# A discretized point-hyperplane incidence bound in $\mathbb{R}^{d}$

Thang Pham University of Science, Vietnam National University, Hanoi. Email:
<EMAIL_ADDRESS>Chun-Yen Shen Department of Mathematics, National
Taiwan University. Email<EMAIL_ADDRESS>Nguyen Pham Minh Tri Ho Chi
Minh University of Education. Email<EMAIL_ADDRESS>
###### Abstract

Let $P$ be a $\delta$-separated $(\delta,s,C_{P})$-set of points in
$B(0,1)\subset\mathbb{R}^{d}$ and $\Pi$ be a $\delta$-separated
$(\delta,t,C_{\Pi})$-set of hyperplanes intersecting $B(0,1)$ in
$\mathbb{R}^{d}$. Define

$I_{C\delta}(P,\Pi)=\\#\\{(p,\pi)\in P\times\Pi\colon p\in\pi(C\delta)\\}.$

Suppose that $s,t\geq\frac{d+1}{2}$, then we have
$I_{C\delta}(P,\Pi)\lesssim\delta|P||\Pi|$. The main ingredient in our
argument is a measure theoretic result due to Eswarathansan, Iosevich, and
Taylor (2011) which was proved by using Sobolev bounds for generalized Radon
transforms. Our result is essentially sharp, a construction will be provided
and discussed in the last section.

##  1 Introduction

We start with the following two definitions.

######  Definition 1.1 ($(\delta,s,C)$ set).

Let $0\leq s<\infty$ and $\delta\in(0,1)$ and a constant $C>0$. Given a metric
space $(X,d)$, a bounded set $E\subset X$ is called $(\delta,s,C)$-set if for
every $\delta\leq r\leq 1$ and for all ball $B\subset X$ of radius $r$, we
have

$|E\cap B|_{\delta}\leq Cr^{s}|E|_{\delta},$

where $|E|_{\delta}$ denotes the $\delta$-covering of $E$ in the space
$(X,d)$.

We note that this definition is slightly different compared to the classical
one introduced by Katz and Tao in [9].

######  Definition 1.2 (Katz-Tao $(\delta,s,C)$-set).

Let $0\leq s<\infty$ and $\delta\in(0,1)$ and a constant $C>0$. Given a metric
space $(X,d)$, a bounded set $E\subset X$ is called Katz-Tao
$(\delta,s,C)$-set if for every $\delta\leq r\leq 1$ and for all ball
$B\subset X$ of radius $r$, we have

$|E\cap B|_{\delta}\leq C\left(\frac{r}{\delta}\right)^{s}.$

Note that if $|E|_{\delta}\sim\delta^{-s}$, then the two above definitions are
equivalent.

Let $P$ be a $\delta$-separated $(\delta,s,C_{P})$-set of points in
$\mathbb{R}^{d}$ and $\Pi$ be a $\delta$-separated $(\delta,t,C_{\Pi})$-set of
hyperplanes intersecting $B(0,1)$ in $\mathbb{R}^{d}$. The number of
discretized point-plane incidences between $P$ and $\Pi$ is defined by

$I_{C\delta}(P,\Pi)=\\#\\{(p,\pi)\in P\times\Pi\colon p\in\pi(C\delta)\\},$

where $\pi(C\delta)$ denotes the $C\delta$ neighborhood of the hyperplane
$\pi$. In this paper, we treat $C_{P}$, $C_{\Pi}$, and other constants as
absolutely positive bounded constants.

Our initial motivation comes from the following recent theorem due to Orponen,
Shmerkin, and Wang [11], and Fu and Ren [4] in two dimensions.

######  Theorem 1.3.

Let $0\leq s,t\leq 2$. Then, for every $\epsilon>0$, there exists
$\delta_{0}=\delta_{0}(\epsilon)>0$ such that the following holds for all
$\delta\in(0,\delta_{0}]$. Let $P\subset B(0,1)\subset\mathbb{R}^{2}$ be a
$\delta$-separated $(\delta,s,\delta^{-\epsilon})$-set of points and
$\mathcal{T}$ be a $\delta$-separated $(\delta,t,\delta^{-\epsilon})$-set of
tubes intersecting $B(0,1)$.

  1. 1.

If $1\geq t\geq s$ or $1\geq s\geq t$, then

$I_{\delta}(P,\mathcal{T})\lesssim|P||\mathcal{T}|\delta^{\frac{st}{s+t}-O(\epsilon)}.$

  2. 2.

If $t\geq 1\geq s\geq t-1$, then

$I_{\delta}(P,\mathcal{T})\lesssim|P||\mathcal{T}|\delta^{\frac{st}{1+s}-O(\epsilon)}$

  3. 3.

If $s\geq 1\geq t\geq s-1$, then

$I_{\delta}(P,\mathcal{T})\lesssim|P||\mathcal{T}|\delta^{\frac{st}{1+t}-O(\epsilon)}.$

  4. 4.

If $t>1$ and $s>1$, then

$I_{\delta}(P,\mathcal{T})\lesssim|P||\mathcal{T}|\delta^{\kappa(s+t-1)-O(\epsilon)},~{}\kappa=\min\\{1/2,~{}1/(s+t-1)\\}.$

Throughout this paper, by $X\lesssim Y$ we mean that $X\leq CY$ for some
constant $C$, and $X\sim Y$ if $X\lesssim Y\lesssim X$.

The main purpose of this paper is to extend Theorem 1.3 to higher dimensions,
namely, in $\mathbb{R}^{d}$ with $d\geq 3$. Our first result is the following.

######  Theorem 1.4.

Let $C>0$, $0\leq s,t\leq d$. There exists $\delta_{0}=\delta_{0}(C,s,t)>0$
such that the following holds for $\delta\in(0,\delta_{0})$. Let $P$ be a
$\delta$-separated $(\delta,s,C_{P})$-set of points in
$B(0,1)\subset\mathbb{R}^{d}$ and $\Pi$ be a $\delta$-separated
$(\delta,t,C_{\Pi})$-set of hyperplanes intersecting $B(0,1)$ in
$\mathbb{R}^{d}$. Define

$I_{C\delta}(P,\Pi)=\\#\\{(p,\pi)\in P\times\Pi\colon p\in\pi(C\delta)\\}.$

Suppose that $s,t>\frac{d+1}{2}$, then we have the sharp estimate that
$I_{C\delta}(P,\Pi)\lesssim\delta|P||\Pi|$.

In the above theorem, the conditions $s>\frac{d+1}{2}$ and $t>\frac{d+1}{2}$
are required in the proof. When $s,t$ are small, using an elementary geometric
argument, we are able to prove the following non-trivial result.

######  Theorem 1.5.

Let $C>0$, $0\leq s,t\leq d$. There exists $\delta_{0}=\delta_{0}(C,s,t)>0$
such that the following holds for $\delta\in(0,\delta_{0})$. Let $P$ be a
$(\delta,s,C_{P})$ set of points in $B(0,1)\subset\mathbb{R}^{d}$ and $\Pi$ be
a $(\delta,t,C_{\Pi})$ set of hyperplanes intersecting $B(0,1)$ in
$\mathbb{R}^{d}$ with $d\geq 3$. We further assume that $s-d+2>0$. Then, for
any $\epsilon>0$, we have

$I_{C\delta}(P,\Pi)\lesssim|P|\cdot|\Pi|\cdot\delta^{f(t)(s-d+2)-\epsilon},$

where

$f(t)=\begin{cases}1/2,~{}&\text{ if }t\geq 1\\\ \dfrac{t}{1+t},~{}&\text{ if
}t<1\end{cases}.$

######  Remark 1.1.

If $t-d+2>0$, then we can use the dual arguments to obtain a similar result.
While Theorem 1.4 is optimal, we do not have any constructions on the
sharpness of this theorem.

#### Main ideas and Comparisons:

We first discuss the main idea in the proof of Theorem 1.3. Observe that if
$P$ is a $(\delta,s,C)$-set, then $P$ can be covered by at most
$|P|\delta^{s-O(\epsilon)}$ Katz-Tao $(\delta,s)$-sets. A detailed proof can
be found in [11, Lemma 3.5]. With this observation, one can apply Theorem 1.4
and Theorem 1.5 due to Fu and Ren in [4]. To prove these two theorems, Fu and
Ren used a geometric argument and some earlier results due to Guth, Solomon,
and Wang [5] and a generalization due to Bradshaw [1].

In higher dimensions $d\geq 3$, to prove Theorem 1.4, we use a completely
different approach. More precisely, we use a measure theoretic result due to
Eswarathansan, Iosevich, and Taylor [3], which was proved by using Sobolev
bounds for generalized Radon transforms. The proof of Theorem 1.5 is a
combination of Cauchy-Schwartz argument and geometric results due to Hera,
Keleti, and Mathe [6].

We also want to add a remark that the approach in the proof of Theorem 1.4 is
similar to the mechanism introduced by Iosevich, Jorati, and Laba [8] when
they studied incidences between a set of “tubes” and a homogeneous set of
points with the same size. Since our proof uses Sobolev bounds for generalized
Radon transforms, it can be generalized to a more general form, i.e. the
equation of hyperplanes $x_{d}=a_{1}x_{1}+\cdots+a_{d-1}x_{d-1}+a_{d}$ can be
replaced by $\Psi(a_{1},\ldots,a_{d},x_{1},\ldots,x_{d})=0$, where the
function $\Psi$ satisfies the Phong-Stein curvature condition (4) below.

If the set $\Pi$ of planes only needs to satisfy the property that it is
$\delta$-separated, a recent work of Dabrowski, Orponen, and Villa [2] for the
case of $(d-1)$-hyperplanes tells us that

$I_{C\delta}(P,\Pi)\lesssim\delta^{-\epsilon}\delta^{\frac{(d-1)(s+1-d)}{2d-1-s}}|P||\Pi|^{\frac{d-1}{2d-1-s}},$
(1)

where $P$ is a $\delta$-separated $(\delta,s,C_{P})$-set with $s>1$.

This result is weaker than Theorem 1.4 when $|\Pi|\leq\delta^{-d}$.

We now compare the estimate (1) and Theorem 1.5. Theorem 1.5 states that if
$s-d+2>0$, then

$I_{C\delta}(P,\Pi)\lesssim|P|\cdot|\Pi|\cdot\delta^{f(t)(s-d+2)-\epsilon},$

where

$f(t)=\begin{cases}1/2,~{}&\text{ if }t\geq 1\\\ \dfrac{t}{1+t},~{}&\text{ if
}t<1\end{cases}.$

Assume $t\geq 1$, then by a direct computation, this bound is stronger than
that of (1) when

$\delta^{-t}\lesssim|\Pi|\leq\delta^{\frac{M(2d-1-s)}{d-s}},$

where

$M=\frac{(d-1)(s+1-d)}{2d-1-s}-\frac{s-d+2}{2}.$

This range is non-empty when $2t\leq s+1<d$.

Assume $t<1$, then Theorem 1.5 is stronger than that of (1) when

$\delta^{-t}\lesssim|\Pi|\leq\delta^{\frac{M^{\prime}(2d-1-s)}{d-s}},$

where

$M^{\prime}=M-\left(\frac{t}{t+1}-\frac{1}{2}\right)(s-d+2).$

This range is non-empty when $d-2<s<d-1$.

##  2 Proof of Theorem 1.4

We first recall some notations that can be found in [2]. Let
$\mathcal{A}(d,n)$ be the set of $n$-dimensional affine subspaces in
$\mathbb{R}^{d}$. The metric $d_{\mathcal{A}}$ defined on $\mathcal{A}(d,n)$
([10, page 53]) is defined by

$d_{\mathcal{A}}(V,W)=||\pi_{V_{0}}-\pi_{W_{0}}||_{op}+|a-b|,$

where $V=V_{0}+a$ and $W=W_{0}+b$, $V_{0},W_{0}\in\mathcal{G}(d,n)$ (the set
of $n$-dimensional subspaces), $a\in V_{0}^{\perp}$, $b\in W_{0}^{\perp}$, and
$||\cdot||_{op}$ is the operator norm.

Let $\gamma_{1}$ and $\gamma_{2}$ be two $(d-1)$-planes defined by

$\gamma_{1}\colon x_{d}=a_{1}x_{1}+\cdots+a_{d-1}x_{d_{1}}+a_{d},$

and

$\gamma_{2}\colon x_{d}=b_{1}x_{1}+\cdots+b_{d-1}x_{d_{1}}+b_{d}.$

The following formula is more useful in practise

$d_{\mathcal{A}}\left(\gamma_{1},\gamma_{2}\right)=\left|\dfrac{(a_{1}\ldots,a_{d-1},-1)}{\left|(a_{1}\ldots,a_{d-1},-1\right|}-\dfrac{(b_{1}\ldots,b_{d-1},-1)}{\left|(b_{1}\ldots,b_{d-1},-1\right|}\right|+\left|\dfrac{a_{d}}{\left|(a_{1}\ldots,a_{d-1},-1\right|}-\dfrac{b_{d}}{\left|(b_{1}\ldots,b_{d-1},-1)\right|}\right|.$

This can be proved by a direct computation or can be found in [7, Lemma 2.5].

For each plane defined by $x_{d}=a_{1}x_{1}+\cdots+a_{d-1}x_{d-1}+a_{d}$, we
call the point $(a_{1},\ldots,a_{d})$ its dual point in $\mathbb{R}^{d}$.

Let $D\colon\mathbb{R}^{d}\to\mathcal{A}(d,d-1)$ be the map defined by

$D\colon(x_{1},\ldots,x_{d})\mapsto\left\\{y_{d}=\sum_{i=1}^{d-1}x_{i}y_{i}+x_{d}\right\\}.$

There are some properties of $D$ we should keep in mind, namely, the
restriction of $D$ to the ball $B(R)$, $0<R<\infty$, is bilipschitz onto its
image, and the bilipschitz constant depends only on $R$ and $d$. Thus, $D$ is
injective. The inverse map of $D$, denoted by
$D^{*}\colon\mathtt{im}(D)\to\mathbb{R}^{d}$, defined by

$D(x_{1},\ldots,x_{d})\mapsto(-x_{1},\ldots,-x_{d-1},x_{d}).$

There exists a dimensional constant $0<r_{d}<1$ such that the restriction of
$D^{*}$ to $B(V_{0},r_{d})$ is bilipschitz onto its image, here
$V_{0}=D(0,\ldots,0)$.

With these notations, the next two lemmas can be proved with standard
arguments.

######  Lemma 2.1.

Let $S$ be a set of $\delta$-separated hyperplanes intersecting the unit ball
$B(0,1)$. Then the set of corresponding dual points is also
$c\delta$-separated for some absolute constant $c$.

###### Proof.

If two hyperplanes $\gamma_{1}$ and $\gamma_{2}$ are $\delta$-separated with
the metric

$\displaystyle
d_{\mathcal{A}}\left(\gamma_{1},\gamma_{2}\right)=\left|\dfrac{(a_{1}\ldots,a_{d-1},-1)}{\left|(a_{1}\ldots,a_{d-1},-1\right|}-\dfrac{(b_{1}\ldots,b_{d-1},-1)}{\left|(b_{1}\ldots,b_{d-1},-1\right|}\right|+\left|\dfrac{a_{d}}{\left|(a_{1}\ldots,a_{d-1},-1\right|}-\dfrac{b_{d}}{\left|(b_{1}\ldots,b_{d-1},-1)\right|}\right|,$

where $\gamma_{1}$ and $\gamma_{2}$ are respectively defined by equations

$(\gamma_{1}):y_{d}=\sum_{i=1}^{d-1}a_{i}y_{i}+a_{d},$

and

$(\gamma_{2}):y_{d}=\sum_{i=1}^{d-1}b_{i}y_{i}+b_{d},$

then we want to prove that the usual Euclidean distance between two points
$(a_{1},\ldots,a_{d})$ and $(b_{1},\ldots,b_{d})$ is at least $c\delta$ for
some absolute constant $c$. For the sake of simplicity, we will denote
$u=(a_{1}\ldots,a_{d-1},-1)$ and $v=(b_{1}\ldots,b_{d-1},-1)$. Then clearly
$|u|,|v|\geq 1$. Since two hyperplanes $\gamma_{1},\gamma_{2}$ are
$\delta$-separated, we have that
$d_{\mathcal{A}}\left(\gamma_{1},\gamma_{2}\right)\geq\delta$. With these
assumptions, we attempt to prove that

$\sqrt{\sum_{i=1}^{d}(a_{i}-b_{i})^{2}}\gtrsim\delta$

It should be noted that, if we view $a_{d},b_{d}$ as vectors
$(0,\ldots,a_{d}),(0,\ldots,b_{d})$ respectively, then the usual Euclidean
distance between two points $(a_{1},\ldots,a_{d})$ and $(b_{1},\ldots,b_{d})$
can be rewritten as $\sqrt{|u-v|^{2}+|a_{d}-b_{d}|^{2}}$.

Thus, the goal is to prove the following inequality

$\displaystyle\left|\dfrac{u}{|u|}-\dfrac{v}{|v|}\right|+\left|\dfrac{a_{d}}{|u|}-\dfrac{b_{d}}{|v|}\right|\lesssim\sqrt{|u-v|^{2}+|a_{d}-b_{d}|^{2}}.$
(2)

Instead of proving 2, we prove a stronger version

$\displaystyle\left|\dfrac{u}{|u|}-\dfrac{v}{|v|}\right|+\left|\dfrac{a_{d}}{|u|}-\dfrac{b_{d}}{|v|}\right|\lesssim|u-v|+|a_{d}-b_{d}|.$
(3)

We first prove

$\left|\dfrac{u}{|u|}-\dfrac{v}{|v|}\right|\leq|u-v|.$

Squaring both sides gives

$|u|^{2}+|v|^{2}-2u\cdot v\geq 2-\dfrac{2u\cdot v}{|u||v|}.$

This is reduced to

$2|u||v|-2u\cdot v\geq 2-\dfrac{2u\cdot v}{|u||v|}.$

We denote $|u||v|=x,u\cdot v=y$, then the above can be represented as

$\displaystyle(x-1)(x-y)\geq 0.$

This inequality is true since $|u|,|v|\geq 1$ and $x\geq y$ by the Cauchy-
Schwarz inequality.

Now we estimate $\left|\frac{a_{d}}{|u|}-\frac{b_{d}}{|v|}\right|$. Since all
hyperplanes intersect $B(0,1)$, one has $\frac{|a_{d}|}{|u|}$ and
$\frac{|b_{d}|}{|v|}$, i.e the distances from the origin to the hyperplanes
$\gamma_{1},\gamma_{2}$, are not larger than 1. Then

$\left|\dfrac{a_{d}}{|u|}-\dfrac{b_{d}}{|v|}\right|=\left|\dfrac{a_{d}|v|-a_{d}|u|+a_{d}|u|-b_{d}|u|}{|u||v|}\right|\leq|u-v|+|a_{d}-b_{d}|.$

This completes the proof. ∎

######  Lemma 2.2.

Let $\Pi$ be a set of $\delta$-separated $(\delta,t,C_{\pi})$-hyperplanes
intersecting $B(0,1)$. Then the set of dual points is $c\delta$-separated
$(\delta,t,c^{\prime})$-set for some absolute constants $c,c^{\prime}>0$.

###### Proof.

We denote $\Pi^{*}$ the set of dual points of hyperplanes $\Pi$. It is
immediate that the set $\Pi^{*}$ is $c\delta$-separated by Lemma 2.1. Hence,
we only need to prove that, for arbitrary ball $B(x,r)$, we have

$|\Pi^{*}\cap B(x,r)|_{\delta}\leq
c^{\prime}r^{t}|\Pi^{*}|_{\delta},\delta\leq r\leq 1,$

for some constant $c^{\prime}$ that will be chosen later. It is sufficient to
prove that

$|\left\\{x^{\prime}\in\Pi^{*}:d(x^{\prime},x)\leq r\right\\}|\leq
c^{\prime}r^{t}|\Pi^{*}|_{\delta},\delta\leq r\leq 1.$

As mentioned earlier, the map $D$ is bilipschitz onto its image, we have
$|\left\\{x^{\prime}\in\Pi^{*}:d(x^{\prime},x)\leq r\right\\}|$ is at most

$\left|\left\\{D(x^{\prime})\in\Pi:d_{\mathcal{A}}(D(x^{\prime}),D(x))\leq
K_{D}r\right\\}\right|\leq C_{\pi}K_{D}^{t}r^{t}|\Pi^{*}|_{\delta},\delta\leq
r\leq 1.$

Choose $c^{\prime}=C_{\pi}K_{D}^{t}$, then the lemma follows. Note that
$K_{D}^{t}$ can be replaced by some constant that does not depend on $t$ since
$t\in(0,d)$. ∎

###  2.1 Sobolev bounds for generalized Radon transforms and consequences

In this section, we recall some known results that make use the boundedness of
general Radon transforms. Let $g\colon\mathbb{R}^{d}\to\mathbb{R}$ be a
Schwartz function, $t\in\mathbb{R}$,
$\psi\colon\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a
smooth cut-off function, and
$\Psi(\mathbf{x},\mathbf{y}):\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
be a smooth function with some suitable assumptions. We define

$T_{\Psi_{t}}g(\mathbf{x}):=\int_{\\{\Psi(\mathbf{x},\mathbf{y})=t\\}}g(\mathbf{y})\psi(\mathbf{x},\mathbf{y})d\sigma_{\mathbf{x},t}(\mathbf{y}),$

where $d\sigma_{\mathbf{x},t}$ is the Lebesgue measure on the set
$\\{\mathbf{y}\colon\Psi(\mathbf{x},\mathbf{y})=t\\}.$

We denote the usual $L^{2}$-Sobolev space of $L^{2}$ functions with $s$
generalized derivatives in $L^{2}(\mathbb{R}^{d})$ by
$L^{2}_{s}(\mathbb{R}^{d})$. The following theorem was proved by Eswarathasan,
Iosevich and Taylor in [3].

######  Theorem 2.3 ([3], Proposition 2.2).

Let $E\subset\mathbb{R}^{d}$ be a compact set with $\dim_{H}(E)=\alpha$, and
$\mu$ be the corresponding Frostman measure on $E$. Assume that $T_{\Psi_{t}}$
maps $L^{2}$ to $L^{2}_{s}$ with constants uniform in a small neighborhood of
$t$, and $d-s<\alpha<d$. Then we have

$\mu\times\mu\\{(\mathbf{x},\mathbf{y})\in E\times
E:t\leq|\Psi(\mathbf{x},\mathbf{y})|\leq t+\epsilon\\}\lesssim\epsilon.$

######  Remark 2.1.

It can be checked from Eswarathasan-Iosevich-Taylor’s proof that the same
result holds for two different sets $E\times F$, namely,

$\mu_{E}\times\mu_{F}\\{(\mathbf{x},\mathbf{y})\in E\times
F:t\leq|\Psi(\mathbf{x},\mathbf{y})|\leq t+\epsilon\\}\lesssim\epsilon,$

if $T_{\Psi_{t}}$ maps $L^{2}$ to $L^{2}_{s}$ with $d-s<\alpha,\beta<d$, where
$\alpha=\dim_{H}(E)$ and $\beta=\dim_{H}(F)$. Notice also that the sets $E$
and $F$ are not required to be A-D regular. Moreover in the proofs, the only
thing that is needed associated with the set $E$ is the existence of a
probably measure $\mu$ supported on $E$ that satisfies $\mu(B(x,r))\lesssim
r^{\alpha}$, the same applies for $F$.

To apply the above theorem in the proof of Theorem 1.4, we need to recall a
celebrated result of Phong and Stein [12] stating that the operator
$T_{\Psi_{t}}$ is uniformly bounded from $L^{2}$ to $L^{2}_{s}$ on a small
neighborhood of $t$ with $s=\frac{d-1}{2}$ if the so-called Phong-Stein
rotational curvature condition

$\det\begin{pmatrix}0&\nabla_{\mathbf{x}}\Psi\\\
-\nabla_{\mathbf{y}}\Psi&\frac{\partial^{2}\Psi}{\partial x_{i}\partial
y_{j}}\end{pmatrix}\neq 0$ (4)

holds on the set
$\\{(\mathbf{x},\mathbf{y})\colon\Psi(\mathbf{x},\mathbf{y})=t\\}$. This and
Remark 2.1 imply the following theorem.

######  Theorem 2.4.

Let $\mu_{E},\mu_{F}$ be probability measures on compact sets
$E,F\subset\mathbb{R}^{d},$ respectively, satisfying

$\mu_{E}(B(x,r))\lesssim r^{\alpha},~{}\mu_{F}(B(x,r))\lesssim r^{\beta},$

for all $x\in\mathbb{R}^{d}$ and $r>0$. Suppose that the Phong-Stein
rotational curvature condition (4) holds for the function $\Psi$, and
$d-(d-1)/2<\alpha,\beta<d.$ Then, for $\epsilon>0,$ we have

$\mu_{E}\times\mu_{F}\\{(\mathbf{x},\mathbf{y})\in E\times
F:|\Psi(\mathbf{x},\mathbf{y})|\leq\epsilon\\}\lesssim\epsilon.$

###  2.2 Proof of Theorem 1.4

Let $\Pi^{*}$ be the set of dual points corresponding to planes in $\Pi$. As
we proved in Lemma 2.2 that this set is $c\delta$-separated
$(\delta,t,c^{\prime})$-set.

We first define two probability measures on $P(\delta)$ and
$\Pi^{*}(c\delta)$, denoted by $\mu_{P}$ and $\mu_{\Pi}$, respectively, as
follows:

$\mu_{P}(X)=\frac{|X\cap P(\delta)|}{|P(\delta)|},$

and

$\mu_{\Pi^{*}}(X)=\frac{|X\cap\Pi^{*}(c\delta)|}{|\Pi^{*}(c\delta)|}.$

Since $P$ is $(\delta,s,C_{P})$-set and $\Pi^{*}$ is
$(c\delta,t,c^{\prime})$-set, we have

$\mu_{P}(B(x,r))\lesssim r^{s},$

and

$\mu_{\Pi^{*}}(B(x,r))\lesssim r^{t},$

for all $x\in\mathbb{R}^{d}$ and $r\geq\delta$.

For $r\leq\delta$, we have $|B(x,r)\cap P(\delta)|\lesssim r^{d}$. On the
other hand, we have $|P(\delta)|\gtrsim\delta^{d-s},$ which follows from the
assumption that $P$ is a $(\delta,s,C_{P})$-set. For $r\leq\delta$ and $s<d$,
we have $r^{d-s}\leq\delta^{d-s}$, this means that

$\mu_{P}(B(x,r))\lesssim\frac{r^{d}}{\delta^{d-s}}\lesssim r^{s}.$

The same holds for $\Pi^{*}$. In other words, the two probability measures
$\mu_{P}$ and $\mu_{\Pi^{*}}$ are Frostman measures with exponents $s$ and
$t$, respectively.

To proceed further, we may assume that all hyperplanes are defined by the
equation of the form

$a_{1}x_{1}+\cdots+a_{d-1}x_{d-1}+a_{d}=x_{d}.$ (5)

This gives that the dual points are of the form $(a_{1},\ldots,a_{d-1},1)$.
Define

$\Psi(x_{1},\ldots,x_{d},a_{1},\ldots,a_{d})=a_{1}x_{1}+\cdots+a_{d-1}x_{d-1}-x_{d}+a_{d}.$

A direct computation shows that this function satisfies the curvature
condition (4). We now observe that $(x_{1},\ldots,x_{d})\in\pi(C\delta)$,
where $\pi$ is defined by (5), if
$|\Psi(x_{1},\ldots,x_{d},a_{1},\ldots,a_{d})|\leq C\delta$. On the other
hand, if $|\Psi(x_{1},\ldots,x_{d},a_{1},\ldots,a_{d})|\leq C\delta$, then
$|\Psi(U,V)|\lesssim\delta$ for all $U\in B((x_{1},\ldots,x_{d}),\delta)$ and
$V\in B((a_{1},\ldots,a_{d}),c\delta)$, when $\delta$ is small enough. This
infers that

$\frac{1}{|P||\Pi|}|I_{\delta}(P,\Pi)|\leq\mu_{P}\times\mu_{\Pi^{*}}\left\\{(U,V)\in
P(\delta)\times\Pi^{*}(c\delta)\colon|\Psi(U,V)|\lesssim\delta\right\\}.$

Therefore, our result is reduced to show the following.

$\mu_{P}\times\mu_{\Pi^{*}}\left\\{(U,V)\in
P(\delta)\times\Pi^{*}(c\delta)\colon|\Psi(U,V)|\lesssim\delta\right\\}\lesssim\delta,$

which follows from Theorem 2.4.

##  3 Proof of Theorem 1.5

In this section, we present an elementary argument to study the incidence
problem.

For $p\in P$, let $I(p)$ be the set of hyperplanes $\pi\in\Pi$ such that
$p\in\pi(C\delta)$. We observe that

$I_{C\delta}(P,\Pi)=\sum_{p\in P}|I(p)|.$

Thus, for $x,y>0$ and $x\geq y$, by the Hölder inequality, we have

$I_{C\delta}(P,\Pi)\leq|P|^{\frac{x}{x+y}}\left(\sum_{p}|I(p)|^{\frac{x+y}{y}}\right)^{\frac{y}{x+y}}.$

This implies that

$I_{C\delta}(P,\Pi)^{x+y}\leq|P|^{x}\left(\sum_{p}|I(p)|^{\frac{x+y}{y}}\right)^{y}.$

To proceed further, we need to estimate the sum $\sum_{p}|I(p)|^{1+x/y}$. We
have

$\displaystyle\sum_{p\in P}|I(p)|^{1+x/y}=\sum_{p\in P}\sum_{\pi\colon
p\in\pi(C\delta)}|I(p)|^{x/y}=\sum_{\pi\in\Pi}J(\pi),$

here $J(\pi)=\sum_{p\in\pi(C\delta)}|I(p)|^{x/y}$, which can be represented as
follows

$\displaystyle J(\pi)=\sum_{p\in\pi(C\delta)}\left(\sum_{i}|I(p)\cap
J_{2^{i}\delta}(\pi)|\right)^{x/y}\lesssim\sum_{p\in\pi(C\delta)}\sum_{i=1}^{\log\delta^{-1}}|I(p)\cap
J_{2^{i}\delta}(\pi)|^{x/y},$

where $J_{2^{i}\delta}(\pi)$ is the set of hyperplanes $\pi^{\prime}$ such
that $d_{\mathcal{A}}(\pi,\pi^{\prime})\sim 2^{i}\delta$.

Since the set of planes is $(\delta,t,C_{\Pi})$, we have

$|J_{2^{i}\delta}(\pi)|\lesssim(2^{i}\delta)^{t}|\Pi|.$

######  Lemma 3.1.

Let $\pi$ and $\pi^{\prime}$ be two hyperplanes in $\mathcal{A}(d,d-1)$.
Assume these two planes both intersect the unit ball $B(0,1)$ and
$d_{\mathcal{A}}(\pi,\pi^{\prime})=w>\delta$, then the intersection
$\pi(\delta)\cap\pi^{\prime}(\delta)\cap B(0,1)$ can be covered by at most
$\delta^{-(d-2)}$ cubes of parameters
$\frac{\delta}{w}\times\delta\times\cdots\times\delta$ in $\mathbb{R}^{d}$.

###### Proof.

The proof is essentially a combination of a number of results from [6]. Let
$e_{0}=(0,\cdots,0)$ and $\left\\{e_{1},\cdots,e_{d}\right\\}$ be the standard
basis of $\mathbb{R}^{d}$. If we denote $H_{0}=\left\langle
e_{d}\right\rangle$ then $H_{0}$ is a line and $\pi\cap H_{0}$ is a point of
the form $(0,\cdots,0,a_{0})$. We also denote $H_{i}=e_{i}+H_{0}$ so that
$\pi\cap H_{i}$ is also a point of the form $(0,\cdots,1,\cdots,a^{i})$. Let
$b_{i}=a_{i}-a_{0}$, then we refer to $a_{0}$ as the vertical intercept and
$b_{i}$ as the slopes of hyperplane $\pi$.

For each hyperplane $\pi$, we associate it to a point
$x=x(\pi)=(a_{0},b_{1},\cdots,b_{d-1})\in\mathbb{R}^{d}$ and note that the map
$\varphi\colon\Pi\mapsto x(\Pi)$ is well-defined and injective. The space
generated by this map is called code space. We endow this space with the
maximum metric, namely,

$||x-x^{\prime}||:=\max\left(|a_{0}-a_{0}^{\prime}|,\max_{i=1,\ldots,d-1}\left(|b_{i}-b_{i}^{\prime}|\right)\right).$

As noted in [6, Remark 4.2], the maximum metric on the code space is strongly
equivalent to the metric between hyperplanes $d_{\mathcal{A}}$, in the sense
that $||\varphi(\pi)-\varphi(\pi^{\prime})||\sim
d_{\mathcal{A}}(\pi,\pi^{\prime})$.

Fix two hyperplanes $\pi,\pi^{\prime}$, and denote their two corresponding
codes by $x(\pi)$ and $x(\pi^{\prime})$. Then we can present the plane $\pi$
by the equation

$a_{0}+b_{1}t_{1}+\cdots+b_{d-1}t_{d-1}=t_{d},(t_{1},\ldots,t_{d-1})\in\mathbb{R}^{d-1},$

where $x(\pi)=(a_{0},b_{1},\cdots,b_{d-1})$. Similarly, we can define
$\pi^{\prime}$ from its code coordinate. As presented in the proof of [6,
Lemma 4.3] that there is a constant $c$ independent on $\delta$ such that:

$\pi(\delta)\subset\pi+(\left\\{0\right\\}\times(-c\delta,c\delta)).$

Therefore, we have

$\pi(\delta)\cap\pi^{\prime}(\delta)\cap\mathcal{C}\subset\left\\{(t,u)\in\mathbb{R}^{d}:u\in
B(t_{d},c\delta)\cap B(t^{\prime}_{d},c\delta)\right\\},$

where $\mathcal{C}=\mathcal{C}_{d-1}\times L^{1}$ and $\mathcal{C}_{d-1}$ is
the convex hull of $(0,\ldots,0),e_{1},\ldots,e_{d-1}$, and $L^{1}$ is the
unit segment centered at the origin in $<e_{d}>$.

If
$|a_{0}-a_{0}^{\prime}|>\max_{i=1,\ldots,d-1}\left(|b_{i}-b_{i}^{\prime}|\right)+D\delta$,
then choosing $D=2c$ implies $B(t_{d},c\delta)\cap B(t^{\prime}_{d},c\delta)$
is empty. This infers that
$\pi(\delta)\cap\pi^{\prime}(\delta)\cap\mathcal{C}=\emptyset$.

If $\max_{i=1,\ldots,d-1}(|b_{i}-b_{i}^{\prime}|)>0$ and $B(t_{d},c\delta)\cap
B(t^{\prime}_{d},c\delta)\neq\emptyset$, we set
$N=\left\\{t\in\mathcal{C}_{d-1}:|t_{d}-t^{\prime}_{d}|<2c\delta\right\\}$,
then we have

$N=\left\\{t\in\mathcal{C}_{d-1}:p_{-}(t)\leq t_{d-1}\leq p_{+}(t)\right\\}$

where

$p_{-}(t)=p_{-}(t_{1},\cdots,t_{d-2})=\dfrac{-2c\delta-(a_{0}-a^{\prime}_{0})-\sum_{i=1}^{d-2}t_{i}(b_{i}-b^{\prime}_{i})}{b_{d-1}-b^{\prime}_{d-1}},$

and

$p_{+}(t)=p_{+}(t_{1},\cdots,t_{d-2})=\dfrac{2c\delta-(a_{0}-a^{\prime}_{0})-\sum_{i=1}^{d-2}t_{i}(b_{i}-b^{\prime}_{i})}{b_{d-1}-b^{\prime}_{d-1}}.$

This implies that $N$ is the intersection of $\mathcal{C}_{d-1}$ and the strip
between two parallel hyperplanes $\left\\{t_{d-1}=p_{-}(t)\right\\}$ and
$\left\\{t_{d-1}=p_{+}(t)\right\\}$. By a direct computation, the distance
between these two hyperplanes is equal to

$d(p_{-}(t),p_{+}(t))=\dfrac{2c\delta}{\sqrt{\sum_{i=1}^{d-2}(b_{i}-b^{\prime}_{i})^{2}}}\lesssim\dfrac{\delta}{\max_{i=1,\ldots,d-1}|b_{i}-b_{i}^{\prime}|}.$

Hence, $N$ is contained in a rectangular box that has the shortest side of
length $d\lesssim\delta/w$ and the other $d-2$ sides of length at most
$diam(\mathcal{C})=\sqrt{2}$.

To conclude the proof, we do a rotation if needed to assume that the
hyperplane $\pi$ has $e_{d}$ as normal vector. This gives

$\pi:f(t)=a_{0}.$

The hyperplane $\pi^{\prime}$ has the equation of the form

$\pi^{\prime}:g(t)=a_{0}^{\prime}+b_{1}^{\prime}t_{1}+\cdots+b_{d-1}^{\prime}t_{d-1}.$

Under these hypotheses, if $(t,u)$ and $(t^{\prime},u^{\prime})$ are two
elements inside $\pi(\delta)\cap\pi^{\prime}(\delta)\cap\mathcal{C}$, then
triangle inequality gives

$|u-u^{\prime}|=|u-f(t)+f(t^{\prime})-u^{\prime}|\leq|u-f(t)|+|u-f(t^{\prime})|<2\delta$

Therefore, the $d$-th coordinate of the intersection part is contained in a
box of dimension $\lesssim\delta$. In other words, the intersection
$\pi(\delta)\cap\pi^{\prime}(\delta)\cap\mathcal{C}$ can be covered by roughly
$\delta^{2-d}$ dyadic boxes of size
$\frac{\delta}{w}\times\underbrace{\delta\times\cdots\delta}_{\textrm{ $d-1$
times}}$. This completes the proof. ∎

######  Lemma 3.2.

Fix $\pi\in\Pi$. For any $\delta<w\ll 1$, we have

$\sum_{p\in\pi(c\delta)}|I(p)\cap
J_{w}(\pi)|\lesssim|J_{w}(\pi)|\cdot|P|\delta^{s}\cdot\frac{1}{w^{\min\\{s,1\\}}}\cdot\frac{1}{\delta^{d-2}}.$

###### Proof.

Fix $\pi\in J_{w}(\pi)$, then it follows from Lemma 3.1 that
$\pi\cap\pi^{\prime}$ is contained in the union of $\delta^{2-d}$ boxes with
parameters $\frac{\delta}{w}\times\delta\times\cdots\times\delta$. This means
that the diameter of each box is $\sim\delta/w$. We now bound the above sum in
two ways:

Since $P$ is $\delta$-separated, each box contains at most
$\lesssim\frac{1}{w}$ elements from $P$. This gives

$\sum_{p\in\pi(c\delta)}|I(p)\cap
J_{w}(\pi)|\lesssim|J_{w}(\pi)|\cdot\frac{1}{w}\cdot\frac{1}{\delta^{d-2}}.$

We also observe that each box is contained in a ball of radius $\delta/w$,
this infers each box contains at most $C_{P}|P|\delta^{s}\frac{1}{w^{s}}$
balls of $P$ since $P$ is $(\delta,s,C_{P})$ set. In total, one has

$\sum_{p\in\pi(c\delta)}|I(p)\cap
J_{w}(\pi)|\lesssim|J_{w}(\pi)|\cdot|P|\delta^{s}\cdot\frac{1}{w^{s}}\cdot\frac{1}{\delta^{d-2}}.$

Note that by assumption we have $|P|\delta^{s}\geq 1$ so that combining these
two estimates, we get

$\sum_{p\in\pi(c\delta)}|I(p)\cap
J_{w}(\pi)|\lesssim|J_{w}(\pi)|\cdot|P|\delta^{s}\cdot\frac{1}{w^{\min\\{s,1\\}}}\cdot\frac{1}{\delta^{d-2}}.$

This completes the proof of the lemma. ∎

We now continue the proof of the incidence estimate. In particular,

$\displaystyle J(\pi)$
$\displaystyle\lesssim\sum_{p\in\pi(C\delta)}\sum_{i}|I(p)\cap
J_{2^{i}\delta}(\pi)|^{x/y}$
$\displaystyle\lesssim\sum_{i}\sum_{p\in\pi(C\delta)}|I(p)\cap
J_{2^{i}\delta}(\pi)|\cdot|J_{2^{i}\delta}(\pi)|^{x/y-1}$
$\displaystyle\lesssim\sum_{i}|(2^{i}\delta)^{t}|\Pi||^{x/y}\cdot|P|\delta^{s}\cdot\frac{1}{(2^{i}\delta)^{\min\\{s,1\\}}}\cdot\frac{1}{\delta^{d-2}}.$

We now fall into the following cases:

If $s\geq 1$ and $tx/y\geq 1$, then

$J(\pi)\lesssim|\Pi|^{x/y}\cdot|P|\delta^{s}\cdot\frac{1}{\delta^{d-2}}\cdot\sum_{i}(2^{i}\delta)^{\frac{tx}{y}-1}\lesssim|\Pi|^{x/y}\cdot|P|\delta^{s}\cdot\frac{1}{\delta^{d-2}}.$

If $s\geq 1$ and $tx/y<1$, then

$J(\pi)\lesssim|\Pi|^{x/y}\cdot|P|\delta^{s}\cdot\frac{1}{\delta^{d-2}}\cdot\sum_{i}(2^{i}\delta)^{\frac{tx}{y}-1}\lesssim|\Pi|^{x/y}\cdot|P|\delta^{s+\frac{tx}{y}-1}\cdot\frac{1}{\delta^{d-2}}.$

If $s<1$ and $tx/y\geq s$, then

$J(\pi)\lesssim|\Pi|^{x/y}\cdot|P|\delta^{s}\cdot\frac{1}{\delta^{d-2}}\cdot\sum_{i}(2^{i}\delta)^{\frac{tx}{y}-s}\lesssim|\Pi|^{x/y}\cdot|P|\delta^{s}\cdot\frac{1}{\delta^{d-2}}.$

If $s<1$ and $tx/y<s$, then

$J(\pi)\lesssim|\Pi|^{x/y}\cdot|P|\delta^{s}\cdot\frac{1}{\delta^{d-2}}\cdot\sum_{i}(2^{i}\delta)^{\frac{tx}{y}-s}\lesssim|\Pi|^{x/y}\cdot|P|\delta^{\frac{tx}{y}-d+2}.$

Plugging these estimates into $I_{C\delta}(P,\Pi)$, one has

Case $1$: If $s\geq 1$ and $tx/y\geq 1$, then

$I_{C\delta}(P,\Pi)\lesssim|P||\Pi|\delta^{\frac{y}{x+y}\left(s-d+2\right)}.$

If $s-d+2<0$ then we only have the trivial upper bound. Thus, this upper bound
is valid in the range $s-d+2>0$. By choosing $x=y$, we obtain the upper bound

$I_{C\delta}(P,\Pi)\lesssim|P||\Pi|\delta^{\frac{1}{2}\left(s-d+2\right)}.$

Case $2$: If $s\geq 1$ and $tx/y<1$, then

$I_{C\delta}(P,\Pi)\lesssim|P||\Pi|\delta^{\frac{y}{x+y}\left(s+\frac{tx}{y}-d+1\right)}.$

We observe that $t\leq 1$ since $x\geq y$. If $s-d+2<0$, then we only have the
trivial upper bound. If $s-d+2>0$, then we can choose $y=xt+\epsilon$,
$\epsilon>0$, to get the upper bound

$I_{C\delta}(P,\Pi)\lesssim|P||\Pi|\delta^{\frac{t}{1+t}\left(s-d+2\right)-\epsilon}.$

Case $3$: If $s<1$ and $tx/y\geq s$, then

$I_{C\delta}(P,\Pi)\lesssim|P||\Pi|\delta^{\frac{y}{x+y}\left(s-d+2\right)}.$

In this case, since $s<1$ and $d\geq 3$, we have $s-d+2$ is always negative,
so only trivial upper bound is obtained.

Case $4$: If $s<1$ and $tx/y<s$, then

$I_{C\delta}(P,\Pi)\lesssim|P||\Pi|\delta^{\frac{y}{x+y}\left(\frac{tx}{y}-d+2\right)}.$

Since $tx/y<s$, $s<1$, and $d\geq 3$, we only have the trivial upper bound in
this case.

Putting these cases together, Theorem 1.5 follows.

##  4 Sharpness of Theorem 1.4

In this section we provide an example to show the sharpness of our result in
Theorem 1.4. The construction is mainly adapted from the two dimensional
example due to Fu and Ren in [4]. For the reader’s convenience, we sketch the
ideas here.

We first recall that Construction 4 in [4] shows that in the plane when
$s+t\geq 3$, the incidence between balls and tubes is
$\sim\delta^{-(s+t-1)}=\delta|P||\mathcal{T}|$. We will see that their example
can be extended to higher dimensions which gives us a sharp upper bound.

Note that the incidences results of Fu and Ren [4] in the plane are applied to
the sets of balls and tubes satisfying Definition 1.2 (Katz-Tao $(\delta,s)$
sets). However, as we mentioned before, when the set is of size
$\sim\delta^{-s}$ then Definitions 1.1 and 1.2 are equivalent. Moreover, in
their sharpness example, what they constructed are actually $(\delta,s,C)$
sets. Therefore, we can extend their construction to higher dimensional
spaces. In the rest, we denote $D=\delta^{-1}$ for convenience.

###  4.1 Construction in $\mathbb{R}^{3}$

The idea is to construct a configuration such that the intersection between
this configuration and the plane $O_{xy}$ is exactly Construction 4 in [4].
Thus, we can reduce to the two-dimensional case when we consider the
intersection between $\delta$-hyperplanes and $\delta$-balls with $O_{xy}$.
Then we move the plane $O_{xy}$ vertically by spacing $2\delta$ (two
consecutive planes have distance $2\delta$) to get the desired bound.

We consider balls of form:

$(x-a_{1})^{2}+(y-a_{2})^{2}+z^{2}=\delta^{2}.$

Clearly these are $\delta$-balls in $\mathbb{R}^{3}$ whose intersection with
$O_{xy}$ are also $\delta$-balls in dimension two. For each tube in
Construction 4 in [4], we just extend the tube vertically to the
$\mathbb{R}^{3}$ which becomes a plane. Thus, if we put balls by spacing
$2\delta$ in $\mathbb{R}^{3}$ (the centers of two consecutive balls have
distance of $2\delta$), then the incidence between these balls and planes is
$\sim\delta^{-(s+t-1)}\cdot\delta^{-1}=\delta\cdot\delta^{-s-1}\cdot\delta^{-t}$
which is equal to $\delta|P||\Pi|$.

The next step is to check that the set of balls and hyperplanes constructed
above satisfy Definition 1.1. Since the hyperplanes are just the tubes
expanding vertically, the number of $\delta$-hyperplanes is equal to that of
$\delta$-tubes, i.e, $\sim D^{t}$. Moreover, as already shown in the paper of
Fu and Ren [4] (page 6) that the set of $\delta$-tubes is a $(\delta,t,C)$ set
which gives that our set of $\delta$-hyperplanes is also a $(\delta,t,C)$ set.

Now for each copy of $O_{xy}$, there are $\sim D^{s}$ balls, as stated in [4],
and there are $\delta^{-1}$ copies in total. So $|P|\sim D^{s+1}$. Moreover,
as shown in [4], for each copy of $O_{xy}$, a ball of radius $w$ in the plane
contains at most $Cw^{s}\delta^{-s}$ $\delta$-balls. So, a ball $B_{w}$ in
$\mathbb{R}^{3}$ contains at most $Cw^{s}\delta^{-s-1}$ $\delta$-balls in $P$.
This gives that

$|\\{p\in P:p\subset B_{w}\\}|\leq Cw^{s}\delta^{-s-1}\sim Cw^{s}|P|.$

Therefore, the set $P$ is a $(\delta,s,C)$-set.

###  4.2 Construction in $\mathbb{R}^{d}$, $d\geq 4$

For dimension $d>3$, the construction is inductively constructed. For
instance, we consider $d=4$. We directly extend each hyperplane from above
3-dimensional construction to a hyperplane in $\mathbb{R}^{4}$ by just adding
a variable $x_{4}$. Precisely, assume a hyperplane in above 3-dimensional
construction is defined by $\\{(x_{1},x_{2},x_{3}):ax_{1}+bx_{2}+cx_{3}=e\\}$,
then the extension in $\mathbb{R}^{4}$ is defined by
$\\{(x_{1},x_{2},x_{3},x_{4}):ax_{1}+bx_{2}+cx_{3}+0x_{4}=e\\}.$ We now
consider balls defined by the equation:

$(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}+x_{3}^{2}+x_{4}^{2}=\delta^{2}.$

These balls are moved by spacing $2\delta$ in $x_{3}$ and $x_{4}$ directions.
Thus, we see that when we fix the last coordinate $x_{4}$, the number of
incidences between these hyperplanes and balls is the same as we constructed
above in 3-dimension which is $\delta\delta^{-s-1}\delta^{-t}$. Moreover, we
have $\delta^{-1}$ number of copies in $x_{4}$ direction which implies that
the number of incidences is
$\delta\delta^{-s-1}\delta^{-t}\delta^{-1}=\delta|P||\Pi|$. It is not
difficult to prove that the sets of balls and hyperplanes are $(\delta,s,C)$
and $(\delta,t,C)$ sets, respectively. For other dimensions $d>4$, the
construction works in the same way. Thus, the number of incidences
$I(P,\Pi)\sim|\\{\text{incidences in the
plane}\\}|\cdot\delta^{2-d}\sim\delta|P||\Pi|$.

For the sharpness of Theorem 1.5, one might think of the same idea by
extending Constructions 1, 2, 3 in [4] to higher dimensions, but it does not
match the upper bounds in Theorem 1.5 at least in the way we tried.

##  5 Acknowledgements

T. Pham would like to thank the Vietnam Institute for Advanced Study in
Mathematics (VIASM) for the hospitality and for the excellent working
condition. C.-Y. Shen was partially supported by NSTC grant
111-2115-M-002-010-MY5.

## References

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  * [4] Y. Fu and K. Ren, Incidence estimates for $\alpha$ dimensional tubes and $\beta$ dimensional balls in $\mathbb{R}^{2}$, arXiv:2111.05093, 2021.
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# Objective Bayesian approach to the Jeffreys-Lindley paradox

Andrew<EMAIL_ADDRESS>[ Department of Physics and
Institute of Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu
210023, China,

###### Abstract

We consider the Jeffreys-Lindley paradox from an objective Bayesian
perspective by attempting to find priors representing complete indifference to
sample size in the problem. This means that we ensure that the prior for the
unknown mean and the prior predictive for the $t$-statistic are independent of
the sample size. If successful, this would lead to Bayesian model comparison
that was independent of sample size and ameliorate the paradox. Unfortunately,
it leads to an improper scale-invariant prior for the unknown mean. We show,
however, that a truncated scale-invariant prior delays the dependence on
sample size, which could be practically significant. Lastly, we shed light on
the paradox by relating it to the fact that the scale-invariant prior is
improper.

##  1 Introduction

The Jeffreys-Lindley (JL) paradox (Jeffreys (1939); Lindley (1957)) exposes a
conflict between Bayesian and frequentist approaches to hypothesis testing
that grows as the sample size increases. Whilst the mathematics of the paradox
are clear, its repercussions and meaning continue to be disputed to this day.
In this note, we emphasise that an objective Bayesian approach partially
ameliorates the paradox. This was previously explored in passing in Fowlie
(2019).

To establish our notation, we briefly review the paradox in sec. 2 (see e.g.,
Shafer (1982); Sprenger (2013); Robert (2014); Cousins (2017) for further
introduction and discussion). In sec. 3 we construct priors representing
complete indifference to the sample size. We show that in this case the
dependence of the Bayes factor on the sample size and thus the JL paradox
vanish. We comment further on this result and conclude in sec. 4.

##  2 The JL paradox

The problem under consideration is as follows. We collect a sample of size $n$
from a normal distribution, $\mathcal{N}(\mu,\sigma^{2})$. We suppose that
$\sigma$ and $n$ are known and that we want to test whether $\mu=0$ ($H_{0}$)
or $\mu\neq 0$ ($H_{1}$). In both frequentist and Bayesian approaches, we may
summarise the data by the sufficient $t$-statistic

$t=\frac{\sqrt{n}\bar{x}}{\sigma},$ (2.1)

where $\bar{x}$ is the sample mean. The likelihood may be written as

$\mathcal{L}=\frac{1}{\sqrt{2\pi}}e^{-\tfrac{1}{2}(t-t_{0})^{2}},$ (2.2)

where $t_{0}=\sqrt{n}\mu/\sigma$. In this form, it depends on $n$ and $\mu$
only through the combination $\sqrt{n}\mu$.

The $t$-statistic is distributed as a standard normal variable,
$\mathcal{N}(0,1)$, under the null hypothesis in which $\mu=0$. In the
frequentist approach, we may thus compute a p-value,

$\text{{p}-value}=2\left(1-\Phi(|t|)\right).$ (2.3)

where $\Phi$ is the cumulative distribution function of a standard normal
distribution. We emphasise that the p-value depends only on $t$ and not on
$n$.

In the Bayesian approach, on the other hand, we compute a Bayes factor (see
e.g., Kass and Raftery (1995)) for the alternative versus the null hypothesis,

$\displaystyle B_{10}={}$
$\displaystyle\frac{p\left(t\,\middle|\,H_{1}\right)}{p\left(t\,\middle|\,H_{0}\right)}$
(2.4) $\displaystyle={}$ $\displaystyle\frac{\int
p\left(t\,\middle|\,\mu\right)p(\mu)\,\text{d}\mu}{p\left(t\,\middle|\,H_{0}\right)}$
(2.5) $\displaystyle={}$ $\displaystyle e^{\frac{1}{2}t^{2}}\int
e^{-\tfrac{1}{2}(t-\sqrt{n}\mu/\sigma)^{2}}p(\mu)\,\text{d}\mu.$ (2.6)

This required that we placed a prior, $p(\mu)$, on the unknown mean in the
alternative model. By a Laplace approximation, we find that for any slowly
varying prior,

$\displaystyle
B_{10}\approx\sqrt{2\pi}e^{\frac{1}{2}t^{2}}\frac{p(\hat{\mu})\sigma}{\sqrt{n}},$
(2.7)

where $\hat{\mu}=t\sigma/\sqrt{n}$ is the maximum of the exponential factor in
the integrand.

We see that whilst the p-value depends only on the observed $t$, the Bayes
factor depends on $t$ and the sample size $n$. For any $t$ for which we would
reject the null hypothesis via the p-value, the Bayes factor could favour the
null by an arbitrary factor by making $n$ sufficiently large. This is the JL
paradox. It concerns changing $n$ and a fixed observation of $t$.

##  3 Objective Bayesian approach

In the JL paradox, $t$ was fixed whereas the sample size, $n$, was changed.
Since the p-value is independent of $n$, for a fair comparison, let us attempt
to construct a prior $p(\mu)$ representing complete indifference to $n$ in the
problem. Specifically, as the problem concerns inferences about $\mu$ in light
of an observation of $t$, the prior should be such that changing $n$ changes
nothing about our prior knowledge of $t$ or $\mu$ in the alternative
hypothesis. The null hypothesis is always indifferent to $n$, as the prior
predictive $p\left(t\,\middle|\,H_{0}\right)$ is independent of $n$.

By construction, for such a prior, the prior predictive for $t$ should be
independent of $n$, and therefore the evidence
$p\left(t\,\middle|\,H_{1}\right)$ and the Bayes factor are independent of
$n$. The JL paradox would thus vanish for such priors, as the frequentist and
Bayesian results would both be independent of $n$. To find such a prior, it
suffices to demand that the prior for the unknown $\mu$, $p(\mu)$ and the
subsequent prior for the unknown $t_{0}$, $q(t_{0})$ are independent of $n$.
By the Jacobian rule and the relation $t_{0}=\sqrt{n}\mu/\sigma$, however, we
require that

$\displaystyle p(\mu)={}$ $\displaystyle p(\sigma/\sqrt{n}t_{0})$ (3.1)
$\displaystyle={}$ $\displaystyle
q(t_{0})\frac{\sqrt{n}}{\sigma}=q(\sqrt{n}\mu/\sigma)\frac{\sqrt{n}}{\sigma}.$
(3.2)

The solution is thus

$p(\mu)=\frac{c}{|\mu|},$ (3.3)

for constant $c$. This is the usual scale-invariant prior (see e.g., Consonni
et al. (2018); Kass and Wasserman (1996) for further discussion of invariant
priors and rules for finding priors), which is improper, reflecting the fact
that complete indifference to scale is impossible.111Since $\sqrt{n}/\sigma$
is positive in our problem, we are in fact free to weight positive and
negative $\mu$ differently, but we won’t pursue that complication here. Note,
however, that the prior predictive for $t$, whilst independent of $n$, is not
scale-invariant.

We may quickly verify that with this prior, the Bayes factor in eq. 2.6 may be
written,

$\displaystyle B_{10}={}$ $\displaystyle
ce^{\tfrac{1}{2}t^{2}}\int_{-\infty}^{\infty}e^{-\tfrac{1}{2}(t-\sqrt{n}\mu/\sigma)^{2}}\frac{d\mu}{|\mu|}$
(3.4) $\displaystyle={}$ $\displaystyle
ce^{\tfrac{1}{2}t^{2}}\int_{-\infty}^{\infty}e^{-\tfrac{1}{2}(t-y)^{2}}\frac{d\\!y}{|y|},$
(3.5)

where we made a change of variable $y=\sqrt{n}\mu/\sigma$ to eliminate $n$.
This is completely independent of the sample size, $n$. It depends, however,
on the arbitrary coefficient $c$ in our improper prior in eq. 3.3 and the
integral diverges. A dependence on scale must be reintroduced when making the
prior proper, as we must modify the form of the prior so that it diverges
slower than $1/|\mu|$ as $\mu\to 0$ and tends to zero faster than $1/|\mu|$ as
$|\mu|\to\infty$. The dependence may, however, be weak.

For example, suppose we give support to a scale-invariant prior on only a
finite region,

$p(\mu)=\begin{cases}\frac{1}{2\log(b/a)}\frac{1}{|\mu|}&a\leq|\mu|\leq b\\\
0&\text{elsewhere},\end{cases}$ (3.6)

where $0<a<b$. This prior is proper. The resulting Bayes factor may be written
as

$B_{10}=\frac{e^{\tfrac{1}{2}t^{2}}}{2\log(b/a)}\int\limits_{\sqrt{n}a/\sigma}^{\sqrt{n}b/\sigma}\left[e^{-\tfrac{1}{2}(t-y)^{2}}+e^{-\tfrac{1}{2}(t+y)^{2}}\right]\frac{d\\!y}{|y|},$
(3.7)

which depends on the sample size only through the integration limits. The
integral converges and the Bayes factor is bounded by $e^{\frac{1}{2}t^{2}}$.
The factor in the square brackets peaks at or near $y=|t|$. So long as that
peak lies within the integration limits, the dependence on the integration
limits, and thus on the sample size, remains weak.

Figure 1: The Bayes factor in eq. 3.4 as a function of sample size $n$ for
$\sigma=1$ and fixed $t=2$ (top) and $3$ (bottom). We show Bayes factors from
five different breadths of scale-invariant prior by different colours. We show
$1/\sqrt{n}$ scaling by a dashed line.

Indeed, we show the Bayes factor in eq. 3.7 as a function of $n$ for fixed
$t=2$ and $3$, and $\sigma=1$ in fig. 1. We show results from five priors of
type eq. 3.6 with different choices of $a$ and $b$ by different colours, e.g.,
from $10^{-1}\leq|\mu|\leq 10^{1}$ (blue). We see that as we increase the
breadth of the prior, the Bayes factor remains constant as $n$ increases for
greater and greater $n$, especially when $t=3$. Once the sample size increases
such that the $\mu$ preferred by the likelihood at fixed $t$ lies outside or
close to the edge of the region supported by the prior, however, the Bayes
factor tends to zero. The JL paradox is thus delayed until the likelihood
favours $\mu$ forbidden by the prior. This delay is practically significant if
we are concerned with realistic experiments in which $n$ cannot be increased
indefinitely.

As emphasised by Bartlett (1957), the Bayes factor is sensitive to the
diffuseness of the prior, e.g., the normalising constant $\log(b/a)$ required
in eq. 3.6. In this problem, however, the Bayes factor may remain finite even
as $\log(b/a)$ diverges, depending how the limit is taken. Let us momentarily
take $t>0$ for simplicity. If we consider,

$p(\mu)=\begin{cases}\frac{1}{4c}\frac{1}{|\mu|}&e^{-c}\leq|\mu|\leq e^{c}\\\
0&\text{elsewhere},\end{cases}$ (3.8)

and take $c\to\infty$, we asymptotically put a quarter of the prior mass
between $\mu=0$ and the maximum likelihood $\mu=t/\sqrt{n}$ for fixed $t$. Of
the other three quarters of the prior mass, half lies at the wrong sign of
$\mu$ and another quarter lies at $\mu>t/\sqrt{n}$. In the region
$0\leq\mu\leq t/\sqrt{n}$, the likelihood is always greater than that for
$\mu=0$ and thus there is a lower bound on the Bayes factor of one quarter.
Thus Bartlett’s paradox is not particularly severe in this case. In practice
the limits $a$ and $b$ in eq. 3.6 should be set by considering realistic
maximum and minimum scales for the unknown mean in a concrete problem with
real information about the meaning of $\mu$.

##  4 Discussion

In the JL paradox, we consider a fixed $t$-statistic but a changing sample
size, $n$. Fixing $t$ but changing $n$ is arguably a change in prior knowledge
about the origin of $t$, since if we were merely collecting more samples,
fixing $t$ would be unrealistic (Robert (2014)). We were thus motivated to
construct priors that are completely indifferent to $n$ because it puts
identical information into the Bayesian calculations and the p-value, as the
latter is independent of $n$. Since the JL paradox concerns an observation of
$t$ and an unknown mean, we desired a prior for the mean that was independent
of $n$ and that resulted in a prior predictive for $t$ that was independent of
$n$. It is self-evident that this would result in a Bayes factor that is
independent of $n$. Thus, for such a prior, the JL paradox would vanish, since
we could no longer exacerbate differences between results from Bayesian and
frequentist tests by increasing $n$.

We showed that the prior representing indifference to $n$ in this problem is a
scale-invariant prior for the unknown mean $\mu$. This isn’t surprising: by
considering a fixed test-statistic but increasing sample size, we are in fact
just re-scaling the problem, and by using a scale-invariant prior, we are
indifferent to that change. This prior is, however, improper, and making it
proper necessarily re-introduces a dependence on scale. The dependence on
scale may, however, be weak until $n$ is increased so much that at fixed $t$
it favours $\mu$ that is so small that it lies in the region in which the
scale-invariant prior was modified. Since a proper prior must diverge slower
than $1/|\mu|$ at small $\mu$, it must disfavour (rather than favour) smaller
scales relative to the scale-invariant prior, and thus favour the null model
when $n$ is sufficiently increased.

We see, then, that a JL paradox is inevitable at sufficiently large $n$,
because there is no scale-invariant proper prior over the entire real line.
Thus, for fixed $t$, not only is it inevitable that the Bayesian result
ultimately depends on sample size, it is also inevitable that it ultimately
favours the null model. In this light, the JL paradox is no longer troubling:
the results from a Bayesian analysis are only independent of the sample size
if we had a specific prior state of knowledge that was indifferent to the
sample size. Such a state of knowledge is not entirely possible, as the
resulting scale-invariant prior is improper. For any other choice of prior,
the result depends on $n$ and thus we find a JL paradox when we compare
against the frequentist result, which is automatically independent of the
sample size. We showed, however, that we may modify the scale-invariant prior
to make it proper and delay the strong dependence on sample size.

Lastly, we can think about this another way. If the Bayesian answer was
troubling, we should carefully think about what question we were asking
(Jaynes, 1980). At first glance, we might think we were asking whether $\mu=0$
or $\mu\neq 0$ — this is the question that the frequentist approach appears to
answer. A proper prior, however, cannot be scale invariant; at the least, it
must disfavour arbitrarily tiny and enormous scales. Thus, we are really
asking, is $\mu=0$ or is it the scale favored by the choice of prior? Asked
this way, it is intuitive that for fixed $t$ the preference for $\mu=0$ must
eventually grow as $n$ increases, as eventually the data favors
$\hat{\mu}=t\sigma/\sqrt{n}$ smaller than that favored by the prior.

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