma(\theta))<s\\}\leq s.$ (1.4)

In particular, Conjecture 1.2 is true.

_Added September 1, 2022:_ There have been several exiting recent developments
related to Theorem 1.3. First, Gan, Guth and Maldague have also recently and
independently resolved Conjecture 1.2, using a different method related to
decoupling. Second, Gan, Guo, Guth, Harris, Maldague and Wang [7] solved a
closely related conjecture of Fässler and Orponen, where projections onto one-
dimensional subspaces of $\mathbb{R}^{3}$ are replaced by projections onto
two-dimensional subspaces. Finally, Harris [9] proved that if
$\mathrm{dim}\,Z>1$, then $\gamma(\theta)\cdot Z$ has positive Lebesgue
measure for almost every $\theta$.

###  1.1. A Furstenberg-type problem for circles

Theorem 1.3 follows from a $L^{p}$ bound for a variant of Wolff’s circular
maximal function, where circles have been replaced by a class of $C^{2}$
curves that are the graphs of we call “cinematic functions,” and the domain of
integration is restricted to a special type of fractal set. To state this
result, we will introduce several definitions. First, we recall the definition
of a $(\delta,\alpha;C)_{1}$-set from [12].

######  Definition 1.4.

Let $0<\alpha\leq 1$, $\delta>0$, $C\geq 1$. A set $E\subset[0,1]$ is called a
$(\delta,\alpha;C)_{1}$-set if

  1. (a)

$E$ is a union of intervals of length $\delta$.

  2. (b)

For every interval $I\subset[0,1]$, we have the Frostman-type non-
concentration condition

$|E\cap I|\leq C\delta^{1-\alpha}|I|^{\alpha}.$ (1.5)

Since (1.5) always holds for intervals of length $\leq\delta$, every
$(\delta,\alpha;C)_{1}$-set is also a $(\delta,\beta;C)_{1}$-set if
$\beta\geq\alpha$. We will be concerned with certain subsets of the plane,
which are fibered products of $(\delta,\alpha;C)_{1}$-sets.

######  Definition 1.5.

Let $0<\alpha,\beta\leq 1$, $\delta>0$, and $C\geq 1$. A set
$E\subset[0,1]^{2}$ is called a
$(\delta,\alpha;C)_{1}\times(\delta,\beta;C)_{1}$ quasi-product if $E$ is
Borel and can be expressed in the form

$E=\bigcup_{a\in A}\\{a\\}\times B_{a},$

where $A$ is a $(\delta,\alpha;C)_{1}$-set and $B_{a}$ is a
$(\delta,\beta;C)_{1}$-set for each $a\in A$.

Our main result is a maximal function estimate, where the domain of
integration is a quasi-product. Our next task is to introduce the class of
objects that can be analyzed using our maximal function estimate. The
following definition was inspired by Sogge’s notion of cinematic curvature
[22], which was considered by Kolasa and Wolff [15] in the context of the
Wolff circular maximal function.

######  Definition 1.6.

Let $I\subset\mathbb{R}$ be a compact interval and let $\mathcal{F}\subset
C^{2}(I)$. We say $\mathcal{F}$ is a _family of cinematic functions_ , with
_cinematic constant_ $K$ and _doubling constant_ $D$, if the following
conditions hold.

  1. (1)

$\mathcal{F}$ has diameter at most $K$ (here and in what follows, we use the
usual metric on $C^{2}(I)$).

  2. (2)

$\mathcal{F}$ is a doubling metric space, with doubling constant at most $D$.

  3. (3)

For all $f,g\in\mathcal{F},$ we have

$\inf_{t}\big{(}|f(t)-g(t)|+|f^{\prime}(t)-g^{\prime}(t)|+|f^{\prime\prime}(t)-g^{\prime\prime}(t)|\big{)}\geq
K^{-1}\|f-g\|_{C^{2}(I)}.$ (1.6)

With these definitions, we can now state the following variant of Wolff’s
circular maximal function theorem.

######  Theorem 1.7.

Let $\varepsilon>0$, $0<\alpha\leq\zeta\leq 1,$ and $D,K\geq 1$. Then the
following is true for all $\delta>0$ sufficiently small (depending on
$D,K,\varepsilon$ only). Let $\mathcal{F}$ be a family of cinematic functions,
with cinematic constant $K$ and doubling constant $D$. Let $E$ be a
$(\delta,\alpha;\delta^{-\varepsilon})_{1}\times(\delta,\alpha;\delta^{-\varepsilon})_{1}$
quasi-product.

Let $F\subset\mathcal{F}$ be a set of functions that satisfies the Frostman-
type non-concentration condition

$\\#(F\cap B)\leq\delta^{-\varepsilon}(r/\delta)^{\zeta}\quad\textrm{for all
balls}\ B\subset C^{2}(I)\ \textrm{of radius}\ r\geq\delta.$ (1.7)

Then

$\int_{E}\Big{(}\sum_{f\in F}{\bf
1}_{f^{\delta}}\Big{)}^{3/2}\leq\delta^{2-\alpha/2-\zeta/2-C\varepsilon}(\\#F),$
(1.8)

where $C=C(D)$ depends only on $D$, and $f^{\delta}$ is the
$\delta$-neighborhood of the graph of $f$.

_Remarks._

  1. 1.

Since $[0,1]^{2}$ is a $(\delta,1;1)_{1}\times(\delta,1;1)_{1}$ quasi-product
and $I,\mathcal{F}$ are bounded, as a special case of Theorem 1.7 we obtain
the estimate

$\Big{\|}\sum_{f\in F}{\bf
1}_{f^{\delta}}\Big{\|}_{3/2}\leq\delta^{-C\varepsilon}(\delta\\#F)^{2/3},$
(1.9)

where $F$ satisfies (1.7) for $\zeta=1$, and the order of parameters are the
same as in the statement of Theorem 1.7.

  2. 2.

As a special case of (1.9), Theorem 1.7 generalizes Wolff’s $L^{3/2}$ circular
maximal function bound from [25]. The (upper) halves of circles with centers
in a small neighborhood of the origin and radii between 1 and 2 form a family
of cinematic functions $\mathcal{C}$ over the interval $[-1/2,1/2]$. If we
paramaterize a circle by it’s center-radius pair $(x,y,r)$, then a set
$C\subset\mathcal{C}$ obeys the non-concentration condition (1.7) if the
corresponding collection of centre-radius pairs $P_{C}\subset\mathbb{R}^{3}$
obeys a corresponding Frostman-type non-concentration condition; namely,

$\\#(P_{C}\cap B)\leq\delta^{-\varepsilon}(r/\delta)\quad\textrm{for all
balls}\ B\subset\mathbb{R}^{3}\ \textrm{of radius}\ r\geq\delta.$ (1.10)

Thus if $C$ is a set of circles with centers in $[0,1]^{2}$ and radii between
1 and 2, whose center-radius pairs obey (1.10), then Theorem 1.7 says that

$\Big{\|}\sum_{c\in C}{\bf
1}_{c^{\delta}}\Big{\|}_{3/2}\leq\delta^{-C\varepsilon}(\delta\\#C)^{2/3}.$
(1.11)

In [25], Wolff proved (1.7) in the special case where the circles in $C$ had
$\delta$-separated radii.

  3. 3.

Theorem 1.7 generalizes the third author’s variable coefficient Wolff circular
maximal function bound [29] from curves given by a $C^{\infty}$ defining
function to curves given by a $C^{3}$ defining function (the latter is the
minimal amount of regularity required to state Sogge’s cinematic curvature
condition for the defining function). A precise statement is slightly
technical, and is given in Appendix A.

  4. 4.

We believe Theorem 1.7 will be useful in other contexts beyond the proof of
Theorem 1.3. For example, Wang and the third author [24] used Theorem 1.7 to
prove a special case of the Kakeya conjecture for sticky Kakeya sets in
$\mathbb{R}^{3}$, and Katz, Wu, and the third author [13] used Theorem 1.7 to
prove a special case of the Kakeya conjecture for $SL_{2}$ Kakeya sets. Chang,
Dosidis, and Kim [3] used a corollary of Theorem 1.7 to analyze a Nikodym-type
maximal function associated to spheres in $\mathbb{R}^{n}$. We remark that in
[24], it is crucial that Theorem 1.3 holds for $C^{2}$ curves; previous
maximal function bounds are insufficient.

Theorem 1.7 will be discussed further in Section 1.3, and the complete proof
will be given in Section 5.

###  1.2. From maximal functions to restricted projections

We will briefly sketch the proof of Theorem 1.3 and describe the connection
between Theorems 1.3 and 1.7. For the purpose of this proof sketch, we will
ignore many technical details. A complete proof is given in Section 2. Let
$Z\subset\mathbb{R}^{3}$ be Borel with $\mathrm{dim}\,Z=\zeta\leq 1$, and let
$0\leq s<\mathrm{dim}\,Z$. Let $\Theta=\\{\theta\in
I\colon\mathrm{dim}\,(Z\cdot\gamma(\theta))<s\\}$, and suppose for
contradiction that $\mathrm{dim}\,\Theta=\alpha>s$.

In [11], Käenmäki, Orponen, and Venieri observed that this problem could be
transformed to a question about curves in the plane. For each
$z\in\mathbb{R}^{3}$, define the plane curve

$\Gamma_{z}=\\{(\theta,z\cdot\gamma(\theta):\theta\in I\\}.$ (1.12)

Then for each $\theta\in\Theta$, the vertical line
$L_{\theta}=\mathbb{R}\times\\{\theta\\}$ satisfies
$\mathrm{dim}\,\big{(}L_{\theta}\cap\bigcup\Gamma_{z}\big{)}<s.$

Next, we discretize the arrangement at a small scale $\delta>0$. The idea
(ignoring some technical details) is that we replace $Z$ by a
$\delta$-separated subset $Z_{\delta}$ of cardinality roughly
$\delta^{-\zeta}$ that satisfies a Frostman-type non-concentration condition,
and we replace $\Theta$ by a $(\delta,\alpha;\delta^{-\varepsilon})_{1}$-set,
which we denote by $\Theta_{\delta}$. For each $z\in Z_{\delta}$, let
$\Gamma^{\delta}_{z}$ be the $\delta$-neighborhood of the curve $\Gamma_{z}$.
Then for most $\theta\in\Theta_{\delta}$, $L_{\theta}\cap\bigcup_{z\in
Z_{\delta}}\Gamma^{\delta}_{z}$ will have one-dimensional Lebesgue measure at
most $\delta^{1-\eta-\varepsilon}$, and it will be contained in a
$(\delta,s,\delta^{-\varepsilon})_{1}$-set. Thus by Hölder’s inequality we
have that

$\int_{\Theta_{\delta}\times\mathbb{R}}\Big{(}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{\delta}_{z}}\Big{)}^{3/2}\geq\delta^{2+\frac{1}{2}s-\frac{3}{2}\zeta-\alpha+\varepsilon}.$
(1.13)

Most of the integrand in (1.13) is supported on a
$(\delta,\alpha;\delta^{-\varepsilon})_{1}\times(\delta,\alpha;\delta^{-\varepsilon})_{1}$
quasi-product. We can use the “escaping great circle” condition (1.3) to show
that the set of curves $\\{\Gamma^{\delta}_{z}\colon z\in Z_{\delta}\\}$ are
the graphs of functions from a family of cinematic functions, and the
Frostman-type non-concentration condition on $Z_{\delta}$ implies the
analogous bound (1.7). Thus we can apply Theorem 1.7 to upper-bound (1.13).
Comparing (1.8) and (1.13) (and recalling that $Z_{\delta}$ has cardinality
roughly $\delta^{-\zeta}$), we conclude that $\alpha\leq s$. This gives us our
desired contradiction and completes the proof.

_Remarks_.

  1. 1.

The main challenge when executing the proof strategy described above is that
we must discretize the set $Z$, the set $\Theta$, and the sets
$L_{\theta}\cap\bigcup\Gamma_{z}$ at a common scale $\delta$. We refer the
reader to Section 2 for details.

  2. 2.

Our bound (1.8) in Theorem 1.7 becomes stronger as the dimension $\alpha$ of
the quasi-product $E$ decreases. With this sensitivity to $E$, Theorem 1.7 is
related to the Furstenberg set problem for curves; see [16] for a related
result on this topic. If we replace the domain of integration in (1.8) by
$\mathbb{R}^{2}$ or $[0,1]^{2}$, then it is possible for the LHS of (1.8) to
be as large as $\delta\\#F$. On the other hand, it is easy to verify that if
the RHS of (1.8) were replaced by the weaker estimate $\delta\\#F$, then this
would not be sufficient to prove the Kaufman-type estimate from Theorem 1.3
(though it would still be strong enough to prove Conjecture 1.2).

###  1.3. Proof sketch of Theorem 1.7

We conclude the introduction with a discussion of the main ideas in Theorem
1.7 and a brief sketch the proof. In [26], Wolff proved that for each
$\varepsilon>0$ and all $\delta>0$ sufficiently small, if $C$ is a set of
circles with $\delta$-separated radii $1\leq r\leq 2$, then

$\Big{\|}\sum_{c\in C}{\bf
1}_{c^{\delta}}\Big{\|}_{p}\leq\delta^{-\varepsilon}(\delta\\#C)^{1/p},\quad
p=3/2,$ (1.14)

where $c^{\delta}$ denotes the $\delta$-neighborhood of the circle $c$.
Wolff’s proof was quite technical, but it has been simplified in several
follow-up works [27, 21]. The main ingredient in these later proofs was a
bound on the number of “tangency rectangles” determined by the arrangement
$C$. Informally, a tangency rectangle is a curvilinear rectangle $R$ of
dimensions $\delta^{1/2}\times\delta$ (other dimensions are possible, but we
will ignore this for now) that is contained in at least two annuli
$c_{1}^{\delta},c_{2}^{\delta}$, with $c_{1},c_{2}\in C$. Heuristically, we
may suppose that every pair of circles $c_{1},c_{2}\in C$ either intersect
transversely, in which case $|c_{1}^{\delta}\cap
c_{2}^{\delta}|\sim\delta^{2}$, or they intersect tangentially, in which case
$|c_{1}^{\delta}\cap c_{2}^{\delta}|\sim\delta^{3/2}$ and the intersection is
localized to a tangency rectangle.

The contribution to the LHS of (1.14) from the first type of intersection is
negligible. Indeed, if every pair of circles intersected transversely, then
(1.14) would hold for $p=2$, which is a stronger estimate. Thus the goal is to
control the number of tangential intersections, or more precisely, the number
of tangency rectangles where these intersections occur. Let $\mathcal{R}$ be a
set of tangency rectangles that cover all the tangential intersections
determined by $C$. For each $R\in\mathcal{R}$, let $m(R)$ be the number
circles $c\in C$ with $R\subset c^{\delta}$. If the rectangles in
$\mathcal{R}$ were disjoint and if the LHS of (1.14) was concentrated on these
rectangles, then we would have a $\ell^{3/2}(\mathcal{R})\to
L^{3/2}(\mathbb{R}^{2})$ bound of the form

$\Big{\|}\sum_{c\in C}{\bf
1}_{c^{\delta}}\Big{\|}_{3/2}\leq\delta\Big{(}\sum_{\mathcal{R}}m(R)^{3/2}\Big{)}^{2/3}.$
(1.15)

We caution that reader that in general, (1.15) is not true as stated. However,
the heuristic (1.15) was made precise by Schlag [21], who used induction on
scales and a bilinear $\implies$ linear argument to prove that (1.14) would
follow from a bilinear version of the weak $\ell^{3/2}$ bound

$\\#\\{R\in\mathcal{R}\colon m(R)\geq k\\}\leq(\\#C/k)^{3/2}.$ (1.16)

Previously in [27], Wolff had proved a suitable bilinear version of (1.16)
using the “cuttings” method developed by researchers in computational geometry
in the late 1980s (see [4, 1]), and these two results combined to give a new
proof of (1.14).

The cuttings method makes crucial use of the property that circles are
algebraic curves. In [28], the third author extended Wolff’s arguments and the
cuttings method to obtain an analogue of (1.14) where circles are replaced by
algebraic curves of bounded degree that satisfy Sogge’s cinematic curvature
condition (see also [15] for earlier work in this direction). In [29], the
third author used the newly developed polynomial partitioning technique of
Guth and Katz [8] to extend (1.14) to smooth curves with cinematic curvature.
All of these techniques, however, require that the curves are algebraic (or in
the case of smooth curves, that they can be accurately approximated by
moderately low degree algebraic curves). These methods do not work, and new
ideas are needed, for curves that are merely $C^{k}$.

The main difficulty in proving Theorem 1.7 is establishing an analogue of
(1.16) for $C^{2}$ curves. We do this by adapting a result due to Marcus and
Tardos [17] from topological graph theory, which concerns objects called
lenses. In brief, if $\mathcal{C}$ is a set of Jordan curves in the plane,
then a _lens_ is a pair of curve-segments $\lambda_{1}\subset\gamma_{1};$
$\lambda_{2}\subset\gamma_{2}$; $\gamma_{1},\gamma_{2}\in\mathcal{C},$ so that
$\gamma_{1}$ and $\gamma_{2}$ intersect at their endpoints, and
$\gamma_{1}\cup\gamma_{2}$ is homeomorphic to $S^{1}$. For example, if two
circles $c_{1},c_{2}$ with well-separated centers share a common tangency
rectangle, then after a small perturbation we expect to find arcs of $c_{1}$
and $c_{2}$ of length roughly $\delta^{1/2}$ that create a lens.

Lemma 10 from [17] says that an arrangement of $n$ closed Jordan curves that
satisfy a certain “pseudo-circle” property determines $O(n^{3/2}\log n)$
pairwise non-overlapping lenses. We use this result to obtain an analogue of
(1.16) by carefully perturbing our arrangement of curves, so that the number
of tangency rectangles in $\mathcal{R}$ is comparable to the number of lenses.
See Proposition 4.1 for a precise statement, and Section 4 for the details of
this argument.

With Proposition 4.1 in hand, it remains to prove Theorem 1.7. We encounter
several difficulties. First, our bilinear analogue of (1.16) is more general
than the corresponding estimates from [27, 28, 29] because it applies to
$C^{2}$ curves rather than algebraic (or smooth) curves. However, our bilinear
analogue of (1.16) requires slightly stronger assumptions, which do not work
well with induction on scales. Thus we need a new bilinear $\implies$ linear
argument that does not use induction on scales. Closely related to this
difficulty is the annoyance that intersections between curves need not be
completely tangential or transverse—there are many intermediate amounts of
tangency or transversality. We handle these difficulties by combining a
$L^{2}$ argument at fine scales with a suitable bilinear version of (1.16) at
coarse scales. In brief, we define a parameter $\Delta$ that measures the
typical amount of tangency between curves; if $\Delta=\delta$ then the main
contribution to the LHS of (1.14) comes from pairs of curves that intersect
completely tangentially, while if $\Delta=1$ then the main contribution comes
from pairs of curves that intersect completely transversely. We then divide
$\mathbb{R}^{2}$ into rectangles of dimensions $\Delta\times\Delta^{1/2}$. If
two curves intersect, then typically this intersection is localized to one
such rectangle, and the curves are tangent at scale $\Delta$. Thus we can use
a bilinear analogue of (1.16) to control the number of tangency rectangles at
scale $\Delta$. Within each rectangle, most pairs of curves intersect with a
precisely prescribed amount of transversality. Thus we can use a $L^{2}$
argument to control the contribution from each rectangle. This is the most
technical part of the paper, and the details are in Section 5.

###  1.4. Notation

To reduce clutter, we will write $A\lesssim B$ or $A=O(B)$ if there is a
constant $C>0$ so that $A\leq CB$. The constant $C$ will be allowed to depend
on several quantities, which are context-dependent. Based on context, $C$ may
depend on any of the quantities $|I|$, $\gamma$, and $\mathrm{dim}\,Z$ from
the statement of Theorem 1.3, and also the quantites $K$, $D$, and
$\varepsilon$ from the statement of Theorem 1.7.

In the arguments that follow, we will frequently introduce a small parameter
$\delta>0$, and we will employ dyadic pigeonholing arguments that replace a
set $X$ with a subset of size roughly $|\log\delta|^{-1}(\\#X)$. We will write
$A\lessapprox B$ or $B\gtrapprox A$ if $A\lesssim|\log\delta|^{C_{0}}B$ for
some absolute constant $C_{0}$ (in practice, we will always have $C_{0}\leq
100$). When we write $A\lessapprox B$, the relevant quantity $\delta$ will
always be apparent from context.

###  1.5. Thanks

The authors would like to thank Orit Raz for many helpful comments and
suggestions throughout the preparation of this manuscript. The authors would
like to thank Alan Chang and Jongchon Kim for comments, suggestions, and
corrections to an earlier version of this manuscript. All authors were
supported by NSERC Discovery grants.

##  2\. From Theorem 1.7 to Theorem 1.3

In this section, we will expand upon the sketch from Section 1.2. The
following version of Theorem 1.7 is a main ingredient of the proof.

######  Proposition 2.1.

Let us fix $\varepsilon>0$ and dimensional parameters $0<\alpha\leq\zeta\leq
1$. Let $I$ be a compact interval and let $\gamma\colon I\to S^{2}$ be a curve
satisfying (1.3). Then there exists a positive constant $\delta_{0}$ depending
only on these quantities, such that the following is true for all
$0<\delta\leq\delta_{0}$.

Let $E$ be a
$(\delta,\alpha;\delta^{-\varepsilon})_{1}\times(\delta,\alpha;\delta^{-\varepsilon})_{1}$
quasi-product. Let $Z_{\delta}\subset B(0,1)\subset\mathbb{R}^{3}$ be a
$\delta$-separated set that satisfies the Frostman-type non-concentration
condition

$\\#(Z_{\delta}\cap
B)\leq\delta^{-\varepsilon}(r/\delta)^{\zeta}\quad\textrm{for all balls}\
B\subset\mathbb{R}^{3}\ \textrm{of radius}\ r\geq\delta.$ (2.1)

Then

$\int_{E}\Big{(}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{\delta}_{z}}\Big{)}^{3/2}\leq\delta^{2-\alpha/2-\zeta/2-C\varepsilon}(\\#Z_{\delta}),$
(2.2)

where $C>0$ is an absolute constant, and $\Gamma^{\delta}_{z}$ is the
$\delta$-neighborhood of the graph $\Gamma_{z}$ defined in (1.12).

###### Proof.

Proposition 2.1 follows from Theorem 1.7, applying the latter to the family of
functins

$F=\bigl{\\{}f_{z}:z\in
Z_{\delta}\bigr{\\}}\subseteq\mathcal{F}=\bigl{\\{}f_{z}:z\in
B(0;1)\bigr{\\}},$

where $f_{z}(\theta)=\gamma(\theta)\cdot z$ and $\gamma\colon I\to S^{2}$ is a
curve satisfying (1.3). In order for Theorem 1.7 to be applicable, we need to
verify that $\mathcal{F}$ is cinematic (with cinematic constant that depends
only on $\gamma$) and that $F$ satisfies a Frostman-type non-concentration
bound (1.7). The verification proceeds as follows. First, we have

$\sup_{f\in\mathcal{F}}\lVert f\rVert_{C^{2}(I)}=\sup_{z\in B(0;1)}\lVert
f_{z}\rVert_{C^{2}(I)}\leq\sup_{\theta\in
I}\bigl{[}|\gamma(\theta)|+|\gamma^{\prime}(\theta)|+|\gamma^{\prime\prime}(\theta)|\bigr{]}|z|\leq||\gamma||_{C^{2}(I)}<\infty,$

and depends only on $\gamma$. Next, if $z_{1},z_{2}\in B(0,1)$ then
$\|f_{z_{1}}-f_{z_{2}}\|_{C^{2}(I)}\leq||\gamma||_{C^{2}(I)}|z_{1}-z_{2}|$.
Furthermore, the non-degeneracy condition (1.3) implies that

$\inf_{\theta\in
I}\Big{[}|f_{z_{1}}(\theta)-f_{z_{2}}(\theta)|+|f^{\prime}_{z_{1}}(\theta)-f^{\prime}_{z_{2}}(\theta)|+|f^{\prime\prime}_{z_{1}}(\theta)-f^{\prime\prime}_{z_{2}}(\theta)|\Big{]}\gtrsim\|f_{z_{1}}-f_{z_{2}}\|_{C^{2}(I)},$

where the implicit constant depends only on $\gamma$. As a consequence,

$\|f_{z_{1}}-f_{z_{2}}\|_{C^{2}(I)}\sim|z_{1}-z_{2}|\text{ for }z_{1},z_{2}\in
B(0;1).$ (2.3)

Thus $\mathcal{F}$ is a doubling metric space (with the same doubling constant
as $\mathbb{R}^{3}$). Finally, identifying $F$ with $Z_{\delta}$, we observe
from (2.3) that the Frostman-type non-concentration condition (1.7) for $F$
follows from the analogous condition (2.1) for the Euclidean set $Z_{\delta}$.
∎

We are now ready to prove Theorem 1.3. Let $\gamma\colon I\to S^{2}$ satisfy
(1.3) and let $Z\subset\mathbb{R}^{3}$ be Borel. After a harmless re-scaling,
we can suppose $Z\subset B(0,1)$ and after possibly replacing $Z$ by a subset,
we may suppose that $\mathrm{dim}\,Z\leq 1$. Let $0\leq s<\mathrm{dim}\,Z$.
Aiming for a contradiction, suppose if possible that (1.4) fails, i.e.
$\mathrm{dim}\,\Theta_{I}>s$, where

$\Theta_{I}=\bigl{\\{}\theta\in
I:\mathrm{dim}\,\bigl{(}Z\cdot\gamma(\theta)\bigr{)}<s\bigr{\\}}.$

Then $\Theta_{I}$ admits a subset $\Theta$ with the propery that
$s<\mathrm{dim}\,\Theta\leq\mathrm{dim}\,Z$. Set
$\alpha=\mathrm{dim}\,\Theta$. We will show that the existence of such a set
$\Theta\subseteq\Theta_{I}$ contradicts the validity of Proposition 2.1.

Define

$\varepsilon=\frac{\alpha-s}{2C+11}>0,$ (2.4)

where $C$ is the constant from (2.2).

###  2.1. Initial discretization

In [2], Bourgain proved a discretized projection theorem, which bounded the
number of exceptional directions where the projection of a
$\delta$-discretized set can be small. Bourgain then showed how this
discretized theorem implies an analogous Hausdorff dimension bound. The next
set of arguments are similar to the reduction from Theorem 4 to Theorem 3 in
[2], but with a twist. The location of the twist will be noted below.

Let $\zeta=\mathrm{dim}\,Z$ and let $\mathbb{P}$ be a probability measure
supported on $Z$ that satisfies the Frostman condition

$\mathbb{P}(B)\lesssim r^{\zeta-\varepsilon}\qquad\textrm{for all balls}\
B\subset\mathbb{R}^{3}\ \textrm{of radius}\ r>0.$ (2.5)

Let $\nu$ be a probability measure supported on $\Theta$ with

$\nu(J)\lesssim|J|^{\alpha-\varepsilon}\ \quad\textrm{for all intervals}\
J\subset\mathbb{R}.$ (2.6)

Let $k_{0}\in\mathbb{Z}$ be chosen sufficiently large, depending on $\gamma$,
$\zeta$, $s$ and $\alpha$. The choice of $k_{0}$ will be determined as
follows: our arguments below will select a number $0<\delta\leq
2^{-\lfloor(1+\varepsilon)^{k_{0}}\rfloor}$ (as in [12, Section 7]; we do not
get to choose $\delta$, but our choice of $k_{0}$ gives an upper bound for
$\delta$). Eventually, we will arrive at the conclusion that a certain
inequality involving $\delta$ is impossible, provided $\delta>0$ is
sufficiently small (depending on $\gamma$, $\zeta$, $s$ and $\alpha$). In
order to arrive at a contradiction, we will select $k_{0}\in\mathbb{Z}$
sufficiently large so that any positive $\delta\leq
2^{-\lfloor(1+\varepsilon)^{k_{0}}\rfloor}$ obeys Proposition 2.1.

For each $\theta\in I$, define
$\rho_{\theta}\colon\mathbb{R}^{3}\to\mathbb{R}$ by
$\rho_{\theta}(z)=z\cdot\gamma(\theta)$. For each
$\theta\in\operatorname{supp}(\nu)\subseteq\Theta$, cover $\rho_{\theta}(Z)$
by a union of sets $\\{G_{\theta,k}\\}_{k=k_{0}}^{\infty}$, where
$G_{\theta,k}$ is a
$(2^{-\lfloor(1+\varepsilon)^{k}\rfloor},s;2^{\varepsilon\lfloor(1+\varepsilon)^{k}\rfloor})_{1}$-set
(in Definition 1.4 a $(\delta,\alpha;C)_{1}$-set must be contained in $[0,1]$,
but it is harmless to apply a translation); see i.e. [12, Lemma 7.5] for
details on how to do this. This is the “twist” alluded to above; in [2],
Bourgain simply covers $\rho_{\theta}(A)$ by unions of $2^{-k}$ intervals; he
does not ask that these unions satisfy the non-concentration condition (1.5).

For each $\theta\in\operatorname{supp}(\nu)$ we have

$1=\mathbb{P}(Z)=\mathbb{P}\big{(}Z\cap\rho_{\theta}^{-1}(\rho_{\theta}(Z)\big{)}=\mathbb{P}\Big{(}Z\cap\bigcup_{k\geq
k_{0}}\rho_{\theta}^{-1}(G_{\theta,k})\Big{)}.$

Thus by integrating over $I$ with respect to $\nu$ we conclude that

$\int_{I}\sum_{k\geq
k_{0}}\mathbb{P}\big{(}Z\cap\rho_{\theta}^{-1}(G_{\theta,k}\big{)}d\nu(\theta)\geq\nu(\Theta)=1.$

In particular, there exists $k\geq k_{0}$ so that

$\int_{I}\mathbb{P}\big{(}Z\cap\rho_{\theta}^{-1}(G_{\theta,k})\big{)}d\nu(\theta)\gtrsim\frac{1}{k^{2}}.$
(2.7)

We will fix this choice of $k$, and for the remiander of this proof, define
$\delta=2^{-\lfloor(1+\varepsilon)^{k}\rfloor}$ and $G_{\theta}=G_{\theta,k}$;
thus the right hand side of (2.7) becomes essentially
$\left(\frac{\log(1+\varepsilon)}{\log\log(1/\delta)}\right)^{2}$, which is
$\geq|\log\delta|^{-2}$ if $\delta$ is sufficiently small relative to
$\varepsilon$.

Recalling the notation introduced in Section 1.4, we will write $A\gtrapprox
B$ to mean $A\gtrsim|\log\delta|^{-C}B$, where $C$ is an absolute constant (in
practice, we will always have $C\leq 100$).

###  2.2. Replacing $\mathbb{P}$ by a sum over curves

Given the choice of $\delta$ above, let us consider a covering of $Z$ using
cubes of sidelength $\delta/2$. After dyadic pigeonholing, we can find a set
$\mathcal{Q}$ of such cubes and a “weight” $w$, so that
$w\leq\mathbb{P}(Q)<2w$ for each $Q\in\mathcal{Q}$, and

$\int_{I}\mathbb{P}\Big{(}\bigcup_{Q\in\mathcal{Q}}Q\cap\rho_{\theta}^{-1}(G_{\theta})\Big{)}d\nu(\theta)\gtrapprox
1.$ (2.8)

Let $G_{\theta}^{\prime}=N_{\delta}(G_{\theta})$; here and henceforth,
$N_{\delta}(X)$ denotes the $\delta$-neighbourhood of the set $X$. Then
$G_{\theta}^{\prime}$ is a $(3\delta,s;3\delta^{-\varepsilon})_{1}$-set, and
if $Q\in\mathcal{Q}$ is a cube that intersects
$\rho_{\theta}^{-1}(G_{\theta})$, then
$Q\subset\rho_{\theta}^{-1}(G_{\theta}^{\prime})$. Thus (2.8) implies that

$\int_{I}w\\#\big{\\{}Q\in\mathcal{Q}\colon
Q\subseteq\rho_{\theta}^{-1}(G_{\theta}^{\prime})\big{\\}}d\nu(\theta)\gtrapprox
1,$

or

$\int_{I}\\#\big{\\{}Q\in\mathcal{Q}\colon\rho_{\theta}(Q)\subseteq
G_{\theta}^{\prime}\big{\\}}d\nu(\theta)\gtrapprox w^{-1}.$ (2.9)

Since $Q$ is a cube of side-length $\delta/2$ and $\gamma(\theta)\in S^{2}$,
we have that $\rho_{\theta}(Q)$ is an interval in $\mathbb{R}$ of length
$\sim\delta$. Thus $\int_{G_{\theta}^{\prime}}{\bf
1}_{\rho_{\theta}(Q)}(y)dy\sim\delta$ for each $Q\in\mathcal{Q}$ with
$\rho_{\theta}(Q)\subset G_{\theta}^{\prime}$. Hence (2.9) implies that

$\int_{I}\int_{G_{\theta}^{\prime}}\sum_{Q\in\mathcal{Q}}{\bf
1}_{\rho_{\theta}(Q)}(y)dyd\nu(\theta)\gtrapprox\delta w^{-1}.$ (2.10)

Note that $\delta^{3}\lesssim w\lesssim\delta^{\zeta-\varepsilon}$, and
$1\lessapprox w\\#\mathcal{Q}\leq 1$. Let
$A=\delta^{\zeta-\varepsilon}w^{-1}\gtrsim 1$, so that
$\\#\mathcal{Q}\gtrapprox A\delta^{-\zeta+\varepsilon}$ and
$w=\delta^{\zeta-\varepsilon}A^{-1}$. Thus the right hand side of (2.10)
becomes $A\delta^{1-\zeta+\varepsilon}$.

Let $Z_{\delta}^{\prime}$ denotes the set of centers of the cubes in
$\mathcal{Q}$; then the Frostman condition (2.5) implies that

$\\#(Z_{\delta}^{\prime}\cap B)\lessapprox w^{-1}\mathbb{P}(B)\lesssim
w^{-1}r^{\zeta-\varepsilon}=A(r/\delta)^{\zeta-\varepsilon}\qquad\textrm{for
all balls}\ B\ \textrm{of radius}\ r\geq\delta.$ (2.11)

Note that if $Q\in\mathcal{Q}$ has center $z\in Z_{\delta}^{\prime}$, then
$Q\subseteq N_{\delta}(z)$ (this is why we required that the cubes in
$\mathcal{Q}$ have side-length $\delta/2$, rather than $\delta$). Thus for
each $\theta\in I$, we have the implication

$y\in\rho_{\theta}(Q)\implies|\gamma(\theta)\cdot z-y|\leq\delta.$

This leads to the inequality

$\sum_{Q\in\mathcal{Q}}{\bf 1}_{\rho_{\theta}(Q)}(y)\leq\sum_{z\in
Z_{\delta}^{\prime}}{\bf 1}_{\Gamma^{\delta}_{z}}(\theta,y).$ (2.12)

Combining (2.10) and (2.12), we obtain

$\int_{I}\int_{G_{\theta}^{\prime}}\sum_{z\in Z_{\delta}^{\prime}}{\bf
1}_{\Gamma^{\delta}_{z}}(\theta,y)dyd\nu(\theta)\gtrapprox
A\delta^{1-\zeta+\varepsilon}.$ (2.13)

Let $Z_{\delta}$ be a subset of $Z^{\prime}_{\delta}$ obtained by randomly and
independently selecting each point of $Z^{\prime}_{\delta}$ with probability
$p=A^{-1}$; we will select the set $Z_{\delta}$ so that the following
properties hold.

$\\#(Z_{\delta}\cap B)\leq|\log\delta|A^{-1}\\#(Z_{\delta}^{\prime}\cap
B)\leq\delta^{-\varepsilon}(r/\delta)^{\zeta}\qquad\textrm{for all balls}\ B\
\textrm{of radius}\ r\geq\delta,$ (2.14)

and

$\int_{I}\int_{G_{\theta}^{\prime}}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{\delta}_{z}}(\theta,y)dyd\nu(\theta)\gtrapprox\delta^{1-\zeta+\varepsilon}.$
(2.15)

Let us briefly discuss the selection of $Z_{\delta}$. Apriori, (2.14) appears
to be an assertion about an uncountable set of balls
$\\{B\subset\mathbb{R}^{3}\colon\operatorname{radius}(B)\geq\delta\\}$.
However, it suffices to establish (2.14) (with the RHS replaced by
$\frac{1}{2}\delta^{-\varepsilon}(r/\delta)^{\zeta}$) for a set of roughly
$\delta^{-3}$ balls: For each $j=0,\dots,|\log\delta|$, let $\mathcal{B}_{j}$
be a set of about $(2^{j}\delta)^{-3}$ balls of radius $2^{j+1}\delta$, so
that every ball of radius $2^{j}\delta$ is contained in at most $O(1)$ balls
from $\mathcal{B}_{j}$. Then it suffices to prove (2.14) for the balls in
$\mathcal{B}=\bigcup_{j=0}^{|\log\delta|}\mathcal{B}_{j}$; this set has
cardinality $\lesssim\delta^{-3}$. If $\delta$ is small enough, then using
Chernoff’s inequality and (2.11) we can ensure that the probability that
(2.14) fails for any particular ball in $\mathcal{B}$ is smaller than
$\delta^{4}$. Applying the union bound over $B\in\mathcal{B}$, we deduce that
with probability at least $1-O(\delta)$, the estimate (2.14) holds for all
balls $B\in\mathcal{B}$ and hence in general for all balls of radius
$r\geq\delta$. Choosing such an outcome with the additional requirement (also
derivable from Chernoff’s inequality) that

$\\#Z_{\delta}>\frac{1}{100A}\\#Z_{\delta}^{\prime},$

the property (2.15) follows from (2.13) and an expected value calculation
(expectation is linear) and Markov’s inequality.

###  2.3. Replacing $\nu$ by an integral over a quasi-product

Let $\mathcal{J}_{0}$ be the set of intervals in $I$ (recall $I$ is the domain
of $\gamma(\theta)$) of the form $[n\delta,(n+1)\delta)$, for
$n\in\mathbb{Z}$. For each $J\in\mathcal{J}_{0}$, let $\theta(J)\in J$ be a
value of $\theta$ that gets within a factor of 2 of maximizing the quantity

$\int_{G_{\theta}^{\prime}}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{\delta}_{z}}(\theta,y)dy.$

For each $J\in\mathcal{J}_{0},$ we have

$\begin{split}\nu(J)\int_{G_{\theta(J)}^{\prime}}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{\delta}_{z}}(\theta(J),y)dy&=\int_{J}\int_{G_{\theta(J)}^{\prime}}\sum_{z\in
Z_{\delta}}{\bf 1}_{\Gamma^{\delta}_{z}}(\theta(J),y)dyd\nu(\theta)\\\
&\geq\frac{1}{2}\int_{J}\int_{G_{\theta}^{\prime}}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{\delta}_{z}}(\theta,y)dyd\nu(\theta).\end{split}$ (2.16)

The first equality is just the statement that $\nu(J)=\int_{J}d\nu(\theta)$,
while the second inequality follows from the definition of $\Theta(J)$. To
simplify notion, define

$K(J)=\int_{G_{\theta(J)}^{\prime}}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{\delta}_{z}}(\theta(J),y)dy,$

so the left hand side of (2.16) is $\nu(J)K(J)$. Summing (2.16) over
$J\in\mathcal{J}_{0}$, we obtain

$\begin{split}\sum_{J\in\mathcal{J}_{0}}\nu(J)K(J)&\geq\frac{1}{2}\sum_{J\in\mathcal{J}_{0}}\int_{J}\int_{G_{\theta}^{\prime}}\sum_{z\in
Z_{\delta}}{\bf 1}_{\Gamma^{\delta}_{z}}(\theta,y)dyd\nu(\theta)\\\
&=\frac{1}{2}\int_{I}\int_{G_{\theta}^{\prime}}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{\delta}_{z}}(\theta,y)dyd\nu(\theta)\\\
&\gtrapprox\delta^{1-\zeta+\varepsilon}.\end{split}$

Observe that
$K(J)\lesssim\delta\\#Z_{\delta}\lessapprox\delta^{1-\zeta+\varepsilon}$ for
each $J\in\mathcal{J}_{0}$, and $\sum_{J\in\mathcal{J}_{0}}\nu(J)=1$. Thus by
dyadic pigeonholing, there is a set $\mathcal{J}_{1}\subset\mathcal{J}_{0}$
and a weight $\delta\lesssim w^{\prime}\leq\delta^{\alpha-\varepsilon}$ so
that $w^{\prime}\leq\nu(J)<2w^{\prime}$ for each $J\in\mathcal{J}_{1}$, and
$1\lessapprox w^{\prime}\\#\mathcal{J}_{1}\leq 1$. We can write
$A^{\prime}=\delta^{\alpha-\varepsilon}w^{\prime-1}\gtrsim 1$, and thus
$\\#\mathcal{J}_{1}\approx A^{\prime}\delta^{-\alpha+\varepsilon}$, and we
have

$\sum_{J\in\mathcal{J}_{1}}\delta^{\alpha-\varepsilon}(A^{\prime})^{-1}K(J)\gtrapprox\delta^{1-\zeta+\varepsilon},$

i.e.

$\sum_{J\in\mathcal{J}_{1}}K(J)\gtrapprox
A^{\prime}\delta^{1-\zeta-\alpha+2\varepsilon}.$

We will now perform a random sampling argument analogous to how we selected
$Z_{\delta}\subset Z_{\delta}^{\prime}$ at the end of Section 2.2. Let
$p=(A^{\prime})^{-1}$, and select $\mathcal{J}\subset\mathcal{J}_{1}$ by
randomly and independently choosing each $J\in\mathcal{J}_{1}$ with
probability $p$. By (2.6), with positive probability, we may choose
$\mathcal{J}$ so that the following two properties hold.

$|\Lambda\cap\bigcup_{J\in\mathcal{J}}J|\leq\delta|\log\delta|(|\Lambda|/\delta)^{\alpha-\varepsilon}\leq\delta^{1-\varepsilon}(|\Lambda|/\delta)^{\alpha-\varepsilon}\qquad\textrm{for
all intervals}\ \Lambda,$ (2.17)

(i.e. $\bigcup_{J\in\mathcal{J}^{\prime\prime}}J$ is a
$(\delta,\alpha-\varepsilon;\delta^{-\varepsilon})_{1}$-set), and

$\sum_{J\in\mathcal{J}}K(J)\gtrapprox\delta^{1-\zeta-\alpha+2\varepsilon}.$
(2.18)

(2.17) and (2.18) are the analogues of (2.14) and (2.15), respectively, and
the argument is the same.

Next, we claim that if $(\theta,y)\in\Gamma^{\delta}_{z}$, and if
$|\theta-\theta^{\prime}|\leq\delta$, then
$\theta^{\prime}\in\Gamma^{L\delta}_{z}$, where

$L=\sup_{z\in Z_{\delta}}\sup_{\theta\in
I}\left|\frac{d}{d\theta}\rho_{\theta}(z)\right|\lesssim 1.$

Thus for each $z\in Z_{\delta}$, each $y\in\mathbb{R}$, and each
$J\in\mathcal{J}$, we have

$\int_{J}{\bf 1}_{\Gamma^{\delta}_{z}}(\theta(J),y)d\theta\leq\int_{J}{\bf
1}_{\Gamma^{L\delta}_{z}}(\theta,y)d\theta.$ (2.19)

Integrating (2.19) over $G_{\theta(J)}^{\prime}$ and summing over $z\in
Z_{\delta}$ (and interchanging the order of integration and summation), we
obtain

$\begin{split}\delta K(J)=&\int_{J}\int_{G_{\theta(J)}^{\prime}}\sum_{z\in
Z_{\delta}}{\bf 1}_{\Gamma^{\delta}_{z}}(\theta(J),y)dyd\theta\\\
&\leq\int_{J}\int_{G_{\theta(J)}^{\prime}}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{L\delta}_{z}}(\theta,y)dyd\theta.\end{split}$

Combining this with (2.18), we conclude that

$\begin{split}\delta^{2-\zeta-\alpha+2\varepsilon}&\lessapprox\sum_{J\in\mathcal{J}}\delta
K(J)\\\
&\leq\sum_{J\in\mathcal{J}^{\prime\prime}}\int_{J}\int_{G_{\theta(J)}^{\prime}}\sum_{z\in
Z_{\delta}}{\bf 1}_{\Gamma^{L\delta}_{z}}(\theta,y)dyd\theta.\end{split}$
(2.20)

Finally, define

$E=\bigcup_{J\in\mathcal{J}}J\times G_{\theta(J)}^{\prime}.$

Then $E$ is a
$(\delta,\alpha-\varepsilon;\delta^{-\varepsilon})_{1}\times(3\delta,s;3\delta^{-\varepsilon})_{1}$
quasi-product, and (2.20) can be re-written as

$\int_{E}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{L\delta}_{z}}\gtrapprox\delta^{2-\zeta-\alpha+2\varepsilon}.$
(2.21)

Since

$|E|\lesssim(\delta^{1-\alpha+\varepsilon}\delta^{-\varepsilon})\times\bigl{(}(3\delta)^{1-s}3\delta^{-\varepsilon}\bigr{)}\sim\delta^{2-\alpha-s-2\varepsilon},$

the relation (2.21) together with Hölder’s inequality implies that

$\int_{E}\Big{(}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{L\delta}_{z}}\Big{)}^{3/2}\gtrapprox\delta^{2+\frac{1}{2}s-\alpha-\frac{3}{2}\zeta+4\varepsilon}.$
(2.22)

###  2.4. Applying Proposition 2.1

Since $E$ is a
$(\delta,\alpha-\varepsilon;\delta^{-\varepsilon})_{1}\times(3\delta,s;3\delta^{-\varepsilon})_{1}$
quasi-product and $s<\alpha$, $E$ can be expressed as a union of $O(1)$ many
$(\delta,\alpha,\delta^{-\varepsilon})_{1}\times(\delta,\alpha,\delta^{-\varepsilon})_{1}$
quasi-products. Thus we can apply Proposition 2.1 with
$\\#Z_{\delta}\lessapprox\delta^{-\zeta-\varepsilon}$ and use the triangle
inequality to conclude that

$\int_{E}\Big{(}\sum_{z\in Z_{\delta}}{\bf
1}_{\Gamma^{L\delta}_{z}}\Big{)}^{3/2}\lessapprox\delta^{2-\frac{1}{2}\alpha-\frac{3}{2}\zeta-C\varepsilon-\varepsilon}.$
(2.23)

Comparing (2.22) and (2.23), we conclude that
$\delta^{s-\alpha}\lessapprox\delta^{-(2C+10)\varepsilon}$. By selecting
$\delta>0$ sufficiently small (depending on $\varepsilon$), we conclude that
$\alpha-s\leq(2C+10)\varepsilon$. But comparing with (2.4), we obtain a
contradiction. This finishes the proof of Theorem 1.3.

##  3\. The geometry of cinematic functions

In this section, we will explore some geometric properties of families of
cinematic functions. Many of the results in this section are inspired by
analogous results in [11, 15, 27]. For the definitions that follow, we will
fix a family of cinematic functions $\mathcal{F}$ over a compact interval $J$,
with cinematic constant $K$ and doubling constant $D$.

For $f\in\mathcal{F}$, we write $f^{\lambda}$ to denote the closed
$\lambda$-vertical neighborhood of the graph of $f$, i.e. the set
$\\{(\theta,y)\in J\times\mathbb{R}\colon|f(\theta)-y|\leq\lambda\\}.$ This
definition differs slightly from the definition of $f^{\delta}$ given in the
statement of Theorem 1.7. This distinction does not matter, however, since
$f^{\lambda}$ is comparable to the $\lambda$-neighborhood of the graph of $f$.

######  Definition 3.1 (Curvilinear rectangles).

Let $0<\delta\leq 1,$ $\delta\leq t\leq 1$, and $f\in\mathcal{F}$. A
$(\delta,t)$-rectangle is defined to be a set of the form

$f^{\delta}(I)=\\{(\theta,y)\in
I\times\mathbb{R}\colon|y-f(\theta)|\leq\delta\\},$

where $I\subset J$ is an interval of length $\sqrt{\delta/t}$.

######  Definition 3.2 (Comparability).

We say two $(\delta,t)$-rectangles $R,R^{\prime}$ are $\lambda$-comparable, if
there is another $(\lambda\delta,t)$-rectangle $R^{\prime\prime}$ that
contains $R\cup R^{\prime}$. Otherwise we say $R,R^{\prime}$ are
$\lambda$-incomparable.

######  Definition 3.3 (Tangency).

We say a $(\delta,t)$-rectangle $R$ is $\lambda$-tangent to the function
$f\in\mathcal{F}$ (or $f$ is $\lambda$-tangent to $R$) if $R\subseteq
f^{\lambda\delta}$. In most cases we take $\lambda=5$, and we will simply
refer to this as $R$ is tangent to $f$.

######  Definition 3.4 (Separation).

Let $\mathcal{W},\mathcal{B}\subset\mathcal{F}$. For $t>0$, we say
$(\mathcal{W},\mathcal{B})$ is $t$-separated if $\|w-b\|\geq t$ for all
$w\in\mathcal{W}$, $b\in\mathcal{B}$.

###  3.1. Preliminary reductions

We start with the following lemma, which is similar to [11, Lemma 3.1].

######  Lemma 3.5.

Let $\kappa>0$. Then there exists a constant $\lambda=\lambda(\kappa,J)>0$
with the following property: if $I\subseteq J$ is an interval of length
$|I|\leq\lambda$, $f,g\in C^{2}(J)$, and $k$ is one of the functions $f-g$,
$f^{\prime}-g^{\prime}$ or $f^{\prime\prime}-g^{\prime\prime}$, then either of
the following alternatives hold (depending on the choice of $I$ and $k$):

  1. (S)

$|k(\theta)|<\kappa\|f-g\|$ for all $\theta\in I$.

  2. (L)

$|k(\theta)|\geq\kappa\|f-g\|/2$ for all $\theta\in I$.

###### Proof.

Let $\overline{B}$ be the closed unit ball in $C^{2}(J)$. Then there is some
$\lambda>0$ such that if $|\theta-\theta^{\prime}|<\lambda$ and
$h\in\overline{B}$, we have $|h(\theta)-h(\theta^{\prime})|<\kappa/2$. Next,
fix $I\subseteq J$ with $|I|<\lambda$, and suppose that the alternative (L)
fails for $k=f-g$. So there is $\theta_{0}\in I$ such that
$|k(\theta_{0})|<\kappa d(f,g)/2$. Then, if $\theta\in I$ is arbitrary, we
have $|\theta-\theta_{0}|\leq\lambda$, and so if we define
$l=k/d(f,g)\in\overline{B}(0,1)$, then

$\displaystyle\frac{|k(\theta)|}{d(f,g)}\leq\frac{|k(\theta_{0})|}{d(f,g)}+\left|\frac{k(\theta)}{d(f,g)}-\frac{k(\theta_{0})}{d(f,g)}\right|<\frac{\kappa}{2}+|l(\theta)-l(\theta_{0})|<\frac{\kappa}{2}+\frac{\kappa}{2}=\kappa.$

Thus the alternative (S) holds for $k$. The proof of the cases
$k=f^{\prime}-g^{\prime}$ and $k^{\prime}=f^{\prime\prime}-g^{\prime\prime}$
are the same. ∎

Recall that we have fixed a family $\mathcal{F}$ of cinematic functions, with
cinematic constant $K$ and doubling constant $D$. We apply the above Lemma
with $\kappa=K/3$, and thus it suffices to prove (1.8) for each individual
subinterval of length $\lambda$. Also, the restriction of the functions in
$\mathcal{F}$ to each such sub-interval is still family of cinematic functions
with cinematic constant $K$ and doubling constant $D$. Considering each sub-
interval in turn, we may assume without loss of generality that for each pair
of functions $f,g\in\mathcal{F}$, either (S) or (L) hold on the entire
interval $J$, when $k$ is one of the functions $f-g$, $f^{\prime}-g^{\prime}$
or $f^{\prime\prime}-g^{\prime\prime}$. A last important observation is that
since $\mathcal{F}$ is a family of cinematic functions, the alternative (L)
holds for at least one of the functions $f-g$, $f^{\prime}-g^{\prime}$ or
$f^{\prime\prime}-g^{\prime\prime}$.

The following lemma is an analogue of [11, Lemma 3.2].

######  Lemma 3.6.

For $f,g\in\mathcal{F}$, the map $\theta\mapsto f(\theta)-g(\theta)$ has at
most two zeros in $J$. Moreover, if $\theta\mapsto
f^{\prime}(\theta)-g^{\prime}(\theta)$ has two zeros in $J$, then the
alternative (L) holds for $\theta\mapsto f(\theta)-g(\theta)$.

###### Proof.

We start with the second claim. Assume $\theta\mapsto
f^{\prime}(\theta)-g^{\prime}(\theta)$ has two zeros in $J$. This implies, by
Rolle’s theroem, that $\theta\mapsto
f^{\prime\prime}(\theta)-g^{\prime\prime}(\theta)$ has a zero in $J$. Now
Lemma 3.5 implies that the alternative (S) holds for $J$ and both
$\theta\mapsto f^{\prime}(\theta)-g^{\prime}(\theta)$ and $\theta\mapsto
f^{\prime\prime}(\theta)-g^{\prime\prime}(\theta)$. Consequently the
alternative (L) holds for $\theta\mapsto f(\theta)-g(\theta)$.

The first claim follows from the second one: if $\theta\mapsto
f(\theta)-g(\theta)$ had three zeros in $J$, then $\theta\mapsto
f^{\prime}(\theta)-g^{\prime}(\theta)$ would have two zeros in $J$ again by
the Rolle’s theorem. But then, by the second claim, $\theta\mapsto
f(\theta)-g(\theta)$ satisfies the alternative (L) in $J$, and hence cannot
have zeros in $J$. ∎

###  3.2. The tangency parameter $\Delta$

Next, we shall define a quantity that measures how close two functions
$f,g\in\mathcal{F}$ are to being tangent. Our definition is similar to the
quantity (3.1) from [11], which in turn was inspired by [15, 27].

######  Definition 3.7.

For $f,g\in\mathcal{F}$, we define

$\Delta(f,g)=\min_{\theta\in
J/2}|f(\theta)-g(\theta)|+|f^{\prime}(\theta)-g^{\prime}(\theta)|,$ (3.1)

where $J/2$ is the interval of length $|J|/2$ with the same midpoint as $J$.

In particular, we have

$\Delta(f,g)\leq\|f-g\|.$ (3.2)

$\Delta(f,g)$ measures the tangency between the graphs of functions $f,g$ over
the interval $J/2$. If $\Delta(f,g)=0$, then $f$ and $g$ are tangent at some
point $\theta\in J/2$. Note that $\Delta(f,g)$ is a pseudo-metric, since it is
symmetric and satisfies the triangle inequality. The restriction to $J/2$ but
not $J$ is needed for technical reasons, as is also the case of [15] and [11].

The next proposition is a close cousin of [15, Lemma 3.3]. It describes the
key geometric information about cinematic functions.

######  Lemma 3.8.

Let $f,g\in\mathcal{F}$, and define $h=f-g$, $t=\|f-g\|$, and
$\Delta=\Delta(f,g)$.

  1. (1)

There exists $c_{1}=c_{1}(K)>0$ so that if $\Delta\leq c_{1}t$, then the
followings are true.

    1. (a)

There is a unique $\theta_{0}\in\frac{3}{5}J$ such that
$h^{\prime}(\theta_{0})=0$. We also have $|h(\theta_{0})|\lesssim\Delta$.

    2. (b)

We have $|h(\theta)|\gtrsim t>0$ for every $\theta\in
J\backslash(\frac{4}{5}J)$. If in addition $h$ changes sign in $J$, then $h$
has exactly 2 zeros $\theta_{1}<\theta_{2}$, both within $\frac{4}{5}J$, such
that $|\theta_{i}-\theta_{0}|\lesssim\sqrt{\Delta/t}$, $i=1,2$.

  2. (2)

For each $0<\delta\leq t/(6K)$, the set

$E_{\delta}=\\{\theta\in J/4:|h(\theta)|\leq\delta\\}$ (3.3)

satisfies the following:

    1. (a)

$E_{\delta}$ is contained in an interval of length
$O(\sqrt{(\Delta+\delta)/t})$.

    2. (b)

$E_{\delta}$ is either a closed interval of length
$O(\delta/\sqrt{(\Delta+\delta)t})$ or the union of two closed intervals of
length $O(\delta/\sqrt{(\Delta+\delta)t})$.

    3. (c)

If in addition there is some $\tilde{\theta}\in J/4$ such that
$|h(\tilde{\theta})|\leq\delta/2$, and $I$ is the interval constituting
$E_{\delta}$ that contains $\tilde{\theta}$, then
$|I|\gtrsim\delta/\sqrt{(\Delta+\delta)t}$.

###### Proof.

Let $\theta_{\Delta}\in J/2$ be such that

$\Delta=|h(\theta_{\Delta})|+|h^{\prime}(\theta_{\Delta})|.$

Thus $|h(\theta_{\Delta})|\leq\Delta$ and
$|h^{\prime}(\theta_{\Delta})|\leq\Delta$.

We first prove Part 1. If we choose $c_{1}<1/(6K)$, then using the definition
of $\Delta(f,g)$, we have the alternative (S) holds for both $h,h^{\prime}$.
Thus the alternative (L) must hold for $h^{\prime\prime}$. Without loss of
generality we may assume $h^{\prime\prime}(\theta)\gtrsim t$.

For Part 1a, using $h^{\prime\prime}(\theta)\gtrsim t$, if $c_{1}$ is chosen
to be small enough, then there is a unique
$\theta_{0}\in\big{(}\frac{1}{2}+O(\Delta/t)\big{)}J\subseteq\frac{3}{5}J$
with $h^{\prime}(\theta_{0})=0$. Also,
$|\theta_{0}-\theta_{\Delta}|\lesssim\Delta/t$, and thus

$\displaystyle|h(\theta_{0})-h(\theta_{\Delta})|\lesssim
t|\theta_{0}-\theta_{\Delta}|\lesssim\Delta.$

Since $|h(\theta_{\Delta})|\leq\Delta$, we also have
$|h(\theta_{0})|\lesssim\Delta$.

For Part 1b, write $J=[a,b]$. Using $|h(\theta_{0})|\lesssim\Delta$, for any
$\theta\in[\theta_{0},b]$ we have

$\displaystyle|h(\theta)|$
$\displaystyle\geq\left|\int_{\theta_{0}}^{\theta}h^{\prime}(s)ds\right|-C\Delta=\int_{\theta_{0}}^{\theta}\left|h^{\prime}(s)\right|ds-C\Delta$
$\displaystyle=\int_{\theta_{0}}^{\theta}\int_{\theta_{0}}^{s}h^{\prime\prime}(u)duds-C\Delta\geq
ct(\theta-\theta_{0})^{2}-C\Delta.$ (3.4)

Here $C,c$ are absolute constants. The same is true for
$\theta\in[a,\theta_{0}]$. If $c_{1}$ is small enough, we then have
$h(\theta)\gtrsim t$ for every $\theta\in J\backslash(\frac{4}{5}J)$. The
statements about the zeros follow from the intermediate value theorem and a
similar application of (3.4).

Now we come to Part 2. First, for Part 2a, if $\Delta>c_{1}t$ then the result
is trivial. If $\Delta\leq c_{1}t$, then the argument of Part 1 applies. Let
$\alpha=\inf E_{\delta}$ and $\beta=\sup E_{\delta}$, and hence
$E_{\delta}\subseteq[\alpha,\beta]$. Using (3.4) we have $\beta-\theta_{0}\leq
C\sqrt{(\Delta+\delta)/t}$ and $\theta_{0}-\alpha\leq
C\sqrt{(\Delta+\delta)/t}$. Thus the result follows.

For Part 2b, we first note that the alternative (L) cannot hold for $h$,
otherwise it will contradict the assumption $\delta<t/(6K)$. Thus (L) holds
for $h^{\prime}$ or $h^{\prime\prime}$. In either case, we see $E_{\delta}$ is
a closed interval or the union of two closed intervals.

For the length estimate, we first consider the easier case $\Delta>c_{1}t$. In
this case, if $\delta>c_{1}t/2$, then the result is trivial since then
$\delta\sim\Delta\sim t$. If $\delta\leq c_{1}t/2$, then for $\theta\in
E_{\delta}$ we have

$c_{1}t<\Delta\leq\min_{\theta\in
E_{\delta}}\big{(}|h(\theta)|+|h^{\prime}(\theta)|\big{)}\leq\delta+\min_{\theta\in
E_{\delta}}|h^{\prime}(\theta)|,$

and thus $|f^{\prime}(\theta)|\gtrsim t$ on $E_{\delta}$, and thus each
interval constituting $E_{\delta}$ has length
$O(\delta/t)=O(\delta/\sqrt{(\Delta+\delta)t})$.

For the harder case $\Delta\leq c_{1}t$, the argument of Part 1 is applicable.
We also have two subcases. First, if $\delta\geq c_{1}\Delta$, then the result
follows from Part 2a, since then
$\sqrt{(\Delta+\delta)/t}\sim\delta/\sqrt{(\Delta+\delta)t}$.

If $\delta<c_{1}\Delta$, then we may further assume $\theta_{0}\in J/2$.
Otherwise, using (3.4) we have $h(\theta)\gtrsim t$ for every $\theta\in J/4$
if $c_{1}$ is small enough. In particular, $E_{\delta}=\varnothing$ and we
have nothing to prove. Thus we have $\theta_{0}\in J/2$.

But by the definition of $\Delta$ and $\theta_{0}$, we have
$|h(\theta_{0})|=|h(\theta_{0})|+|h^{\prime}(\theta_{0})|\geq\Delta$. Thus for
every $\theta\in E_{\delta}\cap[\theta_{0},b]$, we have

$\displaystyle\delta$
$\displaystyle\geq|h(\theta)|\geq\Delta-\int_{\theta_{0}}^{\theta}|h^{\prime}(s)|ds$
$\displaystyle\geq\Delta-\int_{\theta_{0}}^{\theta}\int_{\theta_{0}}^{s}|h^{\prime\prime}(u)|duds\geq\Delta-C^{\prime}t(\theta-\theta_{0})^{2}.$

Here $C^{\prime}$ is an absolute constant. Thus we have
$|\theta-\theta_{0}|\lesssim\delta/\sqrt{(\Delta+\delta)t}$. For $\theta\in
E_{\delta}\cap[a,\theta_{0}]$ the argument is the same. This implies that

$|\theta-\theta_{0}|\gtrsim\sqrt{\Delta/t},\quad\forall\theta\in E_{\delta}.$
(3.5)

Since $h^{\prime\prime}(\theta)\gtrsim t$ in $J$, this further implies that
for $\theta\in E_{\delta}$

$|h^{\prime}(\theta)|=|h^{\prime}(\theta)-h^{\prime}(\theta_{0})|\gtrsim
t|\theta-\theta_{0}|\gtrsim\sqrt{t\Delta}.$ (3.6)

Thus each interval constituting $E_{\delta}$ has length
$O(\delta/\sqrt{t\Delta})=O(\delta/\sqrt{(\Delta+\delta)t})$.

Now we come to Part 2c. Let $I$ be the interval constituting $E_{\delta}$ that
contains $\tilde{\theta}$, and let $\theta_{I}$ be an endpoint of $I$ so that
$|h(\theta_{I})|=\delta$.

If $\Delta>c_{1}t$, then we have

$\delta/2\leq|h(\theta_{I})|-|h(\tilde{\theta})|\lesssim|\theta_{I}-\tilde{\theta}|t,$

and so
$|\theta_{I}-\tilde{\theta}|\gtrsim\delta/t\sim\delta/\sqrt{(\Delta+\delta)t}$.

If $\Delta\leq c_{1}t$, then the argument of Part 1 is applicable. For every
$\theta\in E_{\delta}$, the argument of Part 2a implies that
$|\theta-\theta_{0}|\lesssim\sqrt{(\Delta+\delta)/t}$. Thus

$|h^{\prime}(\theta)|=|h^{\prime}(\theta)-h^{\prime}(\theta_{0})|\lesssim
t|\theta-\theta_{0}|\lesssim\sqrt{(\Delta+\delta)t},\quad\forall\theta\in
E_{\delta}.$

Hence

$\delta/2\leq|h(\theta_{I})|-|h(\tilde{\theta})|\lesssim|\theta_{I}-\tilde{\theta}|\sqrt{(\Delta+\delta)t},$

which implies that
$|\theta_{I}-\tilde{\theta}|\gtrsim\delta/\sqrt{(\Delta+\delta)t}$. In
particular, $|I|\gtrsim\delta/\sqrt{(\Delta+\delta)t}$. ∎

###  3.3. Further geometric estimates

We conclude this section with some simple arguments that relate the quantities
$\|f-g\|$, $\Delta(f,g)$, the volume of $f^{\delta}\cap g^{\delta}$, and the
number of tangency rectangles associated to the intersection.

######  Lemma 3.9.

Let $f,g\in\mathcal{F}$ and let $t>0$. If there is a $(\delta,t)$-rectangle
$R$ over the interval $J/4$ (i.e. the interval $I$ from Definition 3.1 is
contained in $J/4$) that is tangent to both $f$ and $g$, then
$(\Delta(f,g)+10\delta)\|f-g\|\lesssim\delta t$.

###### Proof.

Let $R$ be a $(\delta,t)$-rectangle over an interval $I\subseteq J/4$. By
Definition 3.3 and the triangle inequality, for each $\theta\in I$ we have
$|f(\theta)-g(\theta)|\leq 10\lambda\delta$. Since $I\subseteq J/4$, by Part
2b of Lemma 3.8, we have

$\sqrt{\delta/t}=|I|\lesssim\frac{\delta}{\sqrt{(\Delta(f,g)+10\delta)\|f-g\|}}.\qed$

######  Lemma 3.10.

Let $I\subseteq J/4$ be an interval, let $\lambda\geq 1$, let
$f,g\in\mathcal{F},$ and let $0<\delta\leq\|f-g\|/(6K)$. If
$|f(\theta)-g(\theta)|\leq\delta$ on $I$, then
$|f(\theta)-g(\theta)|\lesssim\lambda^{2}\delta$ on $\lambda I\cap J$.

###### Proof.

Let $k=f-g$, let $t=\|f-g\|$, and let $\Delta=\Delta(f,g)$. Since $I\subseteq
J/4$, by Part 2b of Lemma 3.8 we have
$|I|\lesssim\delta/\sqrt{(\Delta+\delta)t}\leq\sqrt{\delta/t}$.

We will first show that $|k^{\prime}(\theta)|\leq C\lambda\delta|I|^{-1}$ on
$\lambda I\cap J$, for some $C\lesssim 1$. Suppose not. Then there is some
$\theta_{1}\in\lambda I\cap J$ with
$|k^{\prime}(\theta_{1})|>C\lambda\delta|I|^{-1}$. We have
$|k^{\prime\prime}(\theta)|\leq t$ in $J$. Since $|I|\lesssim\sqrt{\delta/t}$,
for any $\theta\in\lambda I\cap J$ we have

$|k^{\prime}(\theta)|\geq C\lambda\delta|I|^{-1}-\lambda t|I|\geq
C\lambda\delta|I|^{-1}/2,$

provided $C\sim 1$ is chosen appropriately. But since $k\in C^{2}(I)$ and
$\lambda>1$, if $C>4$ then it is impossible for $|k(\theta)|\geq
C\lambda\delta|I|^{-1}/2$ and $|k(\theta)|\leq\delta$ to simultaneously hold
for all $\theta\in I$. Hence
$|k^{\prime}(\theta)|\lesssim\lambda\delta|I|^{-1}$ on $\lambda I\cap J$. By
the triangle inequality, we thus have $|k(\theta)|\lesssim\lambda^{2}\delta$
on $\lambda I\cap J$. ∎

######  Definition 3.11.

For $R=f^{\delta}(I)$ where $I\subseteq J$, we define the $\lambda$-dilation
of $R$ to be

$\lambda R=f^{\lambda\delta}(\lambda I\cap J).$ (3.7)

We will record several technical lemmas related to comparability.

######  Lemma 3.12.

Let $f,g\in\mathcal{F}$ and $\lambda\geq 1$. Let $R=f^{\delta}(I)$ and
$R^{\prime}=g^{\delta}(I^{\prime})$ be two $\lambda$-comparable
$(\delta,t)$-rectangles over $J/4$. Let $I^{\prime\prime}$ be the convex hull
of $I\cup I^{\prime\prime}$. Then
$|I^{\prime\prime}|\leq\sqrt{\lambda\delta/t}$, and

$|f(\theta)-g(\theta)|\lesssim\lambda^{3}\delta\quad\textrm{for all}\
\theta\in I^{\prime\prime}.$ (3.8)

In particular, $R^{\prime}\subseteq C\lambda R$ for some $C\sim 1$.

###### Proof.

By definition of comparability, there is some $(\lambda\delta,t)$-rectangle
$R^{\prime\prime}=h^{\lambda\delta}(I_{0})$ such that $R\cup
R^{\prime}\subseteq R^{\prime\prime}$ and $|I_{0}|=\sqrt{\lambda\delta/t}$.
Since $I^{\prime\prime}\subset I_{0}$, we have
$|I^{\prime\prime}|\leq\sqrt{\lambda\delta/t}$.

It remains to prove (3.8). If $\|f-g\|<12K\lambda\delta$ then (3.8) is
immediate and we are done.

Next, suppose $\|f-g\|\geq 12K\lambda\delta.$ We have

$|f(\theta)-g(\theta)|\leq 2\lambda\delta\quad\textrm{for all}\ \theta\in I.$

Using Lemma 3.10 (with $2\lambda\delta$ in place of $\delta$; note that
$2\lambda\delta\leq\|f-g\|/(6K)$), we have

$|f(\theta)-g(\theta)|\lesssim\lambda^{3}\delta\quad\textrm{for all}\
\theta\in I^{\prime\prime}.\qed$

######  Corollary 3.13.

Let $R_{1},R_{2},R_{3}$ be $(\delta,t)$ rectangles over $J/4$. Suppose that
$R_{1}$ is $\lambda_{1}$-comparable to $R_{2}$ and $R_{2}$ is
$\lambda_{2}$-comparable to $R_{3}$. Then $R_{1}$ is $\lambda_{3}$-comparable
to $R_{3}$, for some $\lambda_{3}$ depending on $K$, $\lambda_{1},$ and
$\lambda_{2}$.

######  Lemma 3.14.

Let $0<c<1$. If $C$ is large enough depending on $c$ and $K$, then the
following holds. Let $f,g\in\mathcal{F}$ and $R=f^{\delta}(I)$ and
$R^{\prime}=g^{\delta}(I^{\prime})$ be two $C$-incomparable
$(\delta,t)$-rectangles over subintervals of $J/4$. For $0<c<1$, let
$\tilde{R}=f^{\delta}(\tilde{I})$ and
$\tilde{R}^{\prime}=g^{\delta}(\tilde{I}^{\prime})$ where $\tilde{I}=cI$,
$\tilde{I}^{\prime}=cI^{\prime}$. Then $\tilde{R}$ and $\tilde{R}^{\prime}$
are $100$-incomparable.

###### Proof.

Suppose towards contradiction that $\tilde{R}$ and $\tilde{R}^{\prime}$ are
$100$-comparable. Then

$\mathrm{dist}(c_{I},c_{I^{\prime}})\leq\sqrt{100c\delta/t},$

and thus $\mathrm{diam}(I\cup I^{\prime})\lesssim\sqrt{\delta/t}$. Also, we
have $|f(\theta)-g(\theta)|\leq 100\delta$ for $\theta\in\tilde{I}$, and thus
by Lemma 3.12, we have $|f(\theta)-g(\theta)|\lesssim\delta$ for $\theta\in
I\cup I^{\prime}$. Thus $R,R^{\prime}$ are $C$-comparable for a sufficiently
large $C$, from which a contradiction arises. ∎

######  Lemma 3.15.

Let $\mathcal{R}$ be a family of pairwise $100$-incomparable
$(\delta,t)$-rectangles $R=f^{\delta}(I)$ such that all $f\in\mathcal{F}$ are
contained in a ball $B\subset C^{2}(I)$ of radius $3t$. Suppose on the other
hand that all $R\in\mathcal{R}$ are contained in a
$(\lambda\delta,t)$-rectangle $\tilde{R}$ where $\lambda\geq 100$. Then
$\\#\mathcal{R}\lesssim\lambda^{5/2}$.

###### Proof.

We will show that every point in $\mathbb{R}^{2}$ can be contained in
$\lesssim\lambda$ many rectangles $R\in\mathcal{R}$. Then the lemma follows
from the simple inequality

$(\\#\mathcal{R})\delta^{3/2}t^{-1/2}\lesssim\int_{\tilde{R}}\sum_{R\in\mathcal{R}}{\bf
1}_{R}\lesssim\lambda|\tilde{R}|\lesssim\lambda^{5/2}\delta^{3/2}t^{-1/2}.$

Suppose there is some $(\theta_{0},y_{0})\in\mathbb{R}^{2}$ that is contained
in $N$ pairwise $100$-incomparable $(\delta,t)$-rectangles $R_{i},$
$i=1,\ldots,N$. We want to show $N\lesssim\lambda$.

It suffices to show that for every pair $1\leq i\neq j\leq N$,

$\displaystyle|f^{\prime}_{i}(\theta_{0})-f^{\prime}_{j}(\theta_{0})|\leq
10\lambda\sqrt{\delta t},$ (3.9)
$\displaystyle|f^{\prime}_{i}(\theta_{0})-f^{\prime}_{j}(\theta_{0})|>\sqrt{\delta
t}.$ (3.10)

Suppose $(\theta_{0},y_{0})\in R_{i}\cap R_{j}$. Denote
$h=h_{i,j}=f_{i}-f_{j}$ and $I=I_{i}\cup I_{j}$. Then $I$ is an interval of
length $\sqrt{\delta/t}\leq|I|\leq 2\sqrt{\delta/t}$ and
$|h(\theta_{0})|\leq\delta$.

For (3.9), suppose $|h^{\prime}(\theta_{0})|>10\lambda\sqrt{\delta t}$. Then
since $\|f_{i}-f_{j}\|\leq 6t$, we have in particular $\lVert
h^{\prime\prime}\rVert_{\infty}\leq 6t$, and thus for every $\theta\in I$, we
have

$|h^{\prime}(\theta)|>10\lambda\sqrt{\delta t}-12t\sqrt{\delta/t}\geq
5\lambda\sqrt{\delta t},$

since $\lambda\geq 100$. Since $|h(\theta_{0})|\leq\delta$, this implies that
for some $\theta\in I$ we have

$|h(\theta)|\geq 5\lambda\sqrt{\delta
t}\sqrt{\delta/t}/2-\delta>\lambda\delta,$

contradicting the assumption that all rectangles $R_{i}\subseteq\tilde{R}$.
This establishes (3.9).

For (3.10), we argue similarly. Suppose that
$|h^{\prime}(\theta_{0})|\leq\sqrt{\delta t}$. Then using $\lVert
h^{\prime\prime}\rVert_{\infty}\leq 6t$, we have for every $\theta^{\prime}\in
I$

$|h^{\prime}(\theta)|\leq\sqrt{\delta t}+12t\sqrt{\delta/t}\leq 13\sqrt{\delta
t}.$

Thus for every $\theta\in I$ we have

$|h(\theta)|\leq|h(\theta_{0})|+26\sqrt{\delta t}\sqrt{\delta/t}\leq
100\delta,$

contradicting the assumption that $R_{i},R_{j}$ are $100$-incomparable. This
establishes (3.10). ∎

######  Lemma 3.16.

Let $\mathcal{R}$ be a finite family of pairwise $100$-incomparable
$(\delta,t)$-rectangles $R=f^{\delta}(I)$ such that all $f\in\mathcal{F}$ are
contained in a ball of radius $3t$. If $C\geq 100$, then there is a
$C$-incomparable subcollection $\mathcal{R}^{\prime}\subseteq\mathcal{R}$ such
that $\\#\mathcal{R}^{\prime}\sim\\#\mathcal{R}$.

###### Proof.

We select $\mathcal{R}^{\prime}$ in a greedy way. Start with any
$R_{1}\in\mathcal{R}$, and choose a maximally $C$-incomparable subcollection
$\mathcal{R}_{1}$ containing $R_{1}$. If $\mathcal{R}_{1}=\mathcal{R}$ then we
are done. Otherwise, start with any
$R_{2}\in\mathcal{R}\backslash\mathcal{R}_{1}$, and choose a maximally
$\lambda$-incomparable subcollection $\mathcal{R}_{2}$ containing $R_{2}$.
Note that each $R\in\mathcal{R}_{2}$ is $C$-comparable to every
$R\in\mathcal{R}_{1}$. If $\mathcal{R}_{1}\cup\mathcal{R}_{2}=\mathcal{R}$,
then either $\mathcal{R}_{1}$ or $\mathcal{R}_{2}$ will have cardinality at
least $\\#\mathcal{R}/2$.

Continue in this fashion until it stops at some finite time $N$, which must
occur since $\\#\mathcal{R}$ is finite. Our goal is to show that $N=O(1)$,
from which it follows that at least one of $\mathcal{R}_{i}$, $1\leq i\leq N$
has cardinality $\sim\\#\mathcal{R}$. Call this
$\mathcal{R}_{i}=\mathcal{R}^{\prime}$ and we are done.

Pick any $R_{i}=f_{i}^{\delta}(I_{i})\in\mathcal{R}_{i}$ for each $1\leq i\leq
N$. By construction, they must be mutually $C$-comparable. This implies that
the $\cup_{i}I_{i}$ is contained in an interval $I$ of length at most
$\sqrt{C\delta/t}$, and by Lemma 3.12 there is an $(O(\delta),t)$-rectangle
$R$ that contains all $R_{i}$. Applying Lemma 3.15 with $\lambda=C$ gives
$N=O(1)$, as required. ∎

We end this section with a coarse bound that will be used in some dyadic
pigeonholing argument later.

######  Lemma 3.17.

Let $\lambda\geq 100$. If $\mathcal{R}$ is a collection of pairwise
$\lambda$-incomparable $(\delta,t)$-rectangles of the form $R=f^{\delta}(I)$
where $f\in\mathcal{F}$, then we have a coarse bound
$\\#\mathcal{R}\leq\delta^{-C}$ for some $C=C(D)$ for $\delta$ sufficiently
small (depending on $K$).

###### Proof.

Let $R=f^{\delta}(I)$ and $R^{\prime}=g^{\delta}(I^{\prime})$ where
$f,g\in\mathcal{F}$. It is easy to see that if the centres of $I,I^{\prime}$
are no more than $\sqrt{\delta/t}$-separated and $\|f-g\|\leq\delta$, then
$R,R^{\prime}$ must be $\lambda$-comparable. The result now follows from the
fact that $\mathcal{F}$ has diameter $K$ and doubling constant $D$, and
choosing $\delta\leq K^{-1}$ sufficiently small. ∎

##  4\. Lens counting and a bipartite curve-rectangle tangency bound

In this section, we will use Marcus and Tardos’ lens-counting theorem from
[17] to prove a bipartite version of the weak $\ell^{3/2}$ bound (1.16) from
Section 1.3. The precise statement is as follows. We remark that all
propositions in this section make no assumption on the doubling property in
the definition of cinematic functions.

######  Proposition 4.1.

Let $\mathcal{F}$ be a family of cinematic functions with cinematic constant
$K$. Then there is a constant $C=C(K)>0$ so that the following holds.

Let $A\geq 1$. Let $\delta,t>0$ with $0<\delta\leq At$. Let
$\mathcal{W},\mathcal{B}\subset\mathcal{F}$ be finite sets. Suppose
$\mathcal{W}\cup\mathcal{B}$ has diameter at most $6t,$ and the pair
$(\mathcal{W},\mathcal{B})$ is $t/A$-separated.

Let $\mathcal{R}$ be a collection of pairwise $100$-incomparable
$(\delta,t)$-rectangles of the form $f^{\delta}(I)$ where $f\in\mathcal{F}$
and $I\subseteq J/4$. Suppose each rectangle in $\mathcal{R}$ is tangent to at
least $\mu$ elements of $\mathcal{W}$ and at least $\nu$ elements of
$\mathcal{B}$. Then

$\\#\mathcal{R}\leq
CA^{C}\left(\frac{\\#\mathcal{W}}{\mu}+\frac{\\#\mathcal{B}}{\nu}\right)^{3/2}\log\left(\frac{\\#\mathcal{W}}{\mu}+\frac{\\#\mathcal{B}}{\nu}\right).$
(4.1)

###  4.1. Preliminary reductions

######  Lemma 4.2.

Let $c>0$. Suppose there exists a constant $C_{1}=C_{1}(K)>0$ so that
Proposition 4.1 holds whenever $\delta\leq ct$. Then there is a constant
$C=C(C_{1},K,c)$ so that Proposition 4.1 holds for all $\delta\leq At$.

###### Proof.

If $\delta\leq ct$ then we are done. Suppose instead that $\delta>ct$, and let
$\mathcal{W}$, $\mathcal{B}$, $\mathcal{R}$, $\mu$, and $\nu$ be as in the
statement of the Proposition. Let $C_{1}\sim 1$ be a constant to be specified
below. Apply Lemma 3.16 to find a subcollection
$\mathcal{R}^{\prime}\subseteq\mathcal{R}$ that is pairwise
$C_{1}$-incomparable, with $\\#\mathcal{R}^{\prime}\sim\\#\mathcal{R}$. (By
Lemma 3.17 we may suppose that $\mathcal{R}$ is finite.) Next, shrink the
length of each $R\in\mathcal{R}^{\prime}$ to be $\sqrt{\delta/t^{\prime}}$
where $t^{\prime}=c^{-1}\delta>t$, and so $\delta=ct^{\prime}$. Denote by
$\mathcal{R}^{\prime\prime}$ the resulting collection of
$(\delta,t^{\prime})$-rectangles. If $C_{1}$ is chosen large enough, then
using Lemma 3.14, we have that $\mathcal{R}^{\prime\prime}$ is
$100$-incomparable. Applying Proposition 4.1 with $t^{\prime}$ in place of
$t$, we are done. ∎

Similar arguments yield the following.

######  Lemma 4.3.

Suppose there exists a constant $C_{2}=C_{2}(K)>0$ so that Proposition 4.1
holds whenever $(\mathcal{W},\mathcal{B})$ is $2t$-separated. Then there is a
constant $C=C(C_{2},K)$ so that Proposition 4.1 holds when
$(\mathcal{W},\mathcal{B})$ is $t/A$-separated.

######  Lemma 4.4.

Suppose there exists a constant $C_{3}=C_{3}(K)$ so that Proposition 4.1 holds
whenever $\mathcal{R}$ is $2C_{c}$-incomparable, where $C_{c}$ is as in Lemma
4.9 below. Then there is a constant $C=C(C_{3},K)$ so that Proposition 4.1
holds when $\mathcal{R}$ is $100$-incomparable.

From now on we will assume $0<\delta\leq c_{2}t$ where $c_{2}=c_{2}(K)$ will
be determined in Lemma 4.9 below. We also assume $(\mathcal{W},\mathcal{B})$
is $2t$-separated, and that $\mathcal{R}$ is $2C_{c}$-incomparable.

_Remark._ In the arguments below, we will choose a few constants depending on
$K$. The order of dependence is as follows: $c_{1}$ as in Lemma 3.8; $\lambda$
as in Lemma 4.12; $c_{2}$ and $C_{c}$ as in Lemma 4.9.

###  4.2. Pseudo-circles and lenses

Now we introduce some terminology from Marcus-Tardos [17] that will be used in
the proof of Proposition 4.1.

######  Definition 4.5.

Let $\mathcal{C}$ be a family of closed Jordan plane curves (i.e. the
homeomorphic image of $S^{1}$). We say $\mathcal{C}$ forms a family of pseudo-
circles if for every pair of distinct $c,c^{\prime}\in\mathcal{C}$, the
following conditions are satisfied:

  1. (1)

$\\#(c\cap c^{\prime})\leq 2$.

  2. (2)

If $p\in c\cap c^{\prime}$, then $c$ and $c^{\prime}$ intersect properly at
$p$. That is, for all sufficiently small circles $C(p)$ centered at $p$, the
points in $C(p)\cap c$ and $C(p)\cap c^{\prime}$ appear alternatively
cyclically counterclockwise. (See Figure 1 below).

$c$$c^{\prime}$$c$$c$$c^{\prime}$$c^{\prime}$$c\rightarrow
c^{\prime}\rightarrow c\rightarrow c^{\prime}$proper
intersection$c^{\prime}$$c$$c$$c$$c^{\prime}$$c^{\prime}$$c\rightarrow
c^{\prime}\rightarrow c^{\prime}\rightarrow c$improper
intersection$c^{\prime}$$c$$c$$c$$c^{\prime}$$c^{\prime}$$c\rightarrow
c^{\prime}\rightarrow c\rightarrow c^{\prime}$proper intersection Figure 1.
Proper and improper intersection.

The first condition is key. The second condition was not explicitly stated in
[17], so we included it here for completeness. It is a technical assumption
included to make our settings fit rigorously to the discrete geometry scenario
in [17]. Of course, in the Kakeya type maximal inequality, if two functions
$f,g$ are exactly tangent at some point, then the intersection may be
improper. To avoid this issue, we will slightly perturb the cinematic
functions $f$ by $O(\delta)$, such that no exact tangency could occur. This
will do no harm, since if $f$ and $g$ satisfy (1.6) and also $\|f-g\|\geq
C\delta$ for a sufficiently large constant $C$, then $f+O(\delta)$ and $g$
will continue to satisfy (1.6) (with $K$ replaced by $2K$). See Section 4.5
for the details of this perturbation process.

Now we recall the definition of lenses from [17].

######  Definition 4.6.

Let $\mathcal{C}$ be a set of pseudo-circles. Let $c,c^{\prime}\in\mathcal{C}$
be two curves that intersect exactly twice, at the points $p,p^{\prime}$. This
divides $c,c^{\prime}$ into four closed segments, that intersect only at their
endpoints. We say $\ell\subset c\cup c^{\prime}$ is a _lens_ if it is a simple
closed curve that is the union of two of these segments.

Note that there are exactly $\binom{4}{2}=6$ lenses formed by $c,c^{\prime}$;
two of these are the curves $c$ and $c^{\prime}$ themselves. See [17, Figure
1]. As a technical aside, we only consider “first-generation lenses”, that is,
we ignore lenses formed by lenses. Thus the total number of lenses determined
by an arrangement $\mathcal{C}$ is at most
$4\binom{\\#\mathcal{C}}{2}+\\#\mathcal{C}$.

######  Definition 4.7 (Overlapping lenses).

Let $\mathcal{C}$ be a family of pseudo-circles, and let $\ell,\ell^{\prime}$
be two lenses formed by $\mathcal{C}$. We say $\ell,\ell^{\prime}$ _overlap_
if they share a segment of positive length (see Figure 2).

Note that if two lenses intersect (properly) at only finitely many points,
then they do not overlap.

non-overlapping lensesoverlapping lenses Figure 2. Overlapping and non-
overlapping lenses.

We can now state Lemma 10 from [17].

######  Theorem 4.8.

Let $\mathcal{C}$ be a family of $n$ pseudo-circles. Then every set of non-
overlapping lenses has cardinality $O(n^{3/2}\log n)$.

###  4.3. Lenses and pseudo-parabolas

We return to our case of a finite family $F\subset\mathcal{F}$ of cinematic
functions. Their graphs are not yet pseudo-circles because they are not closed
curves. However, in view of Lemma 3.6, we can extend them to be pseudo-circles
in the following way.

We need a mild technical assumption first. Denote $J=[\alpha,\beta]$. We
require that both sets

$\\{f(\alpha):f\in F\\},\quad\\{f(\beta):f\in F\\}$

consist of distinct numbers, respectively. We also need that a pair of
different cinematic functions $f,g$ never intersect exactly tangentially at
any point in $J$. As mentioned before, this can be guaranteed by a careful
perturbation; see Section 4.5 for details. Also, we note that all curves are
confined in $J\times[-M,M]$ for a finite $M$. We index the curves by
$f_{j},1\leq j\leq\\#F$, and rearrange so that $f_{j}(\alpha)$ is increasing.
We first extend each $f_{j}(\theta)$ to be equal to $f_{j}(\alpha)$ for
$\theta\in[\alpha-j,\alpha]$ and equal to $f_{j}(\beta)$ for
$\theta\in[\beta,\beta+j]$. We then extend by adding the line segments
$\\{\alpha-j\\}\times[-M-j,f_{j}(\alpha)]$ and
$\\{\beta+j\\}\times[-M-j,f_{j}(\beta)]$. Finally, close the loop by adding
the line segment $[\alpha-j,\beta+j]\times\\{-M-j\\}$. In this way we have
extended the graphs of the functions in $F$ to a family $\mathcal{C}$ of
pseudo-circles, with all intersections proper. See Figure 3 below.

$\alpha$$\beta$ Figure 3. Extending the graph of $f\in F$ to a closed curve.

We observe that if two functions $f,g$ intersect exactly twice in the strip
$J\times\mathbb{R}$, then they give rise to exactly one lens $\ell$ that lies
entirely in the strip $J\times\mathbb{R}$. See Figure 3. Denote the collection
of these lenses by $\mathcal{L}$. They are the only lenses we are interested
in.

Now we explain the connection between overlapping lenses and curve tangencies.

######  Lemma 4.9.

If $\delta/t\leq c_{2}\leq c_{1}$ (where $c_{1}$ is as in Part 1 of Lemma 3.8)
for some small enough constant $c_{2}=c_{2}(K)$, then the following holds. Let
$w,b_{1},b_{2}\in\mathcal{F}$. For $i=1,2$, let $t_{i}=\|w-b_{i}\|$. Suppose
$t_{i}\geq t$, and that $w$ and $b_{i}$ intersect exactly twice in $J$, thus
forming a lens $\ell_{i}$ that lies over an interval
$[\alpha_{i},\beta_{i}]\subseteq J$. Let $R_{i}$ be a $(\delta,t)$-rectangle
that is $2\lambda$-tangent to both $w$ and $b_{i}$, where $\lambda=\lambda(K)$
is to be determined in Lemma 4.12.

Suppose that that $\ell_{1}$ and $\ell_{2}$ overlap. Then $R_{1}$ and $R_{2}$
are $C_{c}$-comparable, where $C_{c}$ ($c$ stands for “comparable”) depends on
$\lambda,K$.

###### Proof.

For $i=1,2$, define the interval $I_{i}=[\alpha_{i},\beta_{i}]\subset J$ so
that $R_{i}$ is a rectangle over $I_{i}$. Since $R_{1}$, $R_{2}$ are both
tangent to $w$, it suffices to show that $\mathrm{diam}(I_{1}\cup
I_{2})=O(\sqrt{\delta/t})$.

By Lemma 3.9, we have $\Delta(w,b_{i})\lesssim\delta$. If $\delta/t$ is small
enough, then $\Delta(w,b_{i})\leq c_{1}t$. Thus there is a unique
$\theta^{(i)}_{0}\in\frac{3}{5}J$ such that
$|\alpha_{i}-\theta^{(i)}_{0}|=O(\sqrt{\delta/t_{i})}=O(\sqrt{\delta/t)}$ and
$|\beta_{i}-\theta^{(i)}_{0}|=O(\sqrt{\delta/t})$. Since
$(\alpha_{1},\beta_{1})\cap(\alpha_{2},\beta_{2})\neq\varnothing$, we have
$\theta^{(2)}_{0}-\theta^{(1)}_{0}=O(\sqrt{\delta/t})$.

Now we study $I_{i}$. Using the triangle inequality, for $\theta\in I_{i}$ we
have $|w(\theta)-b_{i}(\theta)|\lesssim\delta$. By Part 2a of Lemma 3.8,
$I_{i}$ is contained in an interval of length $O(\sqrt{\delta/t})$ centered at
$\theta^{(i)}_{0}$. But since
$\theta^{(2)}_{0}-\theta^{(1)}_{0}=O(\sqrt{\delta/t})$, we see that
$\mathrm{diam}(I_{1}\cup I_{2})=O(\sqrt{\delta/t})$, as required. ∎

###  4.4. Rectangle tangency bounds for properly intersecting curves

We first prove a special case of Proposition 4.1, where the curves intersect
properly.

######  Lemma 4.10.

Let $\mathcal{F}$ be a family of cinematic functions, with cinematic constant
$K$. Then there is a constant $C=C(K)>0$ so that the following holds.

Let $\delta,t>0$ with $0<\delta\leq c_{2}t$ (here $c_{2}$ is the quantity from
Lemma 4.9). Let $\mathcal{W},\mathcal{B}\subset\mathcal{F}$ be a pair of
$t$-separated finite sets.

Let $\mathcal{R}$ be a collection of pairwise $C_{c}$-incomparable
$(\delta,t)$-rectangles of the form $f^{\delta}(I)$, where $f\in\mathcal{F}$,
$I\subseteq J/4$ and $C_{c}$ is as in Lemma 4.9. Suppose each rectangle in
$\mathcal{R}$ is tangent to at least one $w_{R}\in\mathcal{W}$ and at least
one $b_{R}\in\mathcal{B}$. Suppose furthermore that the following holds.

  1. A1.

For each $R\in\mathcal{R}$, we have that $w_{R},b_{R}$ intersect exactly twice
in $J$.

  2. A2.

If $J=[\alpha,\beta]$, then the sets

$\\{f(\alpha):f\in\mathcal{W}\cup\mathcal{B}\\},\quad\\{f(\beta):f\in\mathcal{W}\cup\mathcal{B}\\}$

respectively consist of distinct elements.

  3. A3.

For all $f\neq g\in\mathcal{W}\cup\mathcal{B}$, $f,g$ never intersect
tangentially in $J$, that is, the two quantities $f(\theta)-g(\theta)$ and
$f^{\prime}(\theta)-g^{\prime}(\theta)$ are never simultaneously $0$ for any
$\theta\in J$.

Then

$\\#\mathcal{R}\leq
C\left(\\#\mathcal{W}+\\#\mathcal{B}\right)^{3/2}\log\left(\\#\mathcal{W}+\\#\mathcal{B}\right).$
(4.2)

###### Proof.

With the additional assumptions A1–A3, each $R\in\mathcal{R}$ is tangent to a
pair $(w_{R},b_{R})$ intersecting exactly twice in $J$. After extending them
to pseudo-circles in the way described in Section 4.3, we see the pair
$(w_{R},b_{R})$ gives rise to a lens $\ell_{R}$ in the collection
$\mathcal{L}$.

By Lemma 4.9, the lenses in $\mathcal{L}=\\{\ell_{R}:R\in\mathcal{R}\\}$ are
non-overlapping. Since the mapping $R\mapsto\ell_{R}$ is clearly a bijection,
using Theorem 4.8, we conclude that

$\\#\mathcal{R}=\\#\mathcal{L}\lesssim(m+n)^{3/2}\log(m+n).\qed$

###  4.5. Perturbing the curves

Our next step is to remove the additional assumptions A1–A3 in Lemma 4.10 by
perturbing the functions in $\mathcal{W}$ and $\mathcal{B}$. Namely, we would
like to prove the following version of Proposition 4.1.

######  Lemma 4.11.

Let $\mathcal{F}$ be a family of cinematic functions, with cinematic constant
$K$. Then there is a constant $C=C(K)>0$ so that the following holds.

Let $\delta,t>0$ with $0<\delta\leq c_{2}t/2$ (here $c_{2}$ is the quantity
from Lemma 4.9). Let $\mathcal{W},\mathcal{B}\subset\mathcal{F}$ be a pair of
$2t$-separated finite sets.

Let $\mathcal{R}$ be a collection of pairwise $2C_{c}$-incomparable
$(\delta,t)$-rectangles of the form $f^{\delta}(I)$ where $f\in\mathcal{F}$
and $I\subseteq J/4$. Suppose each rectangle in $\mathcal{R}$ is tangent to at
least one element of $\mathcal{W}$ and at least one element of $\mathcal{B}$.

Then

$\\#\mathcal{R}\leq
C\left(\\#\mathcal{W}+\\#\mathcal{B}\right)^{3/2}\log\left(\\#\mathcal{W}+\\#\mathcal{B}\right).$
(4.3)

The reduction from Lemma 4.10 to Lemma 4.11 requires two perturbations. The
first perturbation (which has size roughly $\delta$) ensures that property A1
holds for a considerable fraction of the tangency rectangles in $\mathcal{R}$.
The second perturbation (which is infinitesimally small) ensures that
properties A2 and A3 hold.

######  Lemma 4.12.

Let $0<\delta<c_{1}t$ and let $f,g\in\mathcal{F}$ with $\|f-g\|\geq 2t$.
Suppose $f$ and $g$ are tangent to a common $(\delta,t)$-rectangle $R$ over
$J/4$. If $\lambda=\lambda(K)\geq 1$ is chosen to be large enough, then there
exists $\eta\in\\{-1,0,1\\}$ so that the function $h=f+\eta\lambda\delta$ is
$2\lambda$-tangent to $R$, and the functions $g$ and $h$ intersect exactly
twice in $J$.

###### Proof.

By Part 1b of Lemma 3.8, it suffices to show that for a suitable
$\eta\in\\{-1,0,1\\}$, we have $k=h-g$ changes sign within $J$. We have three
cases. If $f-g$ changes sign in $J$, then we just take $\eta=0$. Otherwise, we
must have $f-g\geq 0$ on $J$ or $f-g\leq 0$ on $J$. In the former case, we
take $\eta=-1$, and so $k=f-\lambda\delta-g$. Also, by Lemma 3.9, we have
$\Delta(f,g)\lesssim\delta$, and thus $\min_{\theta\in
J}f(\theta_{0})-g(\theta_{0})\leq C^{\prime}\delta$ for some
$C^{\prime}=C^{\prime}(K)$, using Part 1a of Lemma 3.8. If we choose
$\lambda>C^{\prime}$, then we have $\min_{\theta\in J}k(\theta_{0})<0$, and
hence $k$ changes sign in $J$.

In the case $f-g\leq 0$ on $J$, we take $\eta=1$ and the argument is similar.
∎

######  Corollary 4.13.

Let $\delta,t>0$ with $0<\delta\leq c_{2}t/2$. Let
$\mathcal{W},\mathcal{B}\subset\mathcal{F}$ be a pair of $2t$-separated finite
sets.

Let $\mathcal{R}$ be a collection of pairwise $2C_{c}$-incomparable
$(\delta,t)$-rectangles of the form $f^{\delta}(I)$ where $f\in\mathcal{F}$
and $I\subseteq J/4$. Suppose each rectangle in $\mathcal{R}$ is tangent to at
least one $w_{R}\in\mathcal{W}$ and at least one $b_{R}\in\mathcal{B}$.

Then there exists $\eta\in\\{-1,0,1\\}$ and a set
$\mathcal{R}^{\prime}\subset\mathcal{R}$ with the following properties.

  * •

$\\#\mathcal{R}^{\prime}\geq\frac{1}{3}\\#\mathcal{R}$.

  * •

If we define $\mathcal{W}^{\prime}=\\{w+\eta\lambda\delta\colon
w\in\mathcal{W}\\}$, then each rectangle in $\mathcal{R}^{\prime}$ is
$2\lambda$-tangent to at least one $w_{R}\in\mathcal{W}^{\prime}$ and at least
one $b_{R}\in\mathcal{B}$.

  * •

For each $R\in\mathcal{R}^{\prime}$, we have that $w_{R},b_{R}$ intersect
exactly twice in $J$.

We are now ready to prove Lemma 4.11. Applying Lemma 4.13 to $\mathcal{W}$ and
$\mathcal{B}$, we obtain a perturbed set $\mathcal{W}^{\prime}$ and a subset
$\mathcal{R}^{\prime}\subset\mathcal{R}$. We claim that
$\mathcal{W}^{\prime}\cup\mathcal{B}$ is a finite subset of a family
$\mathcal{F}^{\prime}$ that has cinematic constant at most $2K$. Indeed, we
can simply define
$\mathcal{F}^{\prime}=(\mathcal{W}-\eta\lambda\delta)\cup\mathcal{B}$ (recall
that $\mathcal{W}\subset C^{2}(J)$, so
$\mathcal{W}-\eta\lambda\delta=\\{f-\eta\lambda\delta\colon
f\in\mathcal{W}\\}$. It is easy to check that since $\mathcal{W}$ and
$\mathcal{B}$ are $2t$-separated, (1.6) continues to hold with $2K$ in place
of $K$ (the only interesting case to check is when
$f\in\mathcal{W}-\eta\lambda\delta$ and $g\in\mathcal{B}$, or vice-versa).
Also, $\mathcal{W}^{\prime}\cup\mathcal{B}$ is at least $t$ separated, if
$c_{2}$ is chosen to be small enough.

Our collection $\mathcal{W}^{\prime}$, $\mathcal{B},$ and
$\mathcal{R}^{\prime}$ satisfies property A1 from the statement of Lemma 4.10.
After applying a second harmless infinitesimal perturbation, we can ensure
that $\mathcal{W}^{\prime}$, $\mathcal{B},$ and $\mathcal{R}^{\prime}$ satisfy
Properties A2 and A3, while maintaining property A1. Lemma 4.11 now follows
from Lemma 4.10.

Finally, in light of the preliminary reductions given by Lemmas 4.2 and 4.3,
Proposition 4.1 follows from Lemma 4.11 via a standard random sampling
argument. See i.e. the proof of Lemma 1.4 from [27].

##  5\. Proof of Theorem 1.7

We have now assembled the necessary ingredients to execute the proof sketch
described in Section 1.3. We will begin with a few preliminary reductions.
First, if $\mathcal{F}\subset C^{2}(J)$ is a family of cinematic functions,
the results from Sections 3 and 4 allow us to control the functions
$f\in\mathcal{F}$ on a slightly shorter interval $J/4$. This is inconvenient,
since the domain of integration in (1.8) might involve the entire interval
$J$. Our first task is to show that Theorem 1.7 will follow from the following
(superficially) weaker version, in which the domain of integration has been
restricted to a slightly smaller set.

######  Lemma 5.1.

Let $\varepsilon>0$, $0<\alpha\leq\zeta\leq 1,$ and $D,K\geq 1$. Then the
following is true for all $\delta>0$ sufficiently small (depending on $D,K,$
and $\varepsilon$). Let $\mathcal{F}\subset C^{2}(J)$ be a family of cinematic
functions, with cinematic constant $K$ and doubling constant $D$. Let $E$ be a
$(\delta,\alpha;\delta^{-\varepsilon})_{1}\times(\delta,\alpha;\delta^{-\varepsilon})_{1}$
quasi-product.

Let $F\subset\mathcal{F}$ be a set of functions that satisfies the Frostman-
type non-concentration condition

$\\#(F\cap B)\leq\delta^{-\varepsilon}(r/\delta)^{\zeta}\quad\textrm{for all
balls}\ B\subset C^{2}(I)\ \textrm{of radius}\ r\geq\delta.$

Let $J^{\prime}=\frac{1}{16}J$ and let
$E^{\prime}=E\cap(J^{\prime}\times\mathbb{R})$. Then

$\int_{E^{\prime}}\Big{(}\sum_{f\in F}{\bf
1}_{f^{\delta}}\Big{)}^{3/2}\leq\delta^{2-\alpha/2-\zeta/2-C_{0}\varepsilon}(\\#F),$
(5.1)

where $C_{0}=C_{0}(D)>0$ depends on $D$.

Using Lemma 5.1 we can prove Theorem 1.7 as follows. Let $J_{1},\ldots,J_{N}$,
$N\sim|\log\delta|$ be a sequence of sub-intervals of $J$, so that
$J\backslash\bigcup_{i=1}^{N}\frac{1}{16}J_{i}$ consists of two intervals of
length $\delta$ (these intervals will be at the two endpoints of $J$). Note
that if $J_{0}$ is an interval of length $\delta$ and if
$E_{0}=E\cap(J_{0}\times\mathbb{R})$, then

$\int_{E_{0}}\Big{(}\sum_{f\in F}{\bf
1}_{f^{\delta}}\Big{)}^{3/2}\lesssim\delta^{2}(\\#F)^{3/2}\leq\delta^{2-\alpha/2-\zeta/2-C_{0}\varepsilon}(\\#F).$
(5.2)

For each index $i$, apply Lemma 5.1 to the restriction
$\mathcal{F}|_{J_{i}}\subset C^{2}(J_{i})$; this is still a family of
cinematic functions, with cinematic constant $K$ and doubling constant $D$.
Using the triangle inequality, we conclude that

$\int_{E}\Big{(}\sum_{f\in F}{\bf
1}_{f^{\delta}}\Big{)}^{3/2}\lesssim|\log\delta|\delta^{2-\alpha/2-\zeta/2-C_{0}\varepsilon}(\\#F).$

Theorem 1.7 now follows, provided we select $C>C_{0}$ and restrict $\delta>0$
sufficiently small.

Next, we remark that when proving Lemma 5.1, we may suppose that $F$ is
$\delta$-separated. This follows from the triangle inequality and observation
that by (1.7) each $\delta$-ball contains at most $\delta^{-\varepsilon}$
elements of $F$.

Finally, we claim that Lemma 5.1 follows from the following restricted weak-
type estimate. For each $x\in\mathbb{R}^{2}$, let $F(x)=\\{f\in F\colon x\in
f^{\delta}\\}.$ For each $x\in\mathbb{R}^{2}$,
$\\#F(x)\leq\\#F\lesssim\delta^{-\varepsilon-\zeta}$. Thus by dyadic
pigeonholing, there is a set $E_{0}\subset E$ and an integer $\mu$ so that
$\mu\leq\\#F(x)<2\mu$ for all $x\in E_{0}$, and
$\mu^{3/2}|E_{0}|\approx\textrm{LHS of}\ \eqref{eqn_L32Bd_generalEprime}$.
Thus in order to prove Lemma 5.1 (and hence Theorem 1.7), it suffices to
establish the estimate

$|E_{0}|\lessapprox\delta^{2-\frac{\alpha}{2}-\frac{\zeta}{2}-C_{1}\varepsilon}\\#F\mu^{-3/2},$
(5.3)

for some $C_{1}=C_{1}(D)$.

###  5.1. Two ends reductions

In this section, we will find quantities called $t$ and $\Delta$, so that for
a typical point $x\in E_{0},$ the functions in $F(x)$ are localized to a ball
in $C^{2}(J)$ of radius $t$, and for a typical pair of functions $f,g\in
F(x)$, we have $\Delta(f,g)\sim\Delta$. Our main tool will be the “two-ends”
reduction. See [23] for an introduction to the topic.

####  5.1.1. Finding the diameter of $F(x)$

Since $F\subset C^{2}(J)$ is bounded, so is the set $F(x)$ for each $x\in
E_{0}$. However, it is possible that $F(x)$ (or a large fraction thereof) has
diameter much smaller than $\operatorname{diam}(F)$ for a typical $x\in
E_{0}$. In this section, we will find the “true” diameter of $F(x)$ for a
typical $x\in E_{0}$.

For $t>0$ and $x\in\mathbb{R}^{2}$, define

$n_{1}(t,x)=\sup_{g\in C^{2}(J)}\\#F(x)\cap B(g,t),$ (5.4)

where $B(g,t)\subset C^{2}(J)$ is the ball centered at $g\in C^{2}(J)$ of
radius $t$. With this, we further define

$n_{2}(x)=\sup_{t\in[\delta,K]}n_{1}(t,x)t^{-\varepsilon}.$ (5.5)

Hence, given $x$, there is some $t(x)\in[\delta,K]$ such that $n_{2}(x)\leq
2n_{1}(t(x),x)t(x)^{-\varepsilon}$. There is also some $g_{x}$ such that

$\\#F(x)\cap B(g_{x},t(x))\geq n_{2}(t(x),x)/2,$ (5.6)

so we have

$n_{2}(x)\leq 4t(x)^{-\varepsilon}\\#F(x)\cap B(g_{x},t(x)).$ (5.7)

For simplicity we denote $B_{x}=B(g_{x},t(x))$.

Since $t(x)\in[\delta,K]$, by a dyadic pigeonholing, there is a Borel set
$E_{1}\subseteq E_{0}$ and a number $t\in[\delta,2K]$ such that
$t(x)\in(t/2,t]$ for every $x\in E_{1}$, and $|E_{1}|\approx|E_{0}|$. It then
suffices to show

$|E_{1}|\lessapprox\delta^{2-\frac{\alpha}{2}-\frac{\zeta}{2}-C_{1}\varepsilon}\\#F\mu^{-3/2}.$
(5.8)

Since $\mathcal{F}$ is doubling, we can cover it by finitely overlapping balls
$B$ of radius $t$, and denote

$F_{B}=F\cap(3B),$

so $\\{F_{B}\\}$ covers $F$ and is finitely overlapping. Also, let $E_{B}$ be
the set of $x\in E_{1}$ such that $B_{x}$ intersects $B$, and thus
$\\{E_{B}\\}$ is also finitely overlapping. Thus it suffices to show for each
$B$

$|E_{B}|\lessapprox\delta^{2-\frac{\alpha}{2}-\frac{\zeta}{2}-C_{1}\varepsilon}\\#F_{B}\mu^{-3/2}.$
(5.9)

Now fix a ball $B$ of radius $t$ and let $x\in E_{B}$. By (5.7), (5.5) and
(5.4), we have

$\displaystyle\\#F(x)\cap B_{x}\gtrsim t^{\varepsilon}n_{2}(x)\gtrsim
t^{\varepsilon}n_{1}(K,x)=t^{\varepsilon}\\#F(x)\sim t^{\varepsilon}\mu.$

By another dyadic pigeonholing, we may find a subset $E^{\prime}_{B}\subseteq
E_{B}$ and a scale $\mu_{1}$ with

$t^{\varepsilon}\mu\lesssim\mu_{1}\leq\mu,$ (5.10)

such that $|E^{\prime}_{B}|\approx|E_{B}|$ and for every $x\in E^{\prime}_{B}$
we have $\\#F(x)\cap B_{x}\sim\mu_{1}$. It suffices to show (5.9) with $E_{B}$
replaced by $E^{\prime}_{B}$.

Now let $x\in E^{\prime}_{B}$. If $B$ intersects $B_{x}$ then we have
$B_{x}\subseteq 3B$. Hence

$\\#F_{B}(x)=\\#F(x)\cap(3B)\cap B_{x}=\\#F(x)\cap B_{x}\sim\mu_{1}.$ (5.11)

From now on we fix such a ball $B$ and aim to prove (5.9). We end with a
useful lemma.

######  Lemma 5.2.

Fix $x\in E^{\prime}_{B}$ and $g\in C^{2}(J)$. If $\delta t^{-1}<\lambda<1$,
then for at most $4\cdot(2\lambda)^{\varepsilon}$ fraction of $f\in F_{B}(x)$
we have $\|f-g\|\leq\lambda t$.

###### Proof.

Let $g\in C^{2}(J)$. By (5.7) and (5.11), we have

$n_{2}(x)\leq 4t(x)^{-\varepsilon}\\#F(x)\cap B_{x}\leq 4\cdot
2^{\varepsilon}t^{-\varepsilon}\\#F_{B}(x).$

But by the definition of $n_{2}$ in (5.5) and $\lambda t\in[\delta,K]$, we
also have

$\displaystyle n_{2}(x)$ $\displaystyle\geq n_{1}(\lambda t,x)(\lambda
t)^{-\varepsilon}\geq\\#F(x)\cap B(g,\lambda
t)\lambda^{-\varepsilon}t^{-\varepsilon}$

Combining the lower and upper bounds of $n_{2}(x)$ gives

$\\#F(x)\cap B(g,\lambda t)\leq 4\cdot(2\lambda)^{\varepsilon}\\#F_{B}(x),$

from which the result follows. ∎

####  5.1.2. Finding the typical tangency of curve-curve intersection

As discussed in Section 1.3, a typical pair of intersecting curves in $F$
might intersect transversely, tangentially, or something in-between. In this
section, we will find the typical degree of tangency when two curves
intersect. We will do this by using a two-ends reduction argument similar to
that in Section 5.1.1. This two-ends reduction will use a much smaller
quantity in place of $\varepsilon$, so that the resulting refinement retains
the two-ends property from Section 5.1.1.

For $x\in E_{B}$ and $\Delta>0$, define

$n_{3}(\Delta,x)=\max_{k\in F_{B}(x)}\\#\\{f\in
F_{B}(x):\Delta(f,k)\leq\Delta\\}.$ (5.12)

Since $F_{B}$ is contained in $3B$ which has diameter $6t$, invoking (5.11)
and the relation $\Delta(f,k)\leq\|f-g\|$, we have

$n_{3}(6t,x)=\\#F_{B}(x).$ (5.13)

We further define

$n_{4}(x)=\sup_{\Delta\in[\delta,6t]}n_{3}(\Delta,x)\Delta^{-\varepsilon^{2}}.$
(5.14)

Given $x$, there is some $\Delta_{x}\in[\delta,6t]$ such that

$n_{4}(x)\leq 2n_{3}(\Delta_{x},x)\Delta_{x}^{-\varepsilon^{2}}.$ (5.15)

There is also some $k_{x}\in F_{B}(x)$ attaining the maximum in (5.12), that
is,

$\\#\\{f\in F_{B}(x):\Delta(f,k_{x})\leq\Delta_{x}\\}=n_{3}(\Delta_{x},x),$
(5.16)

and hence

$n_{4}(x)\leq 2\Delta_{x}^{-\varepsilon^{2}}\\#\\{f\in
F_{B}(x):\Delta(f,k_{x})\leq\Delta_{x}\\}.$ (5.17)

Since $\Delta_{x}\in[\delta,6t]$, by a dyadic pigeonholing, there is a Borel
set $E^{\prime\prime}_{B}\subseteq E^{\prime}_{B}$ and a number
$\Delta\in[\delta,12t]$ such that $\Delta_{x}\in(\Delta/2,\Delta]$ for every
$x\in E^{\prime\prime}_{B}$, and
$|E^{\prime\prime}_{B}|\approx|E^{\prime}_{B}|$. It then suffices to show

$|E^{\prime\prime}_{B}|\lessapprox\delta^{2-\frac{\alpha}{2}-\frac{\zeta}{2}-C_{1}\varepsilon}\\#F_{B}\mu^{-3/2}.$
(5.18)

####  5.1.3. Shadings

At this point, the set $E_{0}$ from (5.3) has been divided into (mostly)
disjoint pieces $E_{B}$. For each $x\in E_{B}$, most of the functions $f\in
F(x)$ are contained in the ball $B$, and only a small fraction of these
functions are contained in any ball substantially smaller than $B$. For a
typical pair of functions $f,g\in F(x)$, $\Delta(f,g)$ has size roughly
$\Delta$. For technical reasons, it will be convenient to restrict the sets
$f^{\delta}$ to a slightly smaller “shading” $Y_{1}(f),$ so that the above
properties hold whenever $x\in Y_{1}(f)$. More precisely, we define

$Y_{1}(f)=\\{x\in E^{\prime\prime}_{B}\cap
f^{\delta}:\Delta(f,k_{x})\leq\Delta\\}.$ (5.19)

In view of (5.17), (5.13) and $\Delta_{x}\leq\Delta$, for every $x\in
E^{\prime\prime}_{B}$ we have

$\\#\\{f\in
F_{B}(x):\Delta(f,k_{x})\leq\Delta\\}\gtrsim(\Delta/t)^{\varepsilon^{2}}\\#F_{B}(x)\sim(\Delta/t)^{\varepsilon^{2}}\mu_{1}.$
(5.20)

On the other hand, we have a trivial bound

$\\#\\{f\in F_{B}(x):\Delta(f,k_{x})\leq\Delta\\}\leq\\#F_{B}(x)\sim\mu_{1}.$

Now we apply another dyadic pigeonholing to find a subset $E_{2}\subseteq
E^{\prime\prime}_{B}$ with $|E_{2}|\approx|E^{\prime\prime}_{B}|$ and a scale
$\mu_{2}$ with

$(\Delta/t)^{\varepsilon^{2}}\mu_{1}\lesssim\mu_{2}\leq\mu_{1},$ (5.21)

such that for every $x\in E_{2}$ we have

$\\#\\{f\in F_{B}(x):\Delta(f,k_{x})\leq\Delta\\}\sim\mu_{2}.$ (5.22)

Equivalently, we have

$\\#\\{f\in F_{B}:x\in Y_{1}(f)\\}=\sum_{f\in F_{B}}{\bf
1}_{Y_{1}(f)}(x)\sim\mu_{2},\quad\forall x\in E_{2}.$ (5.23)

It then suffices to prove

$|E_{2}|\lessapprox\delta^{2-\frac{\alpha}{2}-\frac{\zeta}{2}-C_{1}\varepsilon}(\\#F_{B})\mu^{-3/2}.$
(5.24)

We record a coarse bound that is used in some dyadic pigeonholing argument
later. Since $\\#F_{B}(x)\gtrsim t^{\varepsilon}\mu$ and
$\mu\leq\\#F\lesssim\delta^{-1-\varepsilon}$, the right hand side of (5.24) is
trivially bounded below by $\delta^{3}$, for $\delta$ small enough. If
$|E_{2}|\leq\delta^{3}$ then we are done, and so in the following we may
assume

$|E_{2}|>\delta^{3}.$ (5.25)

###  5.2. Fine-scale rectangles

In this section, we will show that $E_{2}$ can be covered by curvilinear
rectangles of dimensions roughly $\delta\times\delta/\sqrt{t\Delta}$, so that
these rectangles are (mostly) disjoint, and for most pairs $f,g\in F_{B}(x)$,
we have that $Y_{1}(f)\cap Y_{1}(g)$ is localized to roughly one of these
rectangles.

######  Lemma 5.3.

There is a constant $C_{R}=C_{R}(K)$ ($R$ stands for “rectangle”) such that
the following is true. Let

$t^{\prime}=C_{R}t\Delta/\delta.$ (5.26)

For each $x\in E_{2}$, there is a $(\delta,t^{\prime})$-rectangle $R_{x}$ over
$J/8$, such that $x\in R_{x}$, and every $f\in F_{B}$ with $x\in Y_{1}(f)$ is
tangent to $R_{x}$.

###### Proof.

Denote $x=(\theta_{0},y_{0})$ where $x\in J/16$. Let $I$ be the interval of
length $\sqrt{\delta/t^{\prime}}$ centred at $\theta_{0}$. Since
$t\gtrsim\Delta\geq\delta$, we have $I\subseteq J/8$ if $C_{R}$ is large
enough. With this, define the curvilinear rectangle

$R_{x}=f^{\delta}_{k_{x}}(I),$ (5.27)

where $k_{x}$ is as in (5.16). We now show that $R_{x}$ is as required.

First we have $x\in R_{x}$ since $k_{x}\in F_{B}(x)$ by (5.12). Also, let $f$
be such that $x\in Y_{1}(f)$, then in particular, $x\in f^{\delta}$. By the
triangle inequality, $|f(\theta_{0})-k_{x}(\theta_{0})|\leq 2\delta$. By Part
2c of Lemma 3.8 at the scale $4\delta$, we have

$E_{4\delta}=\\{\theta\in J/4:|f(\theta)-k_{x}(\theta)|\leq 4\delta\\}$

contains an interval $I^{\prime}$ obeying $\theta_{0}\in I^{\prime}$ and

$|I^{\prime}|\gtrsim\delta d(f,k_{x})^{-1/2}(\delta+\Delta(f,k_{x}))^{-1/2}.$

Since $x\in Y_{1}(f)$ we have $\Delta(f,k_{x})\leq\Delta$. Also $d(f,k(x))\leq
6t$. If $C_{R}$ is chosen to be large enough, then the interval $I$ containing
$\theta_{0}$ will have length $\sqrt{\delta/t^{\prime}}\leq|I^{\prime}|$, and
thus $I\subseteq I^{\prime}$ and $R_{x}$ is tangent to $f$. ∎

####  5.2.1. Shadings of fine-scale rectangles

For each $(\delta,t^{\prime})$-rectangle $R$, we define its shading as

$Y_{2}(R)=\\{x\in C_{s}R\cap E_{2}:R_{x}\text{ is $C_{s}$-comparable to
}R\\},$ (5.28)

where $C_{s}=C_{s}(K)\geq 100$ ($s$ standing for “shadings”) is a suitable
constant to be determined. Intuitively, $Y_{2}(R)$ is an important subset of
$R$ since it is where the $\delta$-neighbourhoods of the curves in $F_{B}$
overlap greatly.

For technical purposes, we also define a slightly smaller shading of $R$ as

$Y^{\prime}_{2}(R)=\\{x\in R\cap E_{2}:R_{x}\text{ is $100$-comparable to
}R\\}.$

By Lemma 5.3, we have

$E_{2}=\bigcup_{x\in E_{2}}Y^{\prime}_{2}(R_{x}).$ (5.29)

We also record an elementary lemma on the shadings of curvilinear rectangles.

######  Lemma 5.4.

If $C_{s}$ is chosen suitably, then for any pair $(R,R^{\prime})$ of
$100$-comparable $(\delta,t)$-rectangles, we have
$Y^{\prime}_{2}(R^{\prime})\subseteq Y_{2}(R)$.

###### Proof.

If $R,R^{\prime}$ are $100$-comparable, then by Lemma 3.12, we have
$R^{\prime}\subseteq C_{s}R$ if $C_{s}$ is large enough. Thus if $x\in
Y^{\prime}_{2}(R^{\prime})$, then $x\in C_{s}R\cap E_{2}$. Also, $R_{x}$ is
$100$-comparable to $R^{\prime}$, and since $R^{\prime}$ is $100$-comparable
to $R$, Corollary 3.13 implies that $R_{x}$ is $C_{s}$-comparable to $R$ if
$C_{s}$ is large enough. Thus $Y^{\prime}_{2}(R^{\prime})\subseteq Y_{2}(R)$.
∎

####  5.2.2. Finding an incomparable set of fine-scale rectangles

######  Lemma 5.5.

There is a pairwise $100$-incomparable subcollection $\mathcal{R}$ of
$\\{R_{x}:x\in E_{2}\\}$ such that the following holds:

  1. (1)

There is some $\Lambda\in(0,1)$ such that for each $R_{x}\in\mathcal{R}$ we
have $\Lambda\leq|Y_{2}(R_{x})|<2\Lambda$.

  2. (2)

$Y_{2}(R)\cap Y_{2}(R^{\prime})=\varnothing$ for $R\neq
R^{\prime}\in\mathcal{R}$, and

$|E_{2}|\lessapprox\sum_{R\in\mathcal{R}}|Y_{2}(R)|\sim\Lambda\\#\mathcal{R}.$
(5.30)

  3. (3)

We have the coarse bounds

$\\#\mathcal{R}\leq\delta^{-C},\quad\Lambda\geq\delta^{C^{\prime}},$ (5.31)

for $\delta$ sufficiently small (depending on $K,D$), where $C,C^{\prime}$
depend on $D$ only.

###### Proof.

Before we construct $\mathcal{R}$, we first construct a larger collection
$\mathcal{R}^{*}$ in the following greedy way.

Denote $\Lambda_{1}=\sup_{x\in E_{2}}|Y^{\prime}_{2}(R_{x})|$. Since we have
assumed $|E_{2}|>\delta^{3}$ we have $\Lambda_{1}>0$ by (5.29). So we pick
$R_{1}=R_{x_{1}}$ such that $|Y^{\prime}_{2}(R_{1})|>\Lambda_{1}/2$.

To choose $R_{2}$, we consider all $(\delta,t)$-rectangles $R_{x}$ that are
$100$-incomparable to $R_{1}$; denote this collection by $\mathcal{R}^{(1)}$.
Denote $\Lambda_{2}=\sup_{R_{x}\in\mathcal{R}^{(1)}}|Y^{\prime}_{2}(R_{x})|$.
If $\Lambda_{2}=0$, then we just let $\mathcal{R}^{*}=\\{R_{1}\\}$. With this
and Lemma 5.4, we have (here $\sim_{100}$ means $100$-comparable)

$\displaystyle|E_{2}|$ $\displaystyle\leq\left|\bigcup_{x\in
E_{2},R_{x}\sim_{100}R_{1}}Y^{\prime}_{2}(R_{x})\right|\leq|Y_{2}(R_{1})|.$

If $\Lambda_{2}>0$, then we pick $R_{2}\in\mathcal{R}^{(1)}$ such that
$|Y^{\prime}_{2}(R_{2})|>\Lambda_{2}/2$. To choose $R_{3}$, we consider all
$(\delta,t)$-rectangles that are $100$-incomparable to both $R_{1}$ and
$R_{2}$; denote this collection by $\mathcal{R}^{(2)}$. Denote
$\Lambda_{3}=\sup_{R_{x}\in\mathcal{R}^{(2)}}|Y^{\prime}_{2}(R_{x})|$. If
$\Lambda_{3}=0$, we just let $\mathcal{R}^{*}=\\{R_{1},R_{2}\\}$. With this,
we have

$\displaystyle|E_{2}|$ $\displaystyle\leq\left|\bigcup_{x\in
E_{2},R_{x}\sim_{100}R_{1}}Y^{\prime}_{2}(R_{x})\right|+\left|\bigcup_{x\in
E_{2},R_{x}\sim_{100}R_{2}}Y^{\prime}_{2}(R_{x})\right|\leq|Y_{2}(R_{1})|+|Y_{2}(R_{2})|.$

We can continue arguing in this fashion. Note that Lemma 3.17 implies that the
process must stop at some finite time $N<\delta^{-C}$ for some constant
$C=C(D)$, proving the first inequality in (5.31). Thus $\Lambda_{N+1}=0$,
$\Lambda_{N}>0$, $\mathcal{R}^{*}=\\{R_{1},\dots,R_{N}\\}$, and

$|E_{2}|\leq\sum_{n=1}^{N}|Y_{2}(R_{n})|.$

We have $|Y_{2}(R_{n})|\geq\delta^{C+3}$, otherwise $|E_{2}|\leq
N\delta^{C+3}<\delta^{3}$, contradicting the assumption that
$|E_{2}|>\delta^{3}$.

Now for $j\geq 0$, we denote

$\mathcal{R}^{(j)}=\\{R_{n}\in\mathcal{R}^{*}:2^{j-1}\Lambda_{N}\leq|Y_{2}(R_{n})|<2^{j}\Lambda_{N}\\}.$

There are at most $O(|\log\delta|)$ many such $j$’s, since
$\delta^{C+3}\leq|Y_{2}(R_{n})|\leq 1$. By pigeonholing, there is some $j$
such that

$\sum_{n=1}^{N}|Y_{2}(R_{n})|\lesssim|\log\delta|\sum_{n:R_{n}\in\mathcal{R}^{(j)}}|Y_{2}(R_{n})|.$

We now define

$\mathcal{R}=\mathcal{R}^{(j)},\quad\Lambda=2^{j-1}\Lambda_{N}>0.$

Thus

$|E_{2}|\leq\sum_{n=1}^{N}|Y_{2}(R_{n})|\lessapprox\Lambda\\#\mathcal{R}.$

Also, we have $\Lambda\geq\delta^{C+4}$, since $|E_{2}|>\delta^{3}$ and
$\\#\mathcal{R}\leq N<\delta^{-C}$.

∎

We now observe the following. Since $Y_{2}(R)\subseteq C_{s}R\cap E_{2}$, $R$
is over an interval of length $\sim\sqrt{\delta/t^{\prime}}$, and $E_{2}$ is
further contained in a
$(\delta,\alpha;\delta^{-\varepsilon})_{1}\times(\delta,\alpha;\delta^{-\varepsilon})_{1}$
quasi-product, we have

$\Lambda\sim|Y_{2}(R)|\lesssim\delta^{2-\varepsilon}(\sqrt{\delta/t^{\prime}}/\delta)^{\alpha}\sim\delta^{2-\varepsilon}\Delta^{-\alpha/2}t^{-\alpha/2}.$

With this, we write

$\Lambda=\tau\delta^{2-\varepsilon}\Delta^{-\alpha/2}t^{-\alpha/2},$ (5.32)

where

$0<\tau\lesssim 1.$ (5.33)

To establish (5.24), it thus suffices to show

$\\#\mathcal{R}\lessapprox\tau^{-1}\delta^{-\frac{\alpha}{2}-\frac{\zeta}{2}-C_{2}\varepsilon}\Delta^{\alpha/2}t^{\alpha/2}\\#F_{B}\mu^{-3/2},$
(5.34)

for some $C_{2}=C_{2}(D)$.

###  5.3. Coarse-scale rectangles

In this section, we will show that the fine-scale rectangles from Section 5.2
are contained in coarse-scale rectangles of dimensions roughly
$\Delta\times\sqrt{\Delta/t}$. If a fine-scale rectangle $R$ is contained in a
coarse-scale rectangle $\tilde{R}$, then $\tilde{R}$ is (roughly) a dilate of
$R$.

For each $R\in\mathcal{R}$, we define $\tilde{R}(R)$ as follows. If
$R=f^{\delta}(I),$ then $\tilde{R}(R)=f^{\Delta}(\tilde{I})$, where
$\tilde{I}$ is the $\Delta/\delta$ dilation of $I$, and hence it obeys
$|\tilde{I}|=\sqrt{\frac{\Delta}{C_{R}t}}$ where $C_{R}$ is as in Lemma 5.3.
Thus $\tilde{R}(R)$ is a $(\Delta,C_{R}t)$-rectangle, and trivially we have
$R\subseteq\tilde{R}(R)$. Also, since $I\subseteq J/8$, for $C_{R}$ large
enough, we have $\tilde{I}\subseteq J/4$.

######  Lemma 5.6.

Let $R=R_{x}$ and $f\in F_{B}$ with $x\in Y_{1}(f)$, such that $f$ is tangent
to $R$ by Lemma 5.3. If $C_{R}$ was chosen to be sufficiently large, then $f$
is also tangent to $\tilde{R}$.

###### Proof.

It follows from the same proof as that of Lemma 5.3, with $\Delta$ in place of
$\delta$. ∎

Next, we pick a maximally $100$-incomparable subcollection
$\tilde{\mathcal{R}}$ of $\\{\tilde{R}(R):R\in\mathcal{R}\\}$, and we will
denote each member of $\tilde{\mathcal{R}}$ as $\tilde{R}$. In this way we
have partitioned $\mathcal{R}$ into subcollections of the form
$\mathcal{R}(\tilde{R})$ consisting some of those $R\in\mathcal{R}$ such that
the enlarged $\tilde{R}(R)$ of $R$ is $100$-comparable to $\tilde{R}$.

For each $R\in\mathcal{R}(\tilde{R})$, we have $R\subseteq\tilde{R}(R)$, and
since $\tilde{R}(R)$ is $100$-comparable to $\tilde{R}$, Lemma 3.12 shows that
$\tilde{R}(R)\subseteq C\tilde{R}$ for some suitable absolute constant $C\geq
1$. Thus $R\subseteq C\tilde{R}$, and so $R\cap E_{2}\subseteq C\tilde{R}\cap
E_{2}$.

Recall we want to prove (5.34). Using (5.31) and dyadic pigeonholing, we may
find a subcollection $\tilde{\mathcal{R}}_{1}\subseteq\tilde{\mathcal{R}}$
such that for some $M$ we have

$\\#\mathcal{R}(\tilde{R})\sim M$ (5.35)

for each $\tilde{R}\in\tilde{\mathcal{R}}_{1}$. We also define
$\mathcal{R}_{1}=\cup_{\tilde{R}\in\tilde{\mathcal{R}_{1}}}\mathcal{R}(\tilde{R})$,
and so $\\#\mathcal{R}_{1}\approx\\#\mathcal{R}$. As a result, we have
$\\#\mathcal{R}\approx\\#\tilde{\mathcal{R}}_{1}M$. Note that by Lemma 5.5, we
also have $|E_{2}|\approx\Lambda\\#\mathcal{R}_{1}$.

For this reason, in the subsequent argument we will slightly abuse notation
and denote $\mathcal{R}=\mathcal{R}_{1}$ and
$\tilde{\mathcal{R}}_{1}=\tilde{\mathcal{R}}$. Thus we have

$\\#\mathcal{R}\approx\\#\tilde{\mathcal{R}}M,$ (5.36)

and so it suffices to bound $\\#\tilde{\mathcal{R}}$ and $M$.

We will now give an easy upper bound of $M$. Since $E_{2}$ is contained in a
$(\delta,\alpha;\delta^{-\varepsilon})_{1}\times(\delta,\alpha;\delta^{-\varepsilon})_{1}$-quasi
product and $C\tilde{R}$ is a the $O(\Delta)$-neighbourhood of a curve of
length $\sim\sqrt{\Delta/t}$, we have

$|C\tilde{R}\cap
E_{2}|\lesssim\delta^{1-\varepsilon}(\Delta/\delta)^{\alpha}\delta^{1-\varepsilon}(\sqrt{\Delta/t}/\delta)^{\alpha}=\delta^{2-2\alpha-2\varepsilon}\Delta^{3\alpha/2}t^{-\alpha/2}.$

Recall $Y_{2}(R)$ defined as in (5.28). For $R\in\mathcal{R}(\tilde{R})$, we
have $Y_{2}(R)\subseteq C_{s}R\cap E_{2}\subseteq C_{s}\tilde{R}\cap E_{2}$.
But by Lemma 5.5, $Y_{2}(R)$’s are disjoint, and thus we have a measure bound

$M\lesssim\frac{|C_{s}\tilde{R}\cap
E_{2}|}{|Y_{2}(R)|}\lesssim\Lambda^{-1}\delta^{2-2\alpha-2\varepsilon}\Delta^{3\alpha/2}t^{-\alpha/2}=\tau^{-1}\delta^{-2\alpha-2\varepsilon}\Delta^{2\alpha},$
(5.37)

where in the last equality we have used (5.32). Later we will give a second
bound on $M$ using Proposition 4.1.

###  5.4. A bilinear $L^{2}$ bound

In this section, we will analyze the interaction between fine and coarse-scale
rectangles. We will show that for each fine-scale rectangle $R$, there are
about $\mu^{2}$ pairs of functions $f,g$ that are tangent to $R$, so that
$f^{\delta}\cap g^{\delta}$ is localized to $R$. Thus if $N$ functions $f\in
F$ are tangent to a coarse-scale rectangle $\tilde{R},$ then $\tilde{R}$
contains about $N^{2}/\mu^{2}$ fine-scale rectangles. This is the “$L^{2}$
argument” alluded to in the proof sketch from Section 1.3.

We turn to the details. By construction, each $R\in\mathcal{R}$ is indexed by
some $x\in E_{2}$, namely, $R=R_{x}$. We will write $x(R)$ to denote this
value of $x$. Let $f\in F_{B}$ and $R\in\mathcal{R}$. We say the pair $(f,R)$
is good if $x(R)\in Y_{1}(f)$. Note that if $(f,R)$ is good, then $f$ is
tangent to $R$ by Lemma 5.3.

Denote by $\mathcal{G}$ the collection of all good pairs. For each $f\in
F_{B}$, we define

$\mathcal{G}(f)=\\{R\in\mathcal{R}:(f,R)\in\mathcal{G}\\}.$ (5.38)

For each $R\in\mathcal{R}$, we define

$\mathcal{G}(R)=\\{f\in F_{B}:(f,R)\in\mathcal{G}\\}.$ (5.39)

For each $x\in E_{2}$, we have $x\in Y_{1}(f)$ if and only if
$R_{x}\in\mathcal{G}(f)$. Thus by (5.23) and unravelling the definitions
above, for each $R\in\mathcal{R}$ we have

$\\#\mathcal{G}(R)\sim\mu_{2}.$ (5.40)

####  5.4.1. Pigeonholing the good pairs

For each $\tilde{R}\in\tilde{\mathcal{R}}$, we consider the quantity

$\\#\mathcal{G}(f)\cap\mathcal{R}(\tilde{R})=\\#\\{R\in\mathcal{R}(\tilde{R}):(f,R)\in\mathcal{G}\\}.$

By the disjointness of $\mathcal{R}(\tilde{R})$, we have

$\\#\mathcal{G}=\sum_{f\in
F_{B}}\sum_{\tilde{R}\in\tilde{\mathcal{R}}}\\#\mathcal{G}(f)\cap\mathcal{R}(\tilde{R}).$

In view of (5.31), we may apply a dyadic pigeonholing to find an integer $q$
and a set $\mathcal{G}^{\prime}\subset\mathcal{G}$ with
$\\#\mathcal{G}^{\prime}\gtrsim|\log\delta|^{-1}\\#\mathcal{G}$, so that for
each $\tilde{R}\in\tilde{\mathcal{R}}$ and each $f\in F_{B}$, the quantity
$\\#\mathcal{G}^{\prime}(f)\cap\mathcal{R}(\tilde{R})$ is either 0 or $\sim
q$. (Here we define $\mathcal{G}^{\prime}(f)$ and $\mathcal{G}^{\prime}(R)$
similarly as (5.38) and (5.39).)

For each $\tilde{R}\in\tilde{\mathcal{R}}$, define

$F(\tilde{R})=\\{f\in
F_{B}\colon\mathcal{G}^{\prime}(f)\cap\mathcal{R}(\tilde{R})\neq\varnothing\\}.$
(5.41)

For each $f\in F(\tilde{R})$, we have in particular
$\mathcal{G}(f)\cap\mathcal{R}(\tilde{R})\neq\varnothing$, and thus there is
some $R\in\mathcal{R}(\tilde{R})$ such that $(f,R)\in\mathcal{G}$. Hence $f$
is tangent to $R$, and using Lemma 5.6, we also have $f$ is tangent to
$\tilde{R}$.

By construction, $\\#\mathcal{G}^{\prime}(f)\cap\mathcal{R}(\tilde{R})\sim q$
for each $f\in F(\tilde{R})$, and hence

$\\#F(\tilde{R})\sim\frac{1}{q}\\#\\{(f,R)\in\mathcal{G}^{\prime}\colon
R\in\mathcal{R}(\tilde{R})\\}=\frac{1}{q}\sum_{R\in\mathcal{R}(\tilde{R})}\\#\mathcal{G}^{\prime}(R).$
(5.42)

More generally, for every ball $B(g,r)\subseteq C^{2}(J)$ we have

$\\#F(\tilde{R})\cap
B(g,r)\sim\frac{1}{q}\sum_{R\in\mathcal{R}(\tilde{R})}\\#\mathcal{G}^{\prime}(R)\cap
B(g,r).$ (5.43)

By (5.40) and (5.35), for each $\tilde{R}\in\tilde{\mathcal{R}}^{\prime}$ we
have

$\sum_{R\in\mathcal{R}(\tilde{R})}\\#\mathcal{G}^{\prime}(R)\leq\sum_{R\in\mathcal{R}(\tilde{R})}\\#\mathcal{G}(R)\sim\mu_{2}M,$

and hence

$\\#\mathcal{G}^{\prime}=\sum_{\tilde{R}\in\tilde{\mathcal{R}}}\sum_{R\in\mathcal{R}(\tilde{R})}\\#\mathcal{G}^{\prime}(R)\lesssim\mu_{2}M\\#\tilde{\mathcal{R}}.$

On the other hand, we have

$\\#\mathcal{G}^{\prime}\gtrsim|\log\delta|^{-1}\\#\mathcal{G}=|\log\delta|^{-1}\sum_{\tilde{R}\in\tilde{\mathcal{R}}}\sum_{R\in\mathcal{R}(\tilde{R})}\\#\mathcal{G}(R)\sim|\log\delta|^{-1}\mu_{2}M\\#\tilde{\mathcal{R}},$
(5.44)

where the final quasi-inequality used (5.40). Combining the upper and lower
bounds, we see that for a $\gtrsim|\log\delta|^{-1}$ fraction of
$\tilde{R}\in\tilde{\mathcal{R}}$ we have

$\sum_{R\in\mathcal{R}(\tilde{R})}\\#\mathcal{G}^{\prime}(R)\gtrsim|\log\delta|^{-1}\mu_{2}M.$
(5.45)

Denote this subcollection as $\tilde{\mathcal{R}}^{\prime}$. Similarly, for
each $\tilde{R}\in\tilde{\mathcal{R}}^{\prime}$, for a
$\gtrsim|\log\delta|^{-1}$ fraction of $R\in\mathcal{R}(\tilde{R})$ we have

$\\#\mathcal{G}^{\prime}(R)\gtrsim|\log\delta|^{-1}\mu_{2}.$ (5.46)

Denote this subcollection as $\mathcal{R}^{\prime}(\tilde{R})$, which has
cardinality $\gtrsim|\log\delta|^{-1}M$. Denote

$\mathcal{R}^{\prime}=\bigsqcup_{\tilde{R}\in\tilde{\mathcal{R}}^{\prime}}\mathcal{R}^{\prime}(\tilde{R}).$
(5.47)

Thus $\\#\mathcal{R}^{\prime}\gtrsim|\log\delta|^{-2}\\#\mathcal{R}$. Since
each $\\#\mathcal{R}^{\prime}(\tilde{R})\approx M$, it suffices to bound
$\\#\mathcal{R}^{\prime}$.

####  5.4.2. Each fine-scale rectangle has many good pairs

######  Lemma 5.7.

If $\delta>0$ is sufficiently small, then for each $R\in\mathcal{R}^{\prime}$,
there are at least $(\\#\mathcal{G}^{\prime}(R))^{2}/3$ pairs
$(f,g)\in(\mathcal{G}^{\prime}(R))^{2}$ such that
$\delta^{2\varepsilon}t\leq\|f-g\|\leq 6t$ and
$\delta^{\varepsilon}\Delta\leq\Delta(f,g)+\delta\leq 2\Delta$.

###### Proof.

Let $R=R_{x}\in\mathcal{R}^{\prime}$ where $x\in E_{2}$. Fix
$g\in\mathcal{G}^{\prime}(R)$. We deal with $\|f-g\|$ first. If
$t\leq\delta^{1-\varepsilon}$, then we immediately have
$\|f-g\|\geq\delta\geq\delta^{2\varepsilon}t$ for every
$f\in\mathcal{G}^{\prime}(R)$, since the elements of $F$ are
$\delta$-separated.

Next, suppose $t>\delta^{1-\varepsilon}$, and let
$\lambda=\delta^{2\varepsilon}$; by hypothesis we have $\delta
t^{-1}<\lambda<1$. Thus we can apply Lemma 5.2 to conclude that

$\displaystyle\\#\\{f\in\mathcal{G}^{\prime}(R):\|f-g\|\leq\lambda t\\}$
$\displaystyle\leq\\#\\{f\in F_{B}(x):\|f-g\|\leq\lambda t\\}$
$\displaystyle\lesssim\lambda^{\varepsilon}\mu_{1}$
$\displaystyle\lesssim\lambda^{\varepsilon}\delta^{-\varepsilon^{2}}\mu_{2}$
$\displaystyle=\delta^{\varepsilon^{2}}\mu_{2}$
$\displaystyle\lesssim\delta^{\varepsilon^{2}}|\log\delta|\\#\mathcal{G}^{\prime}(R),$

where the third inequality used (5.21) and in the last inequality we have used
(5.46). If $\delta>0$ is sufficiently small (depending on $\varepsilon$), then

$\\#\\{f\in\mathcal{G}^{\prime}(R):\|f-g\|\leq\lambda
t\\}\leq\\#\mathcal{G}^{\prime}(R)/3.$ (5.48)

Next we deal with $\Delta(f,g)$. If $\Delta\leq\delta^{1-\varepsilon}$ then we
immediately have $\Delta(f,g)+\delta\geq\delta^{\varepsilon}\Delta$ for every
$f\in\mathcal{G}^{\prime}(R)$. If $\Delta>\delta^{1-\varepsilon}$, then we
choose

$\sigma=|\log\delta|^{-2},$

and thus
$\sigma^{\varepsilon^{-2}}\Delta\geq\delta^{\varepsilon}\Delta>\delta$ for
$\delta$ small enough. Using (5.14) and (5.15), we have

$2n_{3}(\Delta_{x},x)\Delta_{x}^{-\varepsilon^{2}}\geq n_{4}(x)\geq
n_{3}(\sigma^{\varepsilon^{-2}}\Delta,x)(\sigma^{\varepsilon^{-2}}\Delta)^{-\varepsilon^{2}},$

and thus $n_{3}(\sigma^{\varepsilon^{-2}}\Delta,x)\lesssim\sigma
n_{3}(\Delta_{x},x)$ since $\Delta_{x}\sim\Delta$. Now recalling the
definition of $n_{3}$ in (5.12), taking $k=g\in F_{B}(x)$ shows that

$\displaystyle n_{3}(\sigma^{\varepsilon^{-2}}\Delta,x)$
$\displaystyle\geq\\#\\{f\in
F_{B}(x):\Delta(f,g)\leq\sigma^{\varepsilon^{-2}}\Delta\\}.$

Combining the lower and upper bounds for
$n_{3}(\sigma^{\varepsilon^{-2}}\Delta,x)$ we have

$\begin{split}&\\#\\{f\in\mathcal{G}^{\prime}(R):\Delta(f,g)\leq\sigma^{\varepsilon^{-2}}\Delta\\}\\\
&\leq\\#\\{f\in F_{B}(x):\Delta(f,g)\leq\sigma^{\varepsilon^{-2}}\Delta\\}\\\
&\lesssim\sigma n_{3}(\Delta_{x},x)\\\ &=\sigma\\#\\{f\in
F_{B}(x):\Delta(f,k_{x})\leq\Delta_{x}\\}\\\ &\leq\sigma\mu_{2}\\\
&\lesssim\sigma|\log\delta|\\#\mathcal{G}^{\prime}(R)\\\
&\leq\\#\mathcal{G}^{\prime}(R)/3,\end{split}$ (5.49)

for $\delta$ small enough, where in the fourth line we have used (5.16) and in
the sixth line we have used $\Delta_{x}\leq\Delta$, (5.22) and (5.46).

By (5.48) and (5.49), we conclude that at least a $1/3$ fraction of the
functions $f\in\mathcal{G}^{\prime}(R)$ satisfy
$\|f-g\|\geq\delta^{2\varepsilon}t$ and
$\Delta(f,g)+\delta\geq\delta^{\varepsilon}\Delta$. The corresponding upper
bounds $\|f-g\|\leq 6t$ and $\Delta(f,g)+\delta\leq 2\Delta$ hold for every
$f\in\mathcal{G}^{\prime}(R)$. ∎

######  Lemma 5.8.

For each pair $f,g\in F_{B}\times F_{B}$ such that
$\|f-g\|\geq\delta^{2\varepsilon}t$ and
$\Delta(f,g)+\delta\geq\delta^{\varepsilon}\Delta$, there are
$\lesssim\delta^{-4\varepsilon}$ rectangles $R\in\mathcal{R}^{\prime}$ such
that both $f,g\in\mathcal{G}^{\prime}(R)$.

###### Proof.

If $f\in\mathcal{G}^{\prime}(R)$, then $f\in\mathcal{G}(R)$ and thus $f$ is
tangent to $R$. It then suffices to show that there are at most
$\lesssim\delta^{-3\varepsilon}$ many $R\in\mathcal{R}^{\prime}$ that are
tangent to both $f,g$.

Let $f,g$ with $\|f-g\|\geq\delta^{2\varepsilon}t$ and
$\Delta(f,g)+\delta\geq\delta^{\varepsilon}\Delta$, and suppose
$R=k^{\delta}(I)$ is tangent to both $f$ and $g$. Then over $I$ we have
$|f(\theta)-g(\theta)|\leq 10\delta$. By Part 2b of Lemma 3.8 at the scale
$10\delta$, the union of all $R$ that are tangent to both $f$ and $g$ projects
to the $\theta$-axis into a set of Lebesgue measure

$\lesssim\delta/\sqrt{\|f-g\|(\Delta(f,g)+\delta)}\lesssim\delta^{1-(3/2)\varepsilon}(t\Delta)^{-1/2}\sim\delta^{-(3/2)\varepsilon}\sqrt{\delta/t^{\prime}}.$

Using Lemma 3.15 with $\lambda\lesssim\delta^{-(3/2)\varepsilon}$, we see
there are $\lesssim\delta^{-(15/4)\varepsilon}\leq\delta^{-4\varepsilon}$
$R\in\mathcal{R}^{\prime}$ that are tangent to both $f,g$. ∎

####  5.4.3. A tangency bound on $\\#\mathcal{R}(\tilde{R})$

Now we give another bound on $M$ in addition to (5.37), using the conclusions
in Section 5.4.2. For $\tilde{R}\in\tilde{\mathcal{R}}^{\prime}$, define

$\displaystyle\mathcal{P}(\tilde{R})=\bigcup_{R\in\mathcal{R}^{\prime}(\tilde{R})}\\{(f,g)\in\mathcal{G}^{\prime}(R)^{2}:\|f-g\|\geq\delta^{2\varepsilon}t,\
\Delta(f,g)+\delta\geq\delta^{\varepsilon}\Delta\\}.$

By Lemma 5.7, Lemma 5.8 and (5.46) we have

$\\#\mathcal{P}(\tilde{R})\gtrapprox\delta^{4\varepsilon}\mu_{2}^{2}M.$ (5.50)

On the other hand, we have a trivial bound (recall (5.41))

$\\#\mathcal{P}(\tilde{R})\leq(\\#F(\tilde{R}))^{2}.$ (5.51)

Combining (5.50) and (5.51) and recalling (5.21) and (5.10), we conclude that

$M\lessapprox\delta^{-4\varepsilon}\mu_{2}^{-2}(\\#F(\tilde{R}))^{2}\lessapprox\delta^{-6\varepsilon}\mu^{-2}(\\#F(\tilde{R}))^{2}.$
(5.52)

This is our second bound on $M$. Taking a suitable geometric average of these
bounds, we have

$M\lessapprox\eqref{eqn_M_bound_measure}^{1/4}\eqref{eqn_M_bound_tangency}^{3/4}\lessapprox\tau^{-1/4}\delta^{-\alpha/2-6\varepsilon}\Delta^{\alpha/2}\mu^{-3/2}(\\#F(\tilde{R}))^{3/2}.$
(5.53)

Note that (5.53) holds for every $\tilde{R}\in\tilde{\mathcal{R}}^{\prime}$,
and the LHS is independent of the choice of $\tilde{R}$. Thus if we define

$l=\min_{\tilde{R}\in\tilde{\mathcal{R}}^{\prime}}\\#F(\tilde{R}),$

Then using (5.33) and (5.53), (5.34) would follow from the estimate

$\\#\tilde{\mathcal{R}}^{\prime}\leq\delta^{-C_{3}\varepsilon}\delta^{-\zeta/2}t^{\alpha/2}l^{-3/2}\\#F_{B},$
(5.54)

for some $C_{3}=C_{3}(D)$.

In the next section we will establish this estimate.

###  5.5. Proving Inequality (5.54)

Our goal in this section is to bound the size of
$\tilde{\mathcal{R}}^{\prime}$. Our main tool will be Proposition 4.1.
However, before the proposition can be applied we need to construct suitable
sets $\mathcal{W}$ and $\mathcal{B}$.

####  5.5.1. Non-concentration of functions tangent to $\tilde{R}$

In this section, we will show that for each coarse-scale rectangle
$\tilde{R}$, the functions from $F$ tangent to $\tilde{R}$ cannot be
concentrated inside a ball of diameter much smaller than $t$.

######  Lemma 5.9.

Let $\tilde{R}\in\tilde{\mathcal{R}}^{\prime}$. For every $r>0$ and every
$g\in C^{2}(J)$, we have

$\\#F(\tilde{R})\cap
B(g,r)\lesssim|\log\delta|(r/t)^{\varepsilon}(t/\Delta)^{\varepsilon^{2}}\\#F(\tilde{R}).$
(5.55)

###### Proof.

Recall (5.43) that

$\\#F(\tilde{R})\cap B(g,r)\sim
q^{-1}\sum_{R\in\mathcal{R}(\tilde{R})}\\#\mathcal{G}^{\prime}(R)\cap B(g,r).$

For each $R\in\mathcal{R}(\tilde{R})$, write $R=R_{x}$. using Lemma 5.2 with
$\lambda=r/t$ and (5.11) we have

$\displaystyle\\#\mathcal{G}^{\prime}(R)\cap B(g,r)\leq\\#F_{B}(x)\cap
B(g,r)\lesssim(r/t)^{\varepsilon}\\#F_{B}(x)\sim(r/t)^{\varepsilon}\mu_{1}.$

Thus

$\\#F(\tilde{R})\cap B(g,r)\lesssim
q^{-1}\\#\mathcal{R}(\tilde{R})(r/t)^{\varepsilon}\mu_{1}\sim
q^{-1}M(r/t)^{\varepsilon}\mu_{1}.$

On the other hand, by (5.42) and (5.45), we have

$\displaystyle\\#F(\tilde{R})$ $\displaystyle\sim
q^{-1}\sum_{R\in\mathcal{R}(\tilde{R})}\\#\mathcal{G}^{\prime}(R)$
$\displaystyle\gtrsim q^{-1}M|\log\delta|^{-1}\mu_{2}\gtrsim
q^{-1}M|\log\delta|^{-1}(\Delta/t)^{\varepsilon^{2}}\mu_{1}.$

Thus the result follows. ∎

By Lemma 5.9, we now take

$r=tc^{1/\varepsilon}|\log\delta|^{-1/\varepsilon}(\Delta/t)^{\varepsilon},$
(5.56)

where $c=c(D)$ is a sufficiently small constant to be determined, such that
for every ball $B(g,r)\subseteq C^{2}(J)$ and every
$\tilde{R}\in\tilde{\mathcal{R}}^{\prime}$ we have

$\\#F(\tilde{R})\cap B(g,r)\leq c\\#F(\tilde{R}).$ (5.57)

Note that $t\leq\delta^{-2\varepsilon}r/3$ for $\delta$ small enough.

####  5.5.2. Applying Proposition 4.1

Since $\mathcal{F}$ has doubling constant $D$, there exists $C_{D}\geq 1$ so
that we can cover $3B$ by a set of
$(3t/r)^{C_{D}}\leq\delta^{-2C_{D}\varepsilon}$ balls of radius $r$, so that
these balls are boundedly overlapping (with the bound depending on $D$).
Denote this set of balls by $\mathcal{O}.$

If $c=c(D)$ is chosen to be small enough, then for any
$\tilde{R}\in\tilde{\mathcal{R}}^{\prime}$, by pigeonholing and (5.57), there
are two balls $O_{1}(\tilde{R}),O_{2}(\tilde{R})\in\mathcal{O}$ with their
centres $\geq 10r$ separated, such that for $i=1,2,$ we have

$\\#F(\tilde{R})\cap
O_{i}(\tilde{R})\gtrsim\delta^{2C_{D}\varepsilon}\\#F(\tilde{R})\geq\delta^{2C_{D}\varepsilon}l.$
(5.58)

Since $(\\#\mathcal{O})^{2}\lesssim\delta^{-4C_{D}\varepsilon}$, by
pigeonholing, there is a pair $(O_{1},O_{2})\in\mathcal{O}^{2}$ so that (5.58)
holds for $\gtrsim\delta^{-4C_{D}\varepsilon}\\#\tilde{\mathcal{R}}^{\prime}$
rectangles in $\tilde{\mathcal{R}}^{\prime}$. Fix this choice of $O_{1}$ and
$O_{2}$, and let
$\tilde{\mathcal{R}}^{\prime\prime}\subset\\#\tilde{\mathcal{R}}^{\prime}$ be
the set of rectangles for which (5.58) holds.

To establish (5.54), it suffices to prove that

$\\#\tilde{\mathcal{R}}^{\prime\prime}\lesssim\delta^{-\zeta/2-C_{3}\varepsilon}t^{\alpha/2}l^{-3/2}\\#F_{B},$
(5.59)

for some $C_{3}=C_{3}(D)$.

Let $\mathcal{W}=F_{B}\cap O_{1}(\tilde{R})$ and $\mathcal{B}=F_{B}\cap
O_{2}(\tilde{R})$. Then $(\mathcal{W},\mathcal{B})$ is $r$-separated. Also, by
the discussion right after (5.41), each $(\Delta,C_{R}t)$-rectangle
$\tilde{R}\in\tilde{\mathcal{R}}^{\prime\prime}$ is in particular tangent to
$\geq l$ many $f\in\mathcal{W}$ and $\geq l$ many $f\in\mathcal{B}$.

With this, we can now apply Proposition 4.1 to the family
$\\#\tilde{\mathcal{R}}^{\prime\prime}$ with $\mu=\nu=l$, $C_{R}t$ in place of
$t$, and $A\leq\delta^{-2\varepsilon}$ to get

$\\#\tilde{\mathcal{R}}^{\prime\prime}\lesssim\delta^{-C_{4}\varepsilon}\left(\\#F_{B}/l\right)^{3/2},$
(5.60)

for some $C_{4}=C_{4}(D)$.

By (1.7) with $3t$ in place of $r$, we have

$\\#F_{B}\leq\delta^{-\varepsilon}(t/\delta)^{\zeta}.$ (5.61)

Combining (5.60) and (5.61) and using the condition $\alpha\leq\zeta$, we
obtain (5.59) for some suitable $C_{3}$. This concludes the proof of Theorem
1.7.

######  Remark 5.10.

For some applications, it may be helpful to replace the Frostman condition
(1.7) with the weaker inequality

$\\#(F\cap B)\leq A(r/\delta)^{\zeta}\quad\textrm{for all balls}\ B\subset
C^{2}(I)\ \textrm{of radius}\ r\geq\delta,$ (5.62)

for some $A\geq 1$. By randomly selecting each $f\in F$ with probability $1/A$
and then applying Theorem 1.7, we obtain the following analogue of (1.8).

$\int_{E}\Big{(}\sum_{f\in F}{\bf 1}_{f^{\delta}}\Big{)}^{3/2}\leq
A^{1/2}\delta^{2-\alpha/2-\zeta/2-C\varepsilon}(\\#F).$ (5.63)

##  Appendix A Sogge’s cinematic curvature, and cinematic functions

In this section, we will explain the connection between cinematic functions
(see Definition 1.6) and Sogge’s cinematic curvature condition, as used by
Kolasa and Wolff in [15]. First, we will show that if
$\Phi\colon\mathbb{R}^{4}\to\mathbb{R}$ satisfies the cinematic curvature
condition (for simplicity, we will state this condition in a form that is
convenient to compute with), then there are sets $Y\times
R\subset\mathbb{R}^{3}$ and $I\times J\subset\mathbb{R}^{2}$, so that for each
$(y,r)\in Y\times R$, the curve $\\{x\in I\times J\colon\Phi(x,y)=r\\}$ is the
graph of a function $f_{(y,r)}\colon I\to J$, and $\\{f_{(y,r)}\\}$ is a
family of cinematic functions. The proof of this statement requires careful
computation using the inverse function theorem. These computations are
described below.

###  A.1. Cinematic curvature for normalized defining functions

Let $U=I\times J\times Y\subset\mathbb{R}^{4},$ where $I,J\subset\mathbb{R}$
are open intervals containing 0, and $Y\subset\mathbb{R}^{2}$ is a
neighborhood of 0. Let $\Phi(x,y)\colon U\to\mathbb{R}$, with

$\|\Phi\|_{C^{3}(U)}=C_{0}.$ (A.1)

We will require

$\Phi(x,0)=x_{2}+O(x^{3}),$ (A.2)
$d_{y}\Big{(}\begin{array}[]{c}\partial_{x_{1}}\Phi(x,y)\\\
\partial^{2}_{x_{1}}\Phi(x,y)/|\nabla_{x}\Phi(x,y)|\end{array}\Big{)}\Big{|}_{(x,y)=(0,0)}\quad\textrm{is
invertible}.$ (A.3)

Conditions (A.2) and (A.3) are a formulation of Sogge’s cinematic curvature
condition for the defining function $\Phi$. See, for example Section 3 or the
beginning of Section 5 from [15].

Shrinking $I$ and $Y$ if necessary, by the inverse function theorem we can
find an interval $R$ containing 0 so that for each $(y,r)\in Y\times R$, the
curve $\\{x\in I\times J\colon\Phi(x,y)=r\\}$ is the graph
$x_{2}=f_{y,r}(x_{1})$, where $f_{y,r}\in C^{2}(I)$. We will show that after
possibly further shrinking $I$ and $Y,$ the set

$\mathcal{F}=\\{f_{y,r}\colon(y,r)\in Y\times R\\}\subset C^{2}(I)$ (A.4)

is a cinematic family of functions. Specifically, we will show there exists
$\varepsilon>0$ so that for all $(y,r),(y^{\prime},r^{\prime})\in Y\times R$
and all $x_{1}\in I$, we have

$|f_{y,r}(x_{1})-f_{y^{\prime},r^{\prime}}(x_{1})|+|\dot{f}_{y,r}(x_{1})-\dot{f}_{y^{\prime},r^{\prime}}(x_{1})|+|\ddot{f}_{y,r}(x_{1})-\ddot{f}_{y^{\prime},r^{\prime}}(x_{1})|\geq\varepsilon|(y,r)-(y^{\prime},r^{\prime})|.$
(A.5)

Clearly the metric space $\mathcal{F}$ has bounded diameter, and doubling.

Step 1
After shrinking $Y,$ $I,$ and $J$ if necessary (and re-defining $U=I\times
J\times Y$), we can suppose that

$1/2\leq\partial_{x_{2}}\Phi(x,y)\leq 2\quad\textrm{for all}\ (x,y)\in U,$
(A.6)

and there exists $c>0$ so that

$\Big{|}\det\Big{(}d_{y}\Big{(}\begin{array}[]{c}\partial_{x_{1}}\Phi(x,y)\\\
\partial^{2}_{x_{1}}\Phi(x,y)/|\nabla_{x}\Phi(x,y)|\end{array}\Big{)}\Big{)}\Big{|}\geq
c\quad\textrm{for all}\ (x,y)\in U.$ (A.7)

(A.6) is a consequence of (A.2), while (A.7) is a consequence of (A.3).

Let $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}$ be small positive
numbers. We will choose $\varepsilon_{3}$ sufficiently small depending on $c$
and $C_{0}$. We will choose $\varepsilon_{2}$ sufficiently small depending on
$c,$ $C_{0}$ and $\varepsilon_{3}$. We will choose $\varepsilon_{1}$
sufficiently small depending on $c,$ $C_{0}$ and
$\varepsilon_{2},\varepsilon_{3}$.

After further shrinking $Y,$ $I,$ and $J$ if necessary, we can suppose that

$|\partial_{x_{1}}\Phi(x,y)|\leq\varepsilon_{1},\quad|\nabla^{2}_{x}\Phi(x,y)|\leq\varepsilon_{1}\quad\textrm{for
all}\ (x,y)\in U.$ (A.8)

Step 2 Let $(y,r),\ (y^{\prime},r^{\prime})\in Y\times R$. Let $x_{1}\in I$,
and let $x_{2},x_{2}^{\prime}\in J$ so that $\Phi(x_{1},x_{2},y)=r,\
\Phi(x_{1},x_{2}^{\prime},y^{\prime})=r^{\prime}$. We will define
$x=(x_{1},x_{2})$ and $x^{\prime}=(x_{1},x_{2}^{\prime})$, so
$|x-x^{\prime}|=|x_{2}-x_{2}^{\prime}|.$ Finally, we will adopt the notation
$\Phi_{1}(x,y)=\partial_{x_{1}}\Phi(x,y)$;
$\Phi_{2}(x,y)=\partial_{x_{2}}\Phi(x,y)$;
$\Phi_{11}(x,y)=\partial^{2}_{x_{1}}\Phi(x,y),$ etc.

If $|x_{2}-x_{2}^{\prime}|\geq\frac{1}{4}|r-r^{\prime}|,$ then (A.5) holds
with $\varepsilon=1/4$ and we are done. Thus we can suppose that

$|x_{2}-x_{2}^{\prime}|<\frac{1}{4}|r-r^{\prime}|.$ (A.9)

We have

$|r-r^{\prime}|=|\Phi(x,y)-\Phi(x^{\prime},y^{\prime})|\leq|\Phi(x,y)-\Phi(x^{\prime},y)|+|\Phi(x^{\prime},y)-\Phi(x^{\prime},y^{\prime})|\leq
2|x-x^{\prime}|+C|y-y^{\prime}|.$

Using (A.9) and re-arranging, we conclude that $|r-r^{\prime}|\leq
2C|y-y^{\prime}|$, so in particular

$|(y,r)-(y^{\prime},r^{\prime})|\leq 3C|y-y^{\prime}|.$ (A.10)

Step 2
Let $\varepsilon_{2}$ be a small constant to be determined later. If
$|x_{2}-x_{2}^{\prime}|\geq\varepsilon_{2}|y-y^{\prime}|$, then by (A.10),
(A.5) holds with $\varepsilon=\frac{\varepsilon_{2}}{3C}$ and we are done.
Thus we can suppose that

$|x_{2}-x_{2}^{\prime}|<\varepsilon_{2}|y-y^{\prime}|.$ (A.11)

Next, let $\varepsilon_{3}$ be a small constant to be determined later.
Suppose that

$|\Phi_{1}(x^{\prime},y)-\Phi_{1}(x,y^{\prime})|\geq\varepsilon_{3}|y-y^{\prime}|.$
(A.12)

We will show that if $\varepsilon_{1}$ and $\varepsilon_{2}$ were selected