# Poissonian Pair Correlation for Higher Dimensional Real Sequences

Tanmoy Bera, Mithun Kumar Das, Anirban Mukhopadhyay  1,3The Institute of
Mathematical Sciences, A CI of Homi Bhabha National Institute, CIT Campus,
Taramani, Chennai 600113, India.  2 National Institute of Science Education
and Research, A CI of Homi Bhabha National Institute, Jatni, Khurda, 752050,
India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this article, we examine the Poissonian pair correlation (PPC) statistic
for higher-dimensional real sequences. Specifically, we demonstrate that for
$d\geq 3$, almost all $(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{R}^{d}$, the
sequence $\big{(}\\{x_{n}\alpha_{1}\\},\dots,\\{x_{n}\alpha_{d}\\}\big{)}$ in
$[0,1)^{d}$ has PPC conditionally on the additive energy bound of $(x_{n}).$

This bound is more relaxed compared to the additive energy bound for one
dimension as discussed in [1]. We also establish the metric PPC for
$(n^{\theta_{1}},\ldots,n^{\theta_{d}})$ provided that the $\theta_{i}$’s are
greater than one. More generally, we derive the PPC for
$\big{(}\\{x_{n}^{(1)}\alpha_{1}\\},\dots,\\{x_{n}^{(d)}\alpha_{d}\\}\big{)}\in[0,1)^{d}$
for almost all $(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{R}^{d}.$

###### Key words and phrases:

Pair correlation, Riemann zeta function, Diophantine inequality, exponential
sum

###### 2010 Mathematics Subject Classification:

11K06, 11J83, 11M06, 11J25, 11J71, 42B05, 11L07

##  1\. Introduction

A sequence of real numbers $(a_{k})$ in $[0,1]$ is called uniformly
distributed if for any subinterval $[a,b$] in $[0,1],$

$\lim_{n\rightarrow\infty}\frac{\\#\\{k\leq n:a_{k}\in[a,b]\\}}{n}=b-a.$

A classic example of such a sequence is the Kronecker sequence, defined as
$a_{k}=\\{k\alpha\\}$ (where $\\{\cdot\\}$ denotes the fractional part). This
sequence exhibits uniform distribution if and only if $\alpha$ is irrational.
The concept of uniform distribution of a sequence signifies that the relative
frequency of elements falling into any given subinterval asymptotically
approaches to the length of that subinterval. We refer to [8] for the general
theory of uniform distribution.

When we establish that a sequence follows uniform distribution, it naturally
prompts us to explore its statistical characteristics on more refined scales.
One such statistical property of interest is the pair correlation. For $s>0$,
the pair correlation function is defined as

$R_{2}(s,a_{n},N):=\frac{1}{N}\\#\Big{\\{}1\leq n\neq m\leq
N:\|a_{n}-a_{m}\|\leq\frac{s}{N}\Big{\\}},$

where $\displaystyle\|y\|=\min_{k\in\mathbb{Z}}|y+k|$. We say that $(a_{n})$
in $[0,1]$ exhibit Poissonian pair correlation if

$\displaystyle\lim_{N\rightarrow\infty}R_{2}(s,a_{n},N)=2s.$

This topic has gained significant prominence over the past three decades,
following the pioneering work of Rudnick and Sarnak [14]. While substantial
progress has been made in developing theory for one-dimensional pair
correlation, our understanding remains limited when it comes to higher-
dimensional sequences. At the same time the theory of uniform distribution is
well studied (see the book [7]).

Recently, the concept of pair correlation in higher-dimensional sequences has
been explored in [9, 18, 4]. This exploration involved constraining the test
sets to have their centers at the origin with respect to both the sup-norm and
the $2$-norm. Naturally, a very curious unsolved question is whether the pair
correlation property in higher-dimensional sequences is dependent on the
choice of norm.

For higher-dimensional real sequences, the pair correlation function can be
defined as

######  Definition.

Let $d\geq 1$ be a natural number. Let $s>0$ be a real number and $N$ be a
natural number, $(\boldsymbol{y}_{n})\subseteq[0,1)^{d}$ be a sequence and
$\|\cdot\|$ be a norm in $\mathbb{R}^{d}$. The $d$-dimensional pair
correlation function $R_{2,\|\cdot\|}^{(d)}(s,(\boldsymbol{y}_{n}),N)$ is
defined as follows:

$R_{2,\|\cdot\|}^{(d)}(s,(\boldsymbol{y}_{n}),N):=\frac{1}{N}\\#\Big{\\{}1\leq
n\neq m\leq
N:\|\boldsymbol{y}_{n}-\boldsymbol{y}_{m}\|^{\text{(intdist)}}\leq\frac{s}{N^{1/d}}\Big{\\}},$

where
$\displaystyle\|\boldsymbol{y}\|^{\text{(intdist)}}=\min_{\boldsymbol{k}\in\mathbb{Z}^{d}}\|\boldsymbol{y}+\boldsymbol{k}\|.$
We say that $(\boldsymbol{y}_{n})$ has Poissonian pair correlation with
respect to $\|\cdot\|$-norm (or, $\|\cdot\|$-PPC) if for all $s>0$,

(1.1) $\displaystyle
R_{2,\|\cdot\|}^{(d)}(s,(\boldsymbol{y}_{n}),N)\to\omega_{(d,\|\cdot\|)}s^{d}\quad\mbox{
as }N\rightarrow\infty,$

where $\omega_{(d,\|\cdot\|)}$ is the volume of the unit ball in
$(\mathbb{R}^{d},\|\cdot\|).$

In this article we study the pair correlation function with respect to the
sup-norm. We say $(\boldsymbol{y}_{n})$ has $\infty$-PPC if (1.1) holds for
sup norm. In this case the pair correlation function is denoted by
$R_{2,\infty}^{(d)}(s,(\boldsymbol{y}_{n}),N).$

######  Definition (Metric Poissonian pair correlation (MPPC)).

We say that a sequence $(\boldsymbol{x}_{n})$ in $(\mathbb{R}_{>0})^{d}$
possesses $\infty$-MPPC if the sequence
$(\\{\boldsymbol{x}_{n}\boldsymbol{\alpha}\\})=(\\{x_{n}^{(1)}\alpha_{1}\\},\ldots,\\{x_{n}^{(d)}\alpha_{d}\\})$
exhibits $\infty$-PPC for almost all $\boldsymbol{\alpha}\in\mathbb{R}^{d}$.

For one dimensional real sequence, the additive energy for the set of first
$N$ elements is defined as

$E(X_{N}):=\\#\\{\boldsymbol{n}\in[1,N]^{4}:|x_{n_{1}}+x_{n_{2}}-x_{n_{3}}-x_{n_{4}}|<1\\}.$

It is worth noting that for integer sequences, the condition
‘$|x_{n_{1}}+x_{n_{2}}-x_{n_{3}}-x_{n_{4}}|<1$’ in $E(X_{N})$ is replaced with
‘$x_{n_{1}}+x_{n_{2}}=x_{n_{3}}+x_{n_{4}}$’. For one-dimensional integer
sequences, the study of metric pair correlation has been explored in [3, 5]
conditionally on additive energy bound. In the case of one-dimensional real
sequences, a recent and highly influential set of results has been established
by Aistleitner, El-Baz, and Munsch in [1]. Their contributions are as follows:

###### Theorem A.

Let $(x_{n})$ be a sequence of positive real number for which there exists a
constant $c>0$ such that $x_{n+1}-x_{n}\geq c,$ for all $n\geq 1$. Assume that
there exists some $\delta>0$ such that,

$E(X_{N})\ll N^{\frac{183}{76}-\delta}.$

Then $(x_{n})$ has MPPC.

Furthermore, they consider the quantity $E_{\gamma}(X_{N})$ for
$\gamma\in(0,1]$, given by

(1.2) $\displaystyle
E_{\gamma}(X_{N})=\\#\\{\boldsymbol{n}\in[1,N]^{4}:|x_{n_{1}}+x_{n_{2}}-x_{n_{3}}-x_{n_{4}}|<\gamma\\}.$

Conditioning on this quantity, they have established the following result:

###### Theorem B.

Let $(x_{n})$ be a sequence of positive real number for which there exists a
constant $c>0$ such that $x_{n+1}-x_{n}\geq c,$ $n\geq 1.$ Assume that there
exists some $\delta>0$ such that for all $\eta>0$ we have,

$E_{\gamma}(X_{N})\ll_{\eta,\delta}N^{2+\eta}+\gamma N^{3-\delta}$

as $N\to\infty,$ uniformly for $\gamma\in(0,1].$ Then the sequence $(x_{n})$
has MPPC.

Clearly, this enables us to deduce metric pair correlations in certain cases
where the condition on additive energy in Theorem A does not hold. As an
application of Theorem B, in conjunction with a result by Robert and Sargos
[13, Theorem 2], it has been demonstrated that the sequence $(n^{\theta})$
with $\theta>1$ exhibits the MPPC property (see [1, Theorem 3]). For
$(n^{\theta})$ with $0<\theta<1$, the MPPC property has been established in
[16].

For higher dimensional integer sequences, Hinrichs, Kaltenböck, Larcher,
Stockinger and Ullrich [9] considered the $d$-dimensional sequence
$(x_{n}\boldsymbol{\alpha})\in\mathbb{R}^{d}$ with $(x_{n})\in\mathbb{Z}$.
Then they obtain that $(\\{x_{n}\boldsymbol{\alpha}\\})$ exhibit $\infty$-PPC
for almost all $\boldsymbol{\alpha}$, provided that

$\displaystyle E(X_{N})\ll\frac{N^{3}}{(\log{N})^{1+\epsilon}},\quad\mbox{for
some }\epsilon>0.$

In a recent publication [4], the authors established the $\infty$-MPPC
property for any finite $d$-dimensional integer sequence
$\big{(}x_{n}^{(1)},\dots,x_{n}^{(d)}\big{)}$ with strictly increasing
components. This result is conditional upon the joint additive energy bound
$E(\boldsymbol{X}_{N}^{D})\ll N^{3-\delta}$ for any $\delta>0$, where
$D:=\\{1,\ldots,d\\}$ and the joint additive energy
$E(\boldsymbol{X}_{N}^{D})$ is defined as:

$\displaystyle
E(\boldsymbol{X}_{N}^{D}):=\\#\Big{\\{}\boldsymbol{n}\in[1,N]^{4}:x_{n_{1}}^{(l)}+x_{n_{2}}^{(l)}=x_{n_{3}}^{(l)}+x_{n_{4}}^{(l)},\>1\leq
l\leq d\Big{\\}}.$

In order to obtain our results for higher dimensional real sequences, it is
necessary to define the analogous version of the joint additive energy for
real sequences.

######  Definition.

(Joint additive energy for real sequences) Let
$\boldsymbol{X}_{N}=\\{(x_{n}^{(1)},\dots,x_{n}^{(d)})\in\mathbb{R}^{d}:n\leq
N\\}$ and $\boldsymbol{\gamma}\in(0,1]^{d}$ be real numbers. For
$D^{\prime}\subseteq D:=\\{1,\ldots,d\\}$ we define the quantity
$E_{\boldsymbol{\gamma}}(\boldsymbol{X}_{N}^{D^{\prime}})$ by

(1.3) $\displaystyle
E_{\boldsymbol{\gamma}}(\boldsymbol{X}_{N}^{D^{\prime}})=\\#\\{\boldsymbol{n}\in[1,N]^{4}:|x_{n_{1}}^{(l)}+x_{n_{2}}^{(l)}-x_{n_{3}}^{(l)}-x_{n_{4}}^{(l)}|<\gamma_{l},\>l\in
D^{\prime}\\}.$

The joint additive energy for higher-dimensional real sequences, denoted as
$E(\boldsymbol{X}_{N}^{D})$ (with notation consistent with the integer case),
is defined by
$E(\boldsymbol{X}_{N}^{D}):=E_{\boldsymbol{1}}(\boldsymbol{X}_{N}^{D})$.

Before we state our general result, we state results on two spacial cases. The
first one is as follows:

######  Theorem 1.1.

Let $d\geq 3$ and $(x_{n})$ be a sequence of positive real numbers satisfying
the spacing condition $x_{n+1}-x_{n}\geq c,$ for some $c>0.$ Assume that there
exists some $\lambda>0$ such that

$E(X_{N})\ll
N^{\frac{183}{76}+\frac{16}{76}\left(1-\frac{1}{d}\right)-\lambda}.$

as $N\to\infty.$ Then $(\\{x_{n}\boldsymbol{\alpha}\\})$ has $\infty$-PPC for
almost all $\boldsymbol{\alpha}.$

We observe that when $d=1$, the exponent of the additive energy bound is at
most $\frac{183}{76}\approx 2.4079$. This coincides with the exponent found in
the result by Aistleitner et al. [1] (see Theorem A). However, as soon as we
consider $d\geq 3$, we can relax the additive energy bound by an additional
quantity of $\frac{4}{19}\left(1-\frac{1}{d}\right)$ in the exponent, and as
$d$ increases, the bound can go up to a maximum of $\frac{199}{76}\approx
2.6184$.

Regarding the case when $d=2$, we have not been able to prove this stronger
version, but we establish a weaker result in Theorem 1.3. The second result
aligns with the finding from [1, Theorem 3], where it was demonstrated that
the sequence $(n^{\theta})$ with $\theta>1$ exhibits MPPC. In this context, we
explore its higher-dimensional analogue. If the $\theta_{i}$’s are integers,
this case has already been addressed in [4]. Let
$\theta_{1},\dots,\theta_{d}>1$ be real numbers that are not integers. We
examine the higher-dimensional sequence
$(n^{\theta_{1}},\dots,n^{\theta_{d}})$ and investigate whether it possesses
$\infty$-MPPC. Our findings yield the following affirmative result.

######  Theorem 1.2.

Let $d\geq 4$ and $\theta_{1},\cdots,\theta_{d}>1$ be real numbers that are
not integers. Then $(n^{\theta_{1}},\dots,n^{\theta_{d}})$ has $\infty$-MPPC.

######  Remark 1.

For dimension one, the result holds true, as established by Aistleitner et al.
However, for $d=2$ and $d=3,$ we have not been able to establish the MPPC
result unconditionally. In the case of dimension two, we require
$\theta_{2}\geq\theta_{1}$ and that
$\min\\{2\theta_{2}-\theta_{1},4-\theta_{2}\\}\leq 213/76$. For dimension
three, these conditions apply to any two-dimensional subsequence. Then such
sequences have $\infty$-MPPC.

This theorem is a consequence of our general result, as articulated in Theorem
1.3 below, and it is combined with the joint additive energy bound calculated
in Theorem 2.1. In the case of one dimension, such type of additive energy
bound established by Robert and Sargos [13]. Now, we present the PPC result
for $d$-dimensional real sequences as follows:

######  Theorem 1.3.

Let $(\boldsymbol{x}_{n})$ be a sequence in $\mathbb{R}^{d}$ with positive
components and $d\geq 1$. Also assume that there exist some $c>0$ such that
$x_{n+1}^{(l)}-x_{n}^{(l)}\geq c$ for all $1\leq l\leq d$ and $n\geq 1$. If
for all $D^{\prime}\subseteq D=\\{1,\ldots,d\\}$ of $d^{\prime}$ elements
satisfy the bound

(1.4) $\displaystyle E(\boldsymbol{X}_{N}^{D^{\prime}})\ll
N^{4-\frac{1}{d}\left(2d^{\prime}-\frac{31}{76}\right)-\lambda_{D^{\prime}}}$

for some $\lambda_{D^{\prime}}>0$, then $(\boldsymbol{x}_{n})$ has
$\infty$-MPPC.

######  Remark 2.

Conditionally on the Lindelöf hypothesis, the upper bound of $E(X_{N})$ in
Theorem 1.1 can be relaxed to $E(X_{N})\ll N^{3-\lambda}$ for some
$\lambda>0$. This assumption essentially coincides with the integer case. For
Theorem 1.3, under the Lindelöf hypothesis, the upper bound of
$E(\boldsymbol{X}_{N}^{D^{\prime}})$ can be relaxed to
$E(\boldsymbol{X}_{N}^{D^{\prime}})\ll
N^{4-\frac{1}{d}(2d^{\prime}-1)-\lambda_{D^{\prime}}}$ for all
$D^{\prime}\subseteq D$.

In a recent study, Hofer and Kaltenböck [10] raised the following question:

######  Question.

In the context of Poissonian pair correlations in higher-dimensional settings,
is it necessary for all the component sequences to exhibit Poissonian pair
correlations?

In our work [4], we provide a negative answer to this question when all the
components of a higher-dimensional sequence are integers. The following
corollary further offers a negative response to this question when the
components are real.

######  Corollary 1.4.

For $A\geq 1$, the $2$-dimensional sequence $(n,n\log^{A}{n})$ has
$\infty$-MPPC.

This corollary follows from Theorem 1.3, Corollary 2.5 and the additive energy
bound of $(n\log^{A}n)$ in (2.13). Note that $(\\{n\alpha\\})$ for any
$\alpha\in\mathbb{R}$ does not have PPC and the same for
$(\\{n(\log{n})^{A}\alpha\\})$ is not known.

### Strategy

The proofs of Theorem 1.1 and Theorem 1.3 rely on employing higher-dimensional
Fourier analysis on the torus $\mathbb{T}=[0,1]^{d}$ to detect the indicator
function of the pair correlation function
$R_{2,\infty}^{(d)}(s,(\boldsymbol{y}_{n}),N)$. Subsequently, we exploiting
the property of the $\infty$-norm to derive majorant and minorant functions
for the indicator function, which take the form of truncated trigonometric
series. The key insight here is to demonstrate that the variance of these
majorant and minorant functions with respect to a specific measure on
$\mathbb{R}^{d}$ becomes negligible as $N$ grows sufficiently large. This idea
essentially extends the approach found in [1], which deals with the one-
dimensional case.

By taking advantage of the analysis with respect to the sup-norm for the
sequence of the form $(\\{x_{n}\alpha_{1}\\},\ldots,\\{x_{n}\alpha_{d}\\})$,
we simplify the counting problem from a system of $2d$ variables to a system
of $d$ variables (refer to (5.2) and (5.3)). As a result, Theorem 1.1
establishes the MPPC conditionally on additive energy of the sequence
$(x_{n})$ with a more relaxed additive energy bound than the one presented in
[1].

Once we establish Theorem 1.3, the proof of Theorem 1.2 reduces to demonstrate
that the joint additive energy of $(n^{\theta_{1}},\ldots,n^{\theta_{d}})$ for
$n\leq N$ is at most $N^{2+\epsilon}$ for any $\epsilon>0$. To achieve this,
we make use of the remarkable work by Watt [19] on counting solutions to a
specific system of Diophantine inequalities, coupled with the Van der Corput
estimate of the exponential sum. The remainder of the proof involves a
generalization of the argument found in the proof of [1, Theorem 3].
Furthermore, these tools are instrumental in establishing Corollary 1.4 (refer
to Proposition 2.4).

We list some of the many interesting open problems related to our results in
this paper.

######  Open Problem 1.

Prove or disprove that the notion of pair correlation in higher dimension
depends on the norm.

######  Open Problem 2.

Show that Theorem 1.1 remains valid under the weaker additive energy bound
assumption $E(X_{N})\ll N^{3-\lambda}$ for some $\lambda>0$.

######  Open Problem 3.

Let $d=2$ or $3$ and $\theta_{1},\cdots,\theta_{d}>1$ be real numbers that are
not integers. Show that $(n^{\theta_{1}},\dots,n^{\theta_{d}})$ has
$\infty$-MPPC.

##  2\. Estimating joint additive energy for real sequences

This section is devoted to obtain an upper bound of joint additive energy for
the sequence $(n^{\theta_{1}},\dots,n^{\theta_{d}})$ as well as
$(n,n\log^{A}n)$, where $A\geq 1$. Such a bound is used to obtain Theorem 1.2
and Corollary 1.4.

A result of Robert and Sargos [13, Theorem 2] gives the bound

(2.1) $\displaystyle E_{\gamma}(X_{N})\ll_{\epsilon}N^{2+\epsilon}+\gamma
N^{4-\theta+\epsilon},$

whenever $X_{N}:=\\{n^{\theta}:\theta>1,n\in[N,2N]\\}$ and $\gamma>0$. This
suggest that if $\theta_{i}\geq 2$ for atleast one $i\in D:=\\{1,\ldots,d\\}$
then the trivial upper bound for the joint additive energy for
$\boldsymbol{X}_{N}=\\{(n^{\theta_{1}},\dots,n^{\theta_{d}}):N<n\leq 2N\\}$
must satisfy $E(\boldsymbol{X}_{N}^{D})\ll N^{2+\epsilon}.$ Now we investigate
the upper bound of $E(\boldsymbol{X}_{N}^{D})$ in the case
$1<\theta_{1},\dots,\theta_{d}<2,\>\theta_{1}<\theta_{d}$ in the following
theorem:

######  Theorem 2.1 (Robert-Sargos type).

Let $\boldsymbol{X}_{N}=\\{(n^{\theta_{1}},\dots,n^{\theta_{d}}):N<n\leq
2N\\}$, where $1<\theta_{1},\dots,\theta_{d}<2,\>\theta_{1}<\theta_{d}$.
Assume that $\boldsymbol{\gamma}\in(0,1]^{d}$, then we get

$\displaystyle E_{\boldsymbol{\gamma}}(\boldsymbol{X}_{N}^{D})$
$\displaystyle\ll
N^{2+\epsilon}+N^{2\theta_{d}-(\theta_{1}+\cdots+\theta_{d-1})+\epsilon},$
(2.4) $\displaystyle\ll\left\\{\begin{array}[]{l}N^{2+\epsilon}\text{ if
}d\geq 3\\\
N^{2+\epsilon}+N^{\min\\{2\theta_{2}-\theta_{1},4-\theta_{2}\\}+\epsilon}\text{
if }d=2.\end{array}\right.$

In order to prove the above theorem we need the following two lemmas.

######  Lemma 2.2 (Lemma 2.1 [19]).

Let $K$ be a positive integer. Let $\mathcal{A}\subset\mathbb{R}$ be bounded
and $\boldsymbol{\omega}$ be an $\mathbb{R}^{K}$-valued function defined on
$\mathcal{A}\cap\mathbb{Z}$. Let
$V_{2M}(\mathcal{A},\boldsymbol{\omega},\boldsymbol{\delta})$ be the number of
solutions of the system of Diophantine inequalities,

(2.5)
$\displaystyle\left|\sum_{m=1}^{M}(\omega_{k}(u_{m})-\omega_{k}(v_{m}))\right|<\delta_{k}\text{
for }k=1,\dots,K,$

where $u_{m},v_{m}\in\mathcal{A}\cap\mathbb{Z}$ for $m=1,\dots,M.$ Let
$2\delta_{k}D_{k}=1,$ for $k=1,\dots,K.$ Then

(2.6)
$\displaystyle\delta_{1}\cdots\delta_{K}W_{2M}(\mathcal{A},\boldsymbol{\omega},\boldsymbol{\delta})\ll_{K}V_{2M}(\mathcal{A},\boldsymbol{\omega},\boldsymbol{\delta})\ll_{K}\delta_{1}\cdots\delta_{K}W_{2M}(\mathcal{A},\boldsymbol{\omega},\boldsymbol{\delta}),$

where

$\displaystyle
W_{2M}(\mathcal{A},\boldsymbol{\omega},\boldsymbol{\delta})=\int_{-D_{1}}^{D_{1}}\cdots\int_{-D_{K}}^{D_{K}}|T(\mathcal{A},\boldsymbol{\omega},\boldsymbol{x})|^{2M}dx_{1}\dots
dx_{K}$

and

$\displaystyle
T(\mathcal{A},\boldsymbol{\omega},\boldsymbol{x})=\sum_{u\in\mathcal{A}\cap\mathbb{Z}}e(\boldsymbol{\omega}(u)\cdot\boldsymbol{x})$

######  Lemma 2.3.

[8, Theorem 2.2] Suppose that $f$ is a real valued function with two
continuous derivatives on an interval $I$. Suppose there exits some
$\lambda>0$ and some $\alpha\geq 1$ such that

$\lambda\leq|f^{\prime\prime}(x)|\leq\alpha\lambda$

on $I$. Then

$\sum_{n\in I}e(f(n))\ll\alpha|I|\lambda^{1/2}+\lambda^{-1/2}.$

Now we begin the proof of Theorem 2.1.

###### Proof of Theorem 2.1.

Let $\boldsymbol{\delta}\in\mathbb{R}^{d}$ and set
$\Delta=\delta_{1}\cdots\delta_{d}$. Without loss of generality we assume that
$1<\theta_{1}\leq\theta_{2}\leq\ldots\leq\theta_{d}<2$. In order to get an
upper bound estimate $E_{\boldsymbol{\gamma}}(\boldsymbol{X}_{N}^{D})$, where
$\boldsymbol{X}_{N}=\\{(n^{\theta_{1}},\dots,n^{\theta_{d}}):N<n\leq 2N\\}$,
we start with the following counting problem

$\mathcal{N}(N,\boldsymbol{\delta})=\\#\\{\boldsymbol{n}\in(N,2N]^{4}:\left|n_{1}^{\theta_{i}}+n_{2}^{\theta_{i}}-n_{3}^{\theta_{i}}-n_{4}^{\theta_{i}}\right|<\delta_{i}N^{\theta_{i}},\>1\leq
i\leq d\\}.$

At first we want to obtain the upper bound estimate of
$\mathcal{N}(N,\boldsymbol{\delta})$ as an application of Lemma 2.2. Thus we
take the following setting: $\mathcal{A}=(N,2N]$, $K=d$, $M=2$ and define the
function
$\boldsymbol{\omega}(n)=(n^{\theta_{1}}/N^{\theta_{1}},\dots,n^{\theta_{d}}/N^{\theta_{d}})$
and thus we have
$V_{2M}(\mathcal{A},\boldsymbol{\omega},\boldsymbol{\delta})=\mathcal{N}(N,\boldsymbol{\delta})$.
Further, define

$f(n,\boldsymbol{x})=(n^{\theta_{1}}/N^{\theta_{1}})x_{1}+\cdots+(n^{\theta_{d}}/N^{\theta_{d}})x_{d},$

and
$T(\mathcal{A},\boldsymbol{\omega},\boldsymbol{\delta})=T(N,\boldsymbol{x})$,
where

$T(N,\boldsymbol{x}):=\sum_{N<n\leq 2N}e\left(f(n,\boldsymbol{x})\right).$

Then by Lemma 2.2 we get

(2.7) $\displaystyle\mathcal{N}(N,\boldsymbol{\delta})\ll\Delta
I(N,\boldsymbol{\delta}),$

where

$\displaystyle
I(N,\boldsymbol{\delta}):=\int_{-1/\delta_{1}}^{1/\delta_{1}}\cdots\int_{-1/\delta_{d}}^{1/\delta_{d}}|T(N,\boldsymbol{x})|^{4}dx_{1}\dots
dx_{d}.$

It is clear form the definition of $\mathcal{N}(N,\boldsymbol{\delta})$ that,
if $\delta_{i}^{\prime}\leq\delta_{i}$ for all $i,$ then
$\mathcal{N}(N,\boldsymbol{\delta^{\prime}})\leq\mathcal{N}(N,\boldsymbol{\delta}).$
Let $\delta_{i}=N^{-\theta_{i}}$ for $1\leq i\leq d$ and in this case we
denote $I(N,\boldsymbol{\delta})$ by $I(N,\boldsymbol{\theta})$ and
$\mathcal{N}(N,\boldsymbol{\delta})$ by $\mathcal{N}(N,\boldsymbol{\theta}).$
Now, we want show that

$\mathcal{N}(N,\boldsymbol{\theta})\ll
N^{2}+N^{2\theta_{d}-(\theta_{1}+\cdots+\theta_{d-1})}.$

We see that the range of $i$-th integral in $I(N,\boldsymbol{\theta})$ is
$[-N^{\theta_{i}},N^{\theta_{i}}]$. To obtain the upper bound of
$I(N,\boldsymbol{\theta})$, we break the integrals into

$\displaystyle I(N,\boldsymbol{\theta})$
$\displaystyle=\left(\int_{\|\boldsymbol{x}\|_{\infty}\leq
1}+\sum_{D^{\prime}\subsetneq D}\int_{\begin{subarray}{c}|x_{i}|\leq 1,i\in
D^{\prime}\\\ 1<|x_{i}|\leq N^{\theta_{i}},i\in D\setminus
D^{\prime}\end{subarray}}+\int_{\begin{subarray}{c}1<|x_{i}|\leq
N^{\theta_{i}}\\\ 1\leq i\leq
d\end{subarray}}\right)|T(N,\boldsymbol{x})|^{4}dx_{1}\dots dx_{d}$
$\displaystyle=I_{1}+\sum_{D^{\prime}\subsetneq
D}I_{2}(D^{\prime})+I_{3}\quad\mbox{ (say),}$

where $I_{1}$ and $I_{3}$ represent first and last conditional integral
respectively, and $I_{2}(D^{\prime})$ represents the intermediate integral
with $|x_{i}|\leq 1,i\in D^{\prime}$ and $1<|x_{i}|\leq N^{\theta_{i}},i\in
D\setminus D^{\prime}$. Trivially we get $I_{1}\ll N^{4}.$ Now we want to get
a bound for $I_{3}$. Observe that

$|f^{\prime\prime}(n,\boldsymbol{x})|\asymp\frac{|x_{1}|+\cdots+|x_{d}|}{N^{2}}~{}~{}~{}~{}~{}\mbox{
for }N<n\leq 2N.$

Then by Lemma 2.3 (van der Corput theorem), for $|x_{i}|>1$ we get,

$\displaystyle
T(N,\boldsymbol{x})\ll(|x_{1}|+\cdots+|x_{d}|)^{1/2}+\frac{N}{(|x_{1}|+\cdots+|x_{d}|)^{1/2}}.$

Therefore, by using binomial theorem we get

$\displaystyle I_{3}$ $\displaystyle\ll\int_{\begin{subarray}{c}1<|x_{i}|\leq
N^{\theta_{i}}\\\ 1\leq i\leq d\end{subarray}}\left(\sum_{i\leq
d}|x_{i}|^{2}+N\sum_{i\leq d}|x_{i}|+N^{2}+N^{3}\sum_{i\leq
d}\frac{1}{|x_{i}|}+N^{4}\sum_{i\leq d}\frac{1}{|x_{i}|^{2}}\right)dx_{1}\dots
dx_{d}$ $\displaystyle\ll\sum_{i\leq
d}\left(N^{3\theta_{i}}+N^{1+2\theta_{i}}+N^{2+\theta_{i}}+N^{3}\log
N+N^{4}\right).$

Since $1\leq\theta_{i}<2$ for all $1\leq i\leq d$, we have $2+\theta_{i}<4$
and $3\theta_{i}>1+2\theta_{i}$. This gives us

$\displaystyle I_{3}\ll N^{4}+\sum_{i\leq d}N^{3\theta_{i}}.$

Similarly, one can show that $I_{2}(D^{\prime})\ll N^{4}+\sum_{i\leq
d}N^{3\theta_{i}}.$ Combining bounds for $I_{1},I_{2}(D^{\prime})$ and $I_{3}$
we get

(2.8) $\displaystyle I(N,\boldsymbol{\theta})\ll N^{4}+\sum_{i\leq
d}N^{3\theta_{i}}.$

Thus (2.7) and (2.8) give us

(2.9) $\displaystyle\mathcal{N}(N,\boldsymbol{\theta})\ll
N^{2}+N^{2\theta_{d}-(\theta_{1}+\cdots+\theta_{d-1})},$

since $\theta_{i}$’s are increasing order. If we consider for
$0<\gamma_{i}\leq 1$, $\delta_{i}=\gamma_{i}N^{-\theta_{i}}(\leq
N^{-\theta_{i}})$ for all $i\leq d$, then

$\displaystyle\mathcal{N}(N,\boldsymbol{\boldsymbol{\gamma}})$
$\displaystyle:=\\{\boldsymbol{n}\in(N,2N]:\left|n_{1}^{\theta_{i}}+n_{2}^{\theta_{i}}-n_{3}^{\theta_{i}}-n_{4}^{\theta_{i}}\right|<\gamma_{i},\>1\leq
i\leq d\\}$ (2.10) $\displaystyle\ll\mathcal{N}(N,\boldsymbol{\theta})\ll
N^{2}+N^{2\theta_{d}-(\theta_{1}+\cdots+\theta_{d-1})},\text{ by
}~{}\eqref{bound in theta}$

Let $L$ be an integer and assume that $2^{L}=N$. Define a measure by

$\displaystyle
d\boldsymbol{\mu}_{2\boldsymbol{\gamma}}(\boldsymbol{x})=\prod_{1\leq i\leq
d}\frac{\sin^{2}(\gamma_{i}x_{i})}{\pi\gamma_{i}x_{i}^{2}}dx_{i}.$

Similar to the argument of the proof of [1, Theorem 3], application of
Hölder’s inequality gives

$\displaystyle E_{\boldsymbol{\gamma}}(\boldsymbol{X}_{N})$
$\displaystyle\ll\int_{\mathbb{R}^{d}}\left(\sum_{l=0}^{L}\sum_{2^{l-1}<n\leq
2^{l}}e\left(n^{\theta_{1}}x_{1}+\cdots+n^{\theta_{d}}x_{d}\right)\right)^{4}d\boldsymbol{\mu}_{2\boldsymbol{\gamma}}(\boldsymbol{x})$
$\displaystyle\ll\int_{\mathbb{R}^{d}}\left(\sum_{l=0}^{L}1\right)^{3}\sum_{l=0}^{L}\left|\sum_{2^{l-1}<n\leq
2^{l}}e\left(n^{\theta_{1}}x_{1}+\cdots+n^{\theta_{d}}x_{d}\right)\right|^{4}d\boldsymbol{\mu}_{2\boldsymbol{\gamma}}(\boldsymbol{x})$
$\displaystyle\ll(\log
N)^{3}\sum_{l=0}^{L}\mathcal{N}(2^{l-1},\boldsymbol{\gamma})$
$\displaystyle\ll
N^{2+\epsilon}+N^{2\theta_{d}-(\theta_{1}+\cdots+\theta_{d-1})+\epsilon},\text{
by }~{}\eqref{bound of N(gamma)}.$

Now, for $d\geq 3$ we have $2\theta_{d}-(\theta_{1}+\cdots+\theta_{d-1})<2$
imply that $E_{\boldsymbol{\gamma}}(\boldsymbol{X}_{N})\ll N^{2+\epsilon}$.
For $d=2$, the combination of the result of Robert-Sargos, that is, (2.1) and
the above estimate gives required bound. This proves the Theorem 2.1. ∎

Now we want to estimate the joint additive energy for $\\{(n,n\log^{A}n):n\leq
N\\}$. This we compute as a corollary of the following general result.

######  Proposition 2.4.

Let $F$ be a twice continuously differentiable function such that
$F^{\prime\prime}(n)\not\equiv 0$ and
$|F^{\prime\prime}(n)|\asymp|F^{\prime\prime}(N)|$ for $n\asymp N$. Let
$\boldsymbol{X}_{N}:=\\{(n,F(n)):N\leq n\leq 2N\\}$. For any $M>1$ we have

$\displaystyle
E_{\gamma}(\boldsymbol{X}_{N})\ll\min\Big{\\{}N^{3},\Big{(}N^{3}/M+N^{3}|F^{\prime\prime}(N)|+N\log
M/|F^{\prime\prime}(N)|\Big{)}\Big{\\}}.$

###### Proof.

In order to estimate $E_{\gamma}(\boldsymbol{X}_{N})$, it is enough to
estimate the counting

$\displaystyle\\#\left\\{\underline{n}\in[N,2N]^{4}:n_{1}+n_{2}=n_{3}+n_{4},\left|F(n_{1})+F(n_{2})-F(n_{3})-F(n_{4})\right|<\gamma\right\\}.$

If $n_{1}+n_{2}=n_{3}+n_{4}=k$ then $k\in[2N,4N]$, thus the counting above is
bounded by

$\displaystyle\\#\left\\{k\in[2N,4N]\mbox{ and
}\underline{n}\in[N,2N]^{2}:\left|F(n_{1})+F(k-n_{1})-F(n_{2})-F(k-n_{2})\right|<\gamma\right\\}.$

For $\delta>0$, $\mathcal{N}(N,M,\delta)$ to denote the counting

$\displaystyle\\#\left\\{k\in[2N,4N]\mbox{ and
}\underline{n}\in[N,2N]^{2}:\left|F(n_{1})+F(k-n_{1})-F(n_{2})-F(k-n_{2})\right|<\delta
M\right\\}.$

Then by Lemma 2.2 it is not difficult to obtain

(2.11)
$\displaystyle\mathcal{N}(N,M,\delta)\ll\delta\sum_{k\in[2N,4N]}\int_{0}^{1/2\delta}\bigg{|}\sum_{n\in[N,2N]}e\left(\left(F(n)+F(k-n)\right)y/M\right)\bigg{|}^{2}dy.$

Let us call the integral as $I(k,N,\delta)$ and the exponential sum as
$S(N,y)$. Observe that whenever $k\in[2N,4N]$ and $n\in[N,2N]$, the size of
the second order differentiation of the argument of the exponential function
is of size $\asymp|F^{\prime\prime}(N)|y/M$. Thus by Lemma 2.3,

(2.12) $\displaystyle S(N,y)\ll
N(|F^{\prime\prime}(N)|y/M)^{1/2}+(|F^{\prime\prime}(N)|y/M)^{-1/2}.$

For $\delta=M^{-1}$ we write

$\displaystyle
I(k,N,M^{-1})\leq\int_{0}^{1}|S(N,y)|^{2}dy+\int_{1}^{M}|S(N,y)|^{2}dy$

The first integral trivially bounded by $N^{2}$. For second integral, using
(2.12) we get

$\displaystyle I(k,N,M^{-1})\ll N^{2}+N^{2}|F^{\prime\prime}(N)|M+M\log
M/|F^{\prime\prime}(N)|.$

Replacing above estimate of $I(k,N,M^{-1})$ in (2.11) to deduce

$\displaystyle\mathcal{N}(N,M,M^{-1})\ll
N^{3}/M+N^{3}|F^{\prime\prime}(N)|+N\log M/|F^{\prime\prime}(N)|.$

Since $E_{\gamma}(\boldsymbol{X}_{N})\ll\mathcal{N}(k,N,M^{-1})$ and the
trivial upper bound of $E_{\boldsymbol{\gamma}}(\boldsymbol{X}_{N})$ is $\ll
N^{3}$, the result follows. ∎

Choosing $F(x)=x(\log{x})^{A}$ and $M=N$ in Proposition 2.4 we obtain the
following result.

######  Corollary 2.5.

Let $A\geq 1$ be a real number, $\boldsymbol{X}_{N}=\\{(n,n\log^{A}n):n\leq
N\\}$, $0<\gamma\leq 1$, $\epsilon>0.$ Then

$E_{\gamma}(\boldsymbol{X}_{N})\ll N^{2}(\log{N})^{A+3}.$

Using Lemma 2.2 and Lemma 2.3 it can be shown that for $A\geq 1$ the additive
energy of $(n\log^{A}n)$ is

(2.13) $\displaystyle E(X_{N})\ll N^{3}(\log{N})^{5-A}.$

In the subsequent sections we prove Theorem 1.1 and Theorem 1.3. We
essentially extend the idea of the proof of the results in [1] to higher
dimensions.

##  3\. Preparation of the proofs of Theorem 1.1 and Theorem 1.3

For $\boldsymbol{x}$ in $\mathbb{R}^{d}$ we define the characteristic function
$\chi_{s,N^{1/d}}$ by

$\chi_{s,N^{1/d}}(\boldsymbol{x})=\begin{cases}1&\mbox{ if
}\|\boldsymbol{x}\|_{\infty}^{\text{intdist}}\leq\frac{s}{N^{1/d}},\\\
0&\mbox{ otherwise }.\end{cases}$

Thus, in order to prove Theorem 1.3 we have to show that for almost all
$\boldsymbol{\alpha}$ in $\mathbb{R}^{d}$

(3.1) $\displaystyle\frac{1}{N}\sum_{1\leq m\neq n\leq
N}\chi_{s,N^{1/d}}((\boldsymbol{x}_{n}-\boldsymbol{x}_{m})\boldsymbol{\alpha})\to(2s)^{d}\quad\mbox{
as }N\rightarrow\infty.$

It is clear from the definition of $\chi_{s,N^{1/d}}$ that

(3.2)
$\displaystyle\chi_{s,N^{1/d}}(\boldsymbol{x})=\chi_{s,N^{1/d}}(x^{(1)})\cdots\chi_{s,N^{1/d}}(x^{(d)}).$

Essentially, $\chi_{s,N^{1/d}}(x^{(i)})=1$ if
$\displaystyle\|x^{(i)}\|^{\text{(intdist)}}=\min_{k\in\mathbb{Z}}|x^{(i)}+k|\leq
s/N^{1/d}.$ It is well known that for any $s$ and $N$ there exist
trigonometric polynomials $f_{K,s,N^{1/d}}^{\pm}(x)=\sum_{|j|\leq
K}c_{j}^{\pm}e(jx)$, for a positive integer $K$, such that

(3.3) $\displaystyle f_{K,s,N^{1/d}}^{-}(x)\leq\chi_{s,N^{1/d}}(x)\leq
f_{K,s,N^{1/d}}^{+}(x),$

for all $x\in\mathbb{R}$. Further, we have

(3.4) $\displaystyle
c_{0}^{\pm}=\int_{0}^{1}f_{K,s,N^{1/d}}^{\pm}(x)dx=\frac{2s}{N^{1/d}}\pm\frac{1}{K+1},$

and any non-zero $j$-th Fourier coefficient $c_{j}^{\pm}$ of
$f_{K,s,N^{1/d}}^{\pm}(x)$ satisfies the upper bound

(3.5)
$\displaystyle|c_{j}^{\pm}|\leq\min\left(\frac{2s}{N^{1/d}},\frac{1}{\pi|j|}\right)+\frac{1}{K+1}.$

For more details one can see [12, Chapter 1]. Thus by imposing (3.3) in (3.2)
we obtain

(3.6) $\displaystyle
F_{K,s,N^{1/d}}^{-}(\boldsymbol{x})\leq\chi_{s,N^{1/d}}(\boldsymbol{x})\leq
F_{K,s,N^{1/d}}^{+}(\boldsymbol{x}),$

where

$\displaystyle F_{K,s,N^{1/d}}^{\pm}(\boldsymbol{x})$
$\displaystyle:=f_{K,s,N^{1/d}}^{\pm}(x^{(1)})\cdots
f_{K,s,N^{1/d}}^{\pm}(x^{(d)})$
$\displaystyle=\displaystyle\sum_{\begin{subarray}{c}\boldsymbol{j}\in\mathbb{Z}^{d}\\\
\|\boldsymbol{j}\|_{\infty}\leq
K\end{subarray}}c_{\boldsymbol{j}}^{\pm}e(\boldsymbol{j}\cdot\boldsymbol{x}),$

where $c_{\boldsymbol{j}}:=c_{j_{1}}\cdots c_{j_{d}}$ and
$c_{j_{1}},\dots,c_{j_{d}}$ satisfy (3.4) and (3.5).

In order to establish (3.1) for $\chi_{s,N^{1/d}}$ it is enough to establish
such result for its majorant and minorant $F_{K,s,N^{1/d}}^{\pm}$. Thus we
want to show that for every fixed positive integer $r$ and for every $s>0,$

(3.7) $\displaystyle\frac{1}{N}\sum_{1\leq m\neq n\leq
N}F_{(rN)^{1/d},s,N^{1/d}}^{\pm}((\boldsymbol{x}_{n}-\boldsymbol{x}_{m})\boldsymbol{\alpha})\sim
N\int_{[0,1]^{d}}F_{(rN)^{1/d},s,N^{1/d}}^{\pm}(\boldsymbol{y})dy~{}~{}~{}\mbox{
as }N\rightarrow\infty,$

holds for almost all $\boldsymbol{\alpha}$ in $\mathbb{R}^{d}.$ Then we can
conclude (3.1) by using the fact that
$c_{\mathbb{0}}^{\pm}=(c_{0}^{\pm})^{d}$, (3.4) and tend $r$ to infinity.

###  3.1. Mean estimation

Now we prove (3.7) for $F_{K,s,N^{1/d}}^{+}$ using “variance method”. Similar
arguments works for $F_{K,s,N^{1/d}}^{-}$. From now on $r$ and $s$ are fixed
and we denote $F_{(rN)^{1/d},s,N^{1/d}}^{+}$ by $F_{N}$. Let us consider the
measure

$d\boldsymbol{\mu}(\boldsymbol{x})=\prod_{1\leq i\leq
d}\frac{2\sin^{2}(x_{i}/2)}{\pi x_{i}^{2}}dx_{i}.$

Note that for any $\boldsymbol{\mu}$ almost sure result implies Lebesgue
almost sure result. At first we want to compute the mean value of the quantity
of interest as follows:

$\displaystyle\left|\int_{\mathbb{R}^{d}}\frac{1}{N}\sum_{1\leq m\neq n\leq
N}F_{N}((\boldsymbol{x}_{n}-\boldsymbol{x}_{m})\boldsymbol{\alpha})d\boldsymbol{\mu}(\boldsymbol{\alpha})-N\int_{[0,1]^{d}}F_{N}(\boldsymbol{y})dy\right|$
$\displaystyle\leq\left(N-\frac{N^{2}-N}{N}\right)\int_{[0,1]^{d}}F_{N}(\boldsymbol{y})dy$
$\displaystyle+\frac{1}{N}\sum_{\begin{subarray}{c}\boldsymbol{j}\in\mathbb{Z}^{d}\\\
1\leq\|\boldsymbol{j}\|_{\infty}\leq(rN)^{1/d}\end{subarray}}|c_{\boldsymbol{j}}|\left|\int_{\mathbb{R}^{d}}\sum_{1\leq
m\neq n\leq
N}e(\boldsymbol{j}\cdot(\boldsymbol{x}_{n}-\boldsymbol{x}_{m})\boldsymbol{\alpha})d\boldsymbol{\mu}(\boldsymbol{\alpha})\right|$
$\displaystyle\ll
N^{-1}+N^{-2}\sum_{\begin{subarray}{c}\boldsymbol{j}\in\mathbb{Z}^{d}\\\
1\leq\|\boldsymbol{j}\|_{\infty}\leq(rN)^{1/d}\end{subarray}}\sum_{1\leq m\neq
n\leq
N}\mathds{1}_{\|\boldsymbol{j}(\boldsymbol{x}_{m}-\boldsymbol{x}_{n})\|_{\infty}<1}\ll
N^{-1},$

since $|c_{\boldsymbol{j}}|\ll N^{-1}$ and we used the hypothesis on the
sequence to get that the sum over $\boldsymbol{j},m$ and $n$ is at most $N$.

###  3.2. Variance estimation

Set

$H_{N}(\boldsymbol{x}):=F_{N}(\boldsymbol{x})-\int_{[0,1]^{d}}F_{N}(\boldsymbol{y})d\boldsymbol{y}=\sum_{\boldsymbol{j}\in\mathbb{Z}^{d}\setminus\\{\boldsymbol{0}\\}}c_{\boldsymbol{j}}e(\boldsymbol{j}\cdot\boldsymbol{x}),$

where $c_{\boldsymbol{j}}=0$ for $\|\boldsymbol{j}\|_{\infty}>(rN)^{1/d}$. We
want to estimate the following quantity

$\text{Var}(H_{N},\boldsymbol{\mu}):=\int_{\mathbb{R}^{d}}\left(\frac{1}{N}\sum_{1\leq
m\neq n\leq
N}H_{N}((\boldsymbol{x}_{n}-\boldsymbol{x}_{m})\boldsymbol{\alpha})\right)^{2}d\boldsymbol{\mu}(\boldsymbol{\alpha}).$

Let $D^{\prime}\subseteq D$ with $|D^{\prime}|=d^{\prime}$ and
$D^{\prime}=\\{i_{1},\ldots,i_{d^{\prime}}\\}$ such that
$i_{1}<\dots<i_{d^{\prime}}$. Any element in such $d^{\prime}$-dimensional
vector subspace of $\mathbb{R}^{d}$, which isomorphic to
$\mathbb{R}^{d^{\prime}}$, is denoted by
$\boldsymbol{x}^{D^{\prime}}=(x^{(i_{1})},\dots,x^{(i_{d^{\prime}})})$. Thus

$\displaystyle\text{Var}(H_{N},\boldsymbol{\mu})$
$\displaystyle=\frac{1}{N^{2}}\int_{\mathbb{R}^{d}}\left(\sum_{\begin{subarray}{c}D^{\prime}\subseteq
D\end{subarray}}\sum_{1\leq m\neq n\leq
N}\sideset{}{{}^{\prime}}{\sum}_{\boldsymbol{j}\in\mathbb{Z}^{d^{\prime}}}c_{0}^{d-d^{\prime}}c_{\boldsymbol{j}}e(\boldsymbol{j}\cdot(\boldsymbol{x}_{m}^{D^{\prime}}-\boldsymbol{x}_{n}^{D^{\prime}})\boldsymbol{\alpha})\right)^{2}d\boldsymbol{\mu}(\boldsymbol{\alpha})$
$\displaystyle\ll\frac{1}{N^{2}}\int_{\mathbb{R}^{d}}\left(\sum_{\begin{subarray}{c}D^{\prime}\subseteq
D\end{subarray}}\sum_{1\leq m\neq n\leq
N}\sideset{}{{}^{\prime}}{\sum}_{\boldsymbol{j}\in\mathbb{Z}^{d^{\prime}}}c_{0}^{d-d^{\prime}}c_{\boldsymbol{j}}e(\boldsymbol{j}\cdot(|\boldsymbol{x}_{m}^{D^{\prime}}-\boldsymbol{x}_{n}^{D^{\prime}}|)\boldsymbol{\alpha})\right)^{2}d\boldsymbol{\mu}(\boldsymbol{\alpha}),$

where $|\boldsymbol{x}_{m}^{D^{\prime}}-\boldsymbol{x}_{n}^{D^{\prime}}|$
means the component-wise absolute difference and $\sum^{{}^{\prime}}$ means
the sum over $\boldsymbol{j}\in\mathbb{Z}^{d^{\prime}}$ such that $1\leq
j_{l}\leq(rN)^{1/d}$ for all $l\leq d^{\prime}$. Let $U$ be the smallest
integer such that $2^{U}\geq(rN)^{1/d}$. Now, by dyadic decomposition of the
components of
$\boldsymbol{j}=(j_{1},\ldots,j_{d^{\prime}})\in\mathbb{Z}^{d^{\prime}}$ and
the application of Cauchy-Schwarz inequality gives the upper bound of
$\text{Var}(H_{N},\boldsymbol{\mu})$ as

$\displaystyle\ll\frac{1}{N^{2}}\int_{\mathbb{R}^{d}}$
$\displaystyle\left(\sum_{\begin{subarray}{c}\|\boldsymbol{u}\|_{\infty}\leq
U\\\
\boldsymbol{u}\in\mathbb{N}^{d}\end{subarray}}1\right)\sum_{\begin{subarray}{c}\|\boldsymbol{u}\|_{\infty}\leq
U\\\
\boldsymbol{u}\in\mathbb{N}^{d}\end{subarray}}\left|\sum_{\begin{subarray}{c}D^{\prime}\subseteq
D\end{subarray}}\sum_{1\leq m\neq n\leq
N}\sum_{\begin{subarray}{c}(j_{1},\ldots,j_{d^{\prime}})\in\mathbb{Z}^{d^{\prime}}\\\
2^{u_{i}-1}\leq|j_{i}|<2^{u_{i}}\end{subarray}}c_{0}^{d-d^{\prime}}c_{\boldsymbol{j}}e(\boldsymbol{j}\cdot(|\boldsymbol{x}_{m}^{D^{\prime}}-\boldsymbol{x}_{n}^{D^{\prime}}|)\boldsymbol{\alpha})\right|^{2}d\boldsymbol{\mu}(\boldsymbol{\alpha})$
(3.8) $\displaystyle\ll\frac{(\log
N)^{d}}{N^{4}}\sum_{\begin{subarray}{c}\|\boldsymbol{u}\|_{\infty}\leq U\\\
\boldsymbol{u}\in\mathbb{N}^{d}\end{subarray}}\sum_{\begin{subarray}{c}D^{\prime}\subseteq
D\end{subarray}}\sum_{\begin{subarray}{c}1\leq m\neq n\leq N\\\ 1\leq k\neq
l\leq
N\end{subarray}}\sum_{\begin{subarray}{c}\boldsymbol{j},\boldsymbol{t}\in\mathbb{N}^{d^{\prime}}\\\
2^{u_{i}-1}\leq
j_{i},t_{i}<2^{u_{i}}\end{subarray}}\mathds{1}_{\left\|\boldsymbol{j}(|\boldsymbol{x}_{m}^{D^{\prime}}-\boldsymbol{x}_{n}^{D^{\prime}}|)-\boldsymbol{t}(|\boldsymbol{x}_{k}^{D^{\prime}}-\boldsymbol{x}_{l}^{D^{\prime}}|)\right\|_{\infty}<1}.$

For fixed $\boldsymbol{u}\in\mathbb{N}^{d}$ and $D^{\prime}\subseteq D$,
define the multi-set

$\\{\boldsymbol{z}_{1}^{D^{\prime}},\dots,\boldsymbol{z}_{M}^{D^{\prime}}\\}=\\{|\boldsymbol{x}_{n}^{D^{\prime}}-\boldsymbol{x}_{m}^{D^{\prime}}|:1\leq
n\neq m\leq N\\},$

where $M:=N^{2}-N$. In order to bound (3.2), it is enough to bound the
following sum

(3.9)
$\displaystyle\sum_{\begin{subarray}{c}\boldsymbol{j},\boldsymbol{t}\in\mathbb{N}^{d^{\prime}}\\\
2^{u_{i}-1}\leq j_{i},t_{i}<2^{u_{i}}\end{subarray}}\sum_{1\leq m,n\leq
M}\mathds{1}_{\left\|\boldsymbol{j}\boldsymbol{z}_{m}^{D^{\prime}}-\boldsymbol{t}\boldsymbol{z}_{n}^{D^{\prime}}\right\|_{\infty}<1}.$

##  4\. Proof of Theorem 1.3

To prove Theorem 1.3 we need to bound (3.9). In the subsequent calculations we
consider the subset $D^{\prime}=\\{1,\dots,d^{\prime}\\}\subseteq D,$ because
for other subset of $D$ with cardinality $d^{\prime}$ arguments are same. For
simplicity, we use the notation $\boldsymbol{z}_{n}$ instead of
$\boldsymbol{z}_{n}^{D^{\prime}}\in\mathbb{R}^{d^{\prime}}$ in the remaining
proof.

Let $\beta$ be a function defined by $\beta(2)=-1$ and $\beta(d)=+1$ for
$d>2$. Further define $\Phi(y)=e^{-y^{2}/2}$, then its Fourier transform is
$\widehat{\Phi}=\sqrt{2\pi}\Phi$. Also, assume that $\epsilon>0$ is fixed but
sufficiently small real number. Now we divide the sum over $m,n$ into two
cases based on the size of $\min(z^{(l)}_{m},z^{(l)}_{n})$, where
$z^{(l)}_{m}$ and $z^{(l)}_{n}$ are the $l$-th component of
$\boldsymbol{z}_{m}$ and $\boldsymbol{z}_{n}$, respectively.

######  Case 1.

$\min(z_{m}^{(l)},z_{n}^{(l)})<N^{1/d+\beta(d)\epsilon}$. In this case (3.9)
is bounded by,

(4.1)
$\displaystyle\sum_{\begin{subarray}{c}\boldsymbol{j},\boldsymbol{t}\in\mathbb{N}^{d^{\prime}}\\\
2^{u_{i}-1}\leq
j_{i},t_{i}<2^{u_{i}}\end{subarray}}\sum_{\begin{subarray}{c}1\leq m,n\leq
M\\\ z_{m}^{(l)},z_{n}^{(l)}\leq
4N^{1/{d}+\beta(d)\epsilon}\end{subarray}}\mathds{1}_{\left\|\boldsymbol{j}\boldsymbol{z}_{m}-\boldsymbol{t}\boldsymbol{z}_{n}\right\|_{\infty}<1}.$

For fixed $\boldsymbol{j},\boldsymbol{z}_{m}$ and $\boldsymbol{z}_{n}$ there
are $\ll 1$ many choices for $\boldsymbol{t},$ because of the spacing
assumption. One observe that there are $\ll N^{1+1/d+\beta(d)\epsilon}$ many
$z_{n}$ which are $\leq 4N^{1/d+\beta(d)\epsilon}.$ In this case, the upper
bound of (4.1) is

(4.2) $\displaystyle
N^{2+2/d+2\beta(d)\epsilon}\prod_{l=1}^{d^{\prime}}2^{u_{l}}\ll
N^{2+2/d+d^{\prime}/d+2\beta(d)\epsilon}\leq N^{3+2/d+2\beta(d)\epsilon}.$

######  Case 2.

$\min(z_{m}^{(l)},z_{n}^{(l)})\geq N^{1/d+\epsilon}$ for all $l\in
D^{\prime}.$
For $\boldsymbol{k}\in\mathbb{N}^{d^{\prime}}$, the interval
$[\boldsymbol{k},\boldsymbol{k}+1)$ means the product
$[k_{1},k_{1}+1)\times\ldots\times[k_{d^{\prime}},k_{d^{\prime}}+1)$ and then
define

(4.3) $\displaystyle b_{\boldsymbol{k}}:=\sum_{n\leq
N}\mathds{1}_{\boldsymbol{z}_{n}\in[\boldsymbol{k},\boldsymbol{k}+1)}.$

We split $\mathbb{N}^{d^{\prime}}$ into the intervals $I_{\boldsymbol{h}}$,
where $\boldsymbol{h}\in\mathbb{N}^{d^{\prime}}\cup\\{\mathbb{0}\\}$ and
$I_{\boldsymbol{h}}$ is given by

$\displaystyle
I_{\boldsymbol{h}}:=\prod_{l=1}^{d^{\prime}}\left[\left[\left(1+\frac{1}{T_{l}}\right)^{h_{l}}\right],\left[\left(1+\frac{1}{T_{l}}\right)^{h_{l}+1}\right]\right),$

where $T_{l}:=2^{u_{l}}N^{1/{d}+\epsilon/{2d}}$ for $1\leq l\leq d^{\prime}$
and set

(4.4) $\displaystyle a_{\boldsymbol{h}}:=\left(\sum_{\boldsymbol{k}\in
I_{\boldsymbol{h}}}b_{\boldsymbol{k}}^{2}\right)^{1/2}.$

Using the notation
$\mathbb{y}_{\mathbb{T}}:=\left(\frac{y_{1}}{T_{1}},\ldots,\frac{y_{d^{\prime}}}{T_{d^{\prime}}}\right)$,
we set

$\displaystyle\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right):=\prod_{l=1}^{d^{\prime}}\Phi\left(\frac{y_{l}}{T_{l}}\right).$

Then we introduce the function
$P:\mathbb{R}^{d^{\prime}}\rightarrow\mathbb{C}$, which is given by

(4.5) $\displaystyle P(\boldsymbol{y})=\sum_{\boldsymbol{h}\geq
0}a_{\boldsymbol{h}}\prod_{l=1}^{d^{\prime}}\left(1+\frac{1}{T_{l}}\right)^{ih_{l}y_{l}}.$

Similar to Lemma 2 in [1], we also get its higher dimensional analogue in the
following lemma.

######  Lemma 4.1.

Let us set $T:=T_{1}\ldots T_{d^{\prime}}$ . Then

$\int_{\mathbb{R}^{d^{\prime}}}|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)d\boldsymbol{y}\ll
TE(\boldsymbol{X}_{N}^{D^{\prime}}).$

###### Proof.

Similar to the proof of [1, Lemma 2], expanding the square and apply the
Fourier transform on $\Phi(\boldsymbol{y})$, we get

$\displaystyle\int_{\mathbb{R}^{d^{\prime}}}|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)d\boldsymbol{y}\ll
T\sum_{\mathbb{h_{1}},\mathbb{h_{2}}\in\mathbb{N}^{d^{\prime}}\cup\\{\mathbb{0}\\}}a_{\mathbb{h_{1}}}a_{\mathbb{h_{2}}}\prod_{l=1}^{d^{\prime}}\widehat{\Phi}\left(\frac{h_{1}^{(l)}-h_{2}^{(l)}}{2}\right)\ll
T\sum_{\mathbb{h}\in\mathbb{N}^{d^{\prime}}\cup\\{\mathbb{0}\\}}a_{\mathbb{h}}^{2}.$

The last inequality uses the rapid decay of $\widehat{\Phi}$. Then the
definition (4.4) gives

$\displaystyle\sum_{\mathbb{h}\in\mathbb{N}^{d^{\prime}}\cup\\{\mathbb{0}\\}}a_{\mathbb{h}}^{2}\ll\sum_{\mathbb{k}\in\mathbb{N}^{d^{\prime}}}b_{\boldsymbol{k}}^{2}\ll\sum_{\begin{subarray}{c}1\leq
m,n\leq M\\\ |z_{m}^{(l)}-z_{n}^{(l)}|<1,\>l\in D^{\prime}\end{subarray}}1\leq
E(\mathbb{X}_{N}^{D^{\prime}}).$

Hence the result follows. ∎

######  Lemma 4.2.

$\sum_{\begin{subarray}{c}\boldsymbol{j},\boldsymbol{t}\in\mathbb{Z}^{d^{\prime}}\\\
2^{u_{l}-1}\leq j_{l},t_{l}<2^{u_{l}},\>l\in
D^{\prime}\end{subarray}}\sum_{1\leq m,n\leq
M}\mathds{1}_{\left\|\boldsymbol{j}\boldsymbol{z}_{m}-\boldsymbol{t}\boldsymbol{z}_{n}\right\|_{\infty}<1}\ll\sum_{\begin{subarray}{c}\boldsymbol{j},\boldsymbol{t}\in\mathbb{Z}^{d^{\prime}}\\\
2^{u_{l}-1}\leq j_{l},t_{l}<2^{u_{l}},\>l\in
D^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}\boldsymbol{g},\boldsymbol{h}\in\mathbb{N}^{d^{\prime}}\cup\\{\mathbb{0}\\}\\\
\left|\left(1+\frac{1}{T_{l}}\right)^{h_{l}-g_{l}}-\frac{t_{l}}{j_{l}}\right|\leq\frac{4}{T_{l}},\>l\in
D^{\prime}\end{subarray}}a_{\boldsymbol{g}}a_{\boldsymbol{h}}.$

###### Proof.

For $d^{\prime}=1$, the proof is essentially identical with the proof of [1,
Lemma 3]. For $d^{\prime}>1$, we apply the same method for $d^{\prime}$ many
components in order to get the result. ∎

Let $K:\mathbb{R}\rightarrow\mathbb{R}^{+}$ be a function defined by

(4.6) $\displaystyle
K(\xi)=\frac{\sin^{2}\left(\left(1/d+\delta/{d}\right)\xi\log
N\right)}{\pi\xi^{2}\left(1/d+\delta/{d}\right)\log N},$

for some positive real parameter $\delta$ and positive integer $d$. Its
Fourier transform is given by

(4.7)
$\displaystyle\widehat{K}(v)=\max\left(1-\frac{v}{2\left(1/d+\delta/{d}\right)\log
N},0\right).$

######  Lemma 4.3.

$\displaystyle\sum_{\begin{subarray}{c}\boldsymbol{j},\boldsymbol{t}\in\mathbb{Z}^{d^{\prime}}\\\
2^{u_{l}-1}\leq j_{l},t_{l}<2^{u_{l}},\>l\in
D^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}\boldsymbol{g},\boldsymbol{h}\in\mathbb{N}^{d^{\prime}}\cup\\{\mathbb{0}\\}\\\
\left|\left(1+\frac{1}{T_{l}}\right)^{h_{l}-g_{l}}-\frac{t_{l}}{j_{l}}\right|,\>l\in
D^{\prime}\end{subarray}}a_{\boldsymbol{g}}a_{\boldsymbol{h}}$
$\displaystyle\ll\frac{2^{u_{1}}\cdots
2^{u_{d^{\prime}}}}{T}\int_{\mathbb{R}^{d^{\prime}}}\sum_{\boldsymbol{j},\boldsymbol{t}\in\mathbb{N}^{d^{\prime}}}\prod_{l=1}^{d^{\prime}}\frac{\widehat{K}(\log
j_{l}t_{l})}{(j_{l}t_{l})^{1/2}}\left(\frac{j_{l}}{t_{l}}\right)^{iy_{l}}|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)d\boldsymbol{y}$

###### Proof.

Note that $2^{u_{1}}\cdots
2^{u_{d^{\prime}}}\ll\prod_{l=1}^{d^{\prime}}(j_{l}t_{l})^{1/2}\ll
2^{u_{1}}\cdots 2^{u_{d^{\prime}}}$. This fact combining with the properties
of $\Phi$ as well as $\widehat{K}$, one can obtain this lemma. Also one may
see the proof of [1, Lemma 4] for one dimensional version. ∎

######  Lemma 4.4.

$\displaystyle\frac{2^{u_{1}}\cdots
2^{u_{d^{\prime}}}}{T}\int_{\mathbb{R}^{d^{\prime}}}\sum_{\boldsymbol{j},\boldsymbol{t}\geq
1}\prod_{l=1}^{d^{\prime}}\frac{\widehat{K}(\log
j_{l}t_{l})}{(j_{l}t_{l})^{1/2}}\left(\frac{j_{l}}{t_{l}}\right)^{iy_{l}}|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)d\boldsymbol{y}$
$\displaystyle\ll N^{4-\epsilon
d^{\prime}/4d}+N^{\frac{152d^{\prime}-31}{89d}+\frac{52}{89}+5\epsilon}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}.$

Therefore, in Case $2$, from Lemma 4.2, Lemma 4.3 and Lemma 4.4 the upper
bound of (3.9)

(4.8) $\displaystyle N^{4-\epsilon
d^{\prime}/4d}+N^{\frac{152d^{\prime}-31}{89d}+\frac{52}{89}+5\epsilon}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}.$

In order to prove Lemma 4.4 we we need the following technical lemma, which is
essentially the Lemma 5.3 of [6] or [1, Lemma 6].

######  Lemma 4.5.

Let $\sigma\in(-\infty,1)$ and let $F$ be a holomorphic function in the strip
$y=\Im z\in[\sigma-2,0]$ such that

$\sup_{\sigma-2\leq y\leq 0}|F(x+iy)|\leq\frac{1}{x^{2}+1}.$

Then for all $s=\sigma+it\in\mathbb{C}$ with $t\neq 0,$ we have

$\sum_{j,t\geq 1}\frac{\widehat{K}(\log
jt)}{j^{s}t^{\bar{s}}}=\int_{\mathbb{R}}\zeta(s+iu)\overline{\zeta(s-iu)}F(u)du+2\pi\zeta(1-2it)F(is-i)+2\pi\zeta(1+2it)F(i\bar{s}-i).$

###### Proof of Lemma 4.4.

Let us set

(4.9) $\displaystyle G(y):=\displaystyle\sum_{j,t\geq 1}\frac{\hat{K}(\log
jt)}{(jt)^{1/2}}\left(\frac{j}{t}\right)^{iy},$

and

$G(\boldsymbol{y}):=G(y_{1})\cdots
G(y_{d^{\prime}})=\sum_{\boldsymbol{j},\boldsymbol{t}\geq
1}\prod_{l=1}^{d^{\prime}}\frac{\widehat{K}(\log
j_{l}t_{l})}{(j_{l}t_{l})^{1/2}}\left(\frac{j_{l}}{t_{l}}\right)^{iy_{l}}.$

Applying Lemma 4.5 on each $G(y_{l})$, where $y_{l}\neq 0$, we write

$G(\boldsymbol{y})=\prod_{l=1}^{d^{\prime}}[D_{1}(y_{l})+D_{2}(y_{l})+D_{3}(y_{l})],$

where

$\displaystyle
D_{1}(y_{l}):=\int_{\mathbb{R}}\zeta(1/2+iy_{l}+iu_{l})\zeta(1/2-iy_{l}+iu_{l})K(u_{l})du_{l},$
$\displaystyle D_{2}(y_{l}):=2\pi\zeta(1-2iy_{l})F(-y_{l}-i/2),$
$\displaystyle D_{3}(y_{l}):=2\pi\zeta(1+2iy_{l})F(y_{l}-i/2).$

Then term by term product gives

(4.10) $\displaystyle
G(\boldsymbol{y})=\sum_{\eta_{1},\dots,\eta_{d^{\prime}}\in\\{1,2,3\\}}D_{\eta_{1}}(y_{1})\cdots
D_{\eta_{d^{\prime}}}(y_{d^{\prime}}),\text{ where }y_{l}\neq 0,\>\forall l\in
D^{\prime}.$

From the definition of $P$ in (4.5), we deduce that

(4.11) $\displaystyle|P(\boldsymbol{y})|^{2}\leq$
$\displaystyle\left(\sum_{\boldsymbol{h}\geq
0}a_{\boldsymbol{h}}\right)^{2}\ll\left(\sum_{\boldsymbol{k}\geq
1}b_{\boldsymbol{k}}\right)^{2}\ll
N^{4},\>\>\forall\boldsymbol{y}\in\mathbb{R}^{d^{\prime}}.$

Further, we have

(4.12) $\displaystyle|G(y)|\leq\sum_{\log(jt)\leq 2(1/{d}+\delta/{d})\log
N}\frac{1}{(jt)^{1/2}}\leq\sum_{jt\leq
N^{2(1/d+\delta/{d})}}\frac{1}{(jt)^{1/2}}\ll N^{1/d+\delta/{d}},$ (4.13)
$\displaystyle|K(\pm y-i/2)|\ll N^{1/d+\delta/d}/(y+1)^{2}.$

Let us denote

$\displaystyle
I=\int_{\mathbb{R}^{d^{\prime}}}G(\boldsymbol{y})|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)d\boldsymbol{y}.$

Now we bound the integral $I$ by breaking the range of integrals in several
parts as follows:

(4.14) $\displaystyle I=\sum_{\mathcal{D}\subseteq
D^{\prime}}\int_{\begin{subarray}{c}|y_{l}|\leq 1,\>l\in\mathcal{D}\\\
|y_{l}|>1,\>l\in
D^{\prime}\setminus\mathcal{D}\end{subarray}}G(\boldsymbol{y})|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)d\boldsymbol{y}=\sum_{\mathcal{D}\subseteq
D^{\prime}}I_{\mathcal{D}}~{}~{}~{}~{}~{}~{}\mbox{ (say).}$

For $\mathcal{D}=D^{\prime}$, by employing (4.11) and (4.12) we get
$I_{D^{\prime}}\ll N^{4+d^{\prime}/d+\delta d^{\prime}/d}.$
When $\mathcal{D}=\phi,$ the empty set, using (4.10) we want to estimate the
upper bound of $I_{\phi}.$ Thus

$\displaystyle I_{\phi}\ll$
$\displaystyle\sum_{\begin{subarray}{c}\eta_{i}\in\\{1,2,3\\}\\\
i=1,\ldots,d^{\prime}\end{subarray}}\int_{|y_{l}|>1,\>l\in
D^{\prime}}|D_{\eta_{1}}(y_{1})\cdots
D_{\eta_{d^{\prime}}}(y_{d^{\prime}})||P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)d\boldsymbol{y}=\sum_{\begin{subarray}{c}\eta_{i}\in\\{1,2,3\\}\\\
i=1,\ldots,d^{\prime}\end{subarray}}I_{\eta_{1},\dots,\eta_{d^{\prime}}}~{}~{}~{}\mbox{
(say)}.$

Now we bound $I_{\eta_{1},\dots,\eta_{d^{\prime}}}$ in three sub cases
depending on the values of $\eta_{1},\dots,\eta_{d^{\prime}}$ as follows:

  * (i)

$\eta_{1}=\cdots=\eta_{d^{\prime}}=1,$

  * (ii)

$\eta_{1},\ldots,\eta_{d^{\prime}}\in\\{2,3\\},$

  * (iii)

for some $l,k\in D^{\prime}$ $\eta_{l}=1$ and $\eta_{k}=2$ or $3$.

#### Subcase 1:

Assume $\eta_{1}=\cdots=\eta_{d^{\prime}}=1$. In this case, we want to
estimate

$\displaystyle I_{1,\dots,1}=\int_{|y_{l}|>1,\>l\in
D^{\prime}}|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)\int_{\mathbb{R}^{d^{\prime}}}\prod_{l=1}^{d^{\prime}}|\zeta(1/2+iy_{l}+iu_{l})\zeta(1/2-iy_{l}+iu_{l})K(u_{l})|d\boldsymbol{u}d\boldsymbol{y}.$

Let us denote the inner integral ( integral over $\mathbb{R}^{d^{\prime}}$) in
the above inequality by $N_{d^{\prime}}(\boldsymbol{y},\mathbb{T})$. Then we
rewrite it as

(4.15) $\displaystyle
N_{d^{\prime}}(\boldsymbol{y},\mathbb{T})=\sum_{\widetilde{D}\subseteq
D^{\prime}}\int_{\begin{subarray}{c}|u_{l}|\leq T_{l},\>l\in\widetilde{D}\\\
|u_{l}|>T_{l},\>l\in
D^{\prime}\setminus\widetilde{D}\end{subarray}}\prod_{l=1}^{d^{\prime}}|\zeta(1/2+iy_{l}+iu_{l})\zeta(1/2-iy_{l}+iu_{l})K(u_{l})|d\boldsymbol{u}.$

For $\widetilde{D}=\phi$, using the trivial convexity bound of Riemann zeta
function, $|\zeta(1/2+it)|\ll|t|^{1/4}$ and the upper bound $K(u_{l})\ll
u_{l}^{-2}$, the contribution to $N_{d^{\prime}}(\boldsymbol{y},\mathbb{T})$
is

$\displaystyle\ll\prod_{l=1}^{d^{\prime}}\left(|y_{l}|^{1/2}/T_{l}+1/T_{l}^{1/2}\right).$

Consequently the contribution to $I_{1,\dots,1}$ in this case is given by $\ll
N^{4}T^{1/2}$, which is negligible.

Next, we treat the case $\widetilde{D}=D^{\prime}$. In this case, from (4.15)
we see that the range of the integral of any variable $u_{l}$ is
$[-T_{l},T_{l}]$. We would use the following moment estimate of Riemann zeta
function due to Ivić (see [11, Theorem 8.3]).

(4.16) $\displaystyle\int_{0}^{T}|\zeta(1/2+it)|^{\frac{178}{13}}dt\ll
T^{\frac{29}{13}+\epsilon}.$

Let $A,B$ be two integers such that $\frac{1}{A}+\frac{1}{A}+\frac{1}{B}=1.$
Then by Hölder’s inequality on the integration over $y_{l}$’s, gives the upper
bound

$\displaystyle\ll\int_{|u_{l}|\leq T_{l},\>l\in
D^{\prime}}\prod_{l=1}^{d^{\prime}}K(u_{l})\left(\int_{\mathbb{R}}|\zeta(1/2+iy_{l}+iu_{l})|^{A}\Phi(y_{l}/T_{l})dy_{l}\right)^{1/A}\times$
$\displaystyle\left(\int_{\mathbb{R}}|\zeta(1/2-iy_{l}+iu_{l})|^{A}\Phi(y_{l}/T_{l})dy_{l}\right)^{1/A}\times|P(0)|^{2(1-1/B)}\left(\int_{\mathbb{R}^{d^{\prime}}}|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)d\boldsymbol{y}\right)^{1/B}d\boldsymbol{u}$
$\displaystyle\ll\prod_{l=1}^{d^{\prime}}\left(\int_{0}^{T_{l}}|\zeta(1/2+it)|^{A}dt\right)^{2/A}\times
N^{8/A}T^{1/B}E^{*}(\boldsymbol{X}_{N}^{D^{\prime}})^{1/B},$

where the last inequality uses the estimates $\int_{|u_{l}|\leq
T_{l}}K(u_{l})du_{l}\ll 1$ and Lemma 4.1.Taking $A=178/13$ and employing
(4.16) we get the contribution to the upper bound of $I_{1,\dots,1}$

$\displaystyle\ll
T^{105/89+\epsilon}N^{52/89}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}.$

Now, the remaining cases are when $\widetilde{D}$ is any proper non-trivial
subset of $D^{\prime}$. The associate contribution to the upper bound of
$I_{1,\ldots,1}$ is given by,

$\displaystyle\ll$ $\displaystyle\int_{|u_{l}|\leq
T_{l},\>l\in\widetilde{D}}\prod_{l\in\widetilde{D}}|K(u_{l})|\int_{\mathbb{R}^{d^{\prime}}}P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)\times$
$\displaystyle\prod_{l\in
D^{\prime}\setminus\widetilde{D}}\bigg{(}\frac{|y_{l}|^{1/2}}{T_{l}}+\frac{1}{T_{l}^{1/2}}\bigg{)}\times\prod_{l\in\widetilde{D}}|\zeta(1/2+iy_{l}+iu_{l})\zeta(1/2-iy_{l}+iu_{l})|d\boldsymbol{y}\prod_{l\in\widetilde{D}}du_{l},$

where we use the estimate like the case $\widetilde{D}=\phi$ and interchanges
of integrals. By using Holder’s inequality on the integration over
$\boldsymbol{y}$ (similar to the case $\widetilde{D}=D^{\prime}$), the above
estimate

$\displaystyle\ll$ $\displaystyle\int_{\begin{subarray}{c}|u_{l}|\leq
T_{l},\\\
l\in\widetilde{D}\end{subarray}}\prod_{l=1}^{d^{\prime}}K(u_{l})\left(\int_{\mathbb{R}^{d^{\prime}}}\prod_{l\in\widetilde{D}}|\zeta(1/2+iy_{l}+iu_{l})|^{A}\prod_{l\in
D^{\prime}\setminus\widetilde{D}}\bigg{(}\frac{|y_{l}|^{1/2}}{T_{l}}+\frac{1}{T_{l}^{1/2}}\bigg{)}^{A/2}\Phi\Big{(}\frac{y_{l}}{T_{l}}\Big{)}dy_{l}\right)^{1/A}\times$
$\displaystyle\left(\int_{\mathbb{R}^{d^{\prime}}}\prod_{l\in\widetilde{D}}|\zeta(1/2-iy_{l}+iu_{l})|^{A}\prod_{l\in
D^{\prime}\setminus\widetilde{D}}\bigg{(}\frac{|y_{l}|^{1/2}}{T_{l}}+\frac{1}{T_{l}^{1/2}}\bigg{)}^{A/2}\Phi\Big{(}\frac{y_{l}}{T_{l}}\Big{)}dy_{l}\right)^{1/A}\times$
$\displaystyle|P(0)|^{2(1-1/B)}\left(\int_{\mathbb{R}^{d^{\prime}}}|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)d\boldsymbol{y}\right)^{1/B}\prod_{l\in\widetilde{D}}du_{l}.$

Note that $\int_{|u_{l}|\leq T_{l}}K(u_{l})du_{l}\ll 1$ for any $l\in
D^{\prime}$ and also we estimate

$\displaystyle\int_{\mathbb{R}}\left(|y_{l}|^{1/2}/T_{l}+1/T_{l}^{1/2}\right)^{A/2}\Phi(y_{l}/T_{l})dy_{l}\ll
T_{l}^{1-A/4}\ll 1\mbox{ if }A\geq 4.$

Employing these notes, Lemma 4.1 as well as choosing $A=178/13$ and hence
using (4.16), we reduce the above estimate to the following upper bound

$\displaystyle\prod_{l\in
D^{\prime}\setminus\widetilde{D}}T_{l}^{2/A-1/2}\prod_{l\in\widetilde{D}}T_{l}^{2\theta/A+\epsilon}N^{8/A}T^{1/B}E(\boldsymbol{X}_{N}^{D^{\prime}})^{1/B}\ll
T^{105/89+\epsilon}N^{52/89}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}.$

Combining the estimates for any subset $\widetilde{D}$ of $D^{\prime}$, we
conclude that

$\displaystyle I_{1\dots 1}\ll
N^{4}T^{1/2}+T^{105/89+\epsilon}N^{52/89}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}.$

#### Subcase 2:

Let $\eta_{1},\ldots,\eta_{d^{\prime}}\in\\{2,3\\}$ . Using (4.11), (4.13) and
the bound $|\zeta(1\pm it)|\ll\log|t|$,

$\displaystyle I_{\eta_{1},\dots,\eta_{d^{\prime}}}\ll\int_{|y_{l}|>1,\>l\in
D^{\prime}}N^{4}N^{d^{\prime}/d+\delta
d^{\prime}/d}\prod_{l=1}^{d^{\prime}}\frac{\log
y_{l}}{(y_{l}+1)^{2}}d\boldsymbol{y}\ll N^{4+d^{\prime}/d+\delta
d^{\prime}/d}.$

#### Subcase 3:

There exist some $l,k\in D^{\prime}$ such that $\eta_{l}=1$ and $\eta_{k}=2$
or $3$. Let $D^{\prime\prime}$ be the subset of $D^{\prime}$ such that $l\in
D^{\prime\prime}$ imply $\eta_{l}=1$. Set
$d^{\prime\prime}=|D^{\prime\prime}|\geq 1$, then
$d^{\prime}-d^{\prime\prime}=|D^{\prime}\setminus D^{\prime\prime}|\geq 1$ and
hence $l\in D^{\prime}\setminus D^{\prime\prime}$ means
$\eta_{l}\in\\{2,3\\}$. In this case the upper bound of
$I_{\eta_{1},\dots,\eta_{d^{\prime}}}$ is

$\displaystyle\int_{|y_{l}|>1,\>l\leq
d^{\prime}}|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)$
$\displaystyle\int_{\mathbb{R}^{d^{\prime\prime}}}\prod_{l\in
D^{\prime\prime}}|\zeta(1/2+iy_{l}+iu_{l})\zeta(1/2-iy_{l}+iu_{l})K(u_{l})|d\boldsymbol{u}$
$\displaystyle\times\prod_{l\in D^{\prime}\setminus
D^{\prime\prime}}|2\pi\zeta(1\pm 2iy_{l})K(\pm y_{l}-i/2)|d\boldsymbol{y}.$

Now the estimates (4.13) and $|\zeta(1\pm it)|\ll\log|t|$ gives

$\displaystyle I_{\eta_{1},\dots,\eta_{d^{\prime}}}\ll
N^{(1/d+\delta/d)(d^{\prime}-d^{\prime\prime})}\int_{|y_{l}|>1,\>l\in
D^{\prime}}|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)\prod_{l\in
D^{\prime}\setminus D^{\prime\prime}}\frac{\log|y_{l}|}{y_{l}^{2}+1}N_{l\in
D^{\prime\prime}}(y_{l},T_{l})d\boldsymbol{y},$

where $N_{l\in D^{\prime\prime}}(y_{l},T_{l})$ is defined by

$\displaystyle N_{l\in
D^{\prime\prime}}(y_{l},T_{l}):=\int_{\mathbb{R}^{d^{\prime\prime}}}\prod_{l\in
D^{\prime\prime}}|\zeta(1/2+iy_{l}+iu_{l})\zeta(1/2-iy_{l}+iu_{l})K(u_{l})|d\boldsymbol{u}.$

Similar to $N_{d^{\prime}}(\boldsymbol{y},\mathbb{T})$ in (4.15), we break the
integrals of $N_{l\in D^{\prime\prime}}(y_{l},T_{l})$ as

$\displaystyle\sum_{\widetilde{D}^{\prime}\subseteq
D^{\prime\prime}}\int_{\begin{subarray}{c}|u_{l}|\leq
T_{l},\>l\in\widetilde{D}^{\prime}\\\ |u_{l}|>T_{l},\>l\in
D^{\prime\prime}\setminus\widetilde{D}^{\prime}\end{subarray}}.$

The remaining computation of $I_{\eta_{1},\dots,\eta_{d^{\prime}}}$ in this
case is very similar to the Subcase 1, where we computed $I_{1,\dots,1}$. Let
$J_{\phi},J_{D^{\prime\prime}}$ and $J_{\widetilde{D}^{\prime}}$ be the
contributions to $I_{\eta_{1},\dots,\eta_{d^{\prime}}}$ according to the
choices $\widetilde{D}^{\prime}=\phi$,
$\widetilde{D}^{\prime}=D^{\prime\prime}$ and $\widetilde{D}^{\prime}\neq\phi$
and $D^{\prime\prime}$, respectively. Then we get

$\displaystyle J_{\phi}\ll
N^{(1/d+\delta/d)(d^{\prime}-d^{\prime\prime})}N^{4}\prod_{l\in
D^{\prime\prime}}T_{l}^{1/2}.$

Following Subcase 1, as an application of Holder’s inequality with
$\frac{1}{A}+\frac{1}{A}+\frac{1}{B}=1$, we obtain

$\displaystyle J_{D^{\prime\prime}},J_{\widetilde{D}^{\prime}}\ll
N^{(1/d+\delta/d)(d^{\prime}-d^{\prime\prime})}N^{8/A}T^{1/B}E(\boldsymbol{X}_{N}^{D^{\prime}})^{1/B}\prod_{l\in
D^{\prime\prime}}T_{l}^{2\theta/A+\epsilon},$

where $A=178/13$ and $\theta=29/13$. Combining the above estimates we have

$\displaystyle I_{\eta_{1},\dots,\eta_{d^{\prime}}}\ll
N^{4+(1/d+\delta/d)(d^{\prime}-d^{\prime\prime})}\prod_{l\in
D^{\prime\prime}}T_{l}^{1/2}+N^{52/89+d^{\prime}/d-d^{\prime\prime}/d+\delta(d^{\prime}-d^{\prime\prime})/d}T^{76/89}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}\prod_{l\in
D^{\prime\prime}}T_{l}^{29/89+\epsilon}.$

Choose $\delta=\epsilon/4$ and we note that $2^{u_{l}}\ll N^{1/d}$. This gives
$T_{l}=2^{u_{l}}N^{1/d+\epsilon/2d}\ll N^{2/d+\epsilon/2d}$ , for all $l\in
D^{\prime}$. Hence we have

(4.17) $\displaystyle T\ll N^{2d^{\prime}/d+d^{\prime}\epsilon/2d}\mbox{ and
}\prod_{l\in D^{\prime\prime}}T_{l}\ll
N^{2d^{\prime\prime}/d+d^{\prime\prime}\epsilon/2d}.$

Combining the estimates from Subcese 1, 2 and 3, with $\delta=\epsilon/4$, and
using (4.17), we get

$\displaystyle I_{\phi}\ll$ $\displaystyle N^{4+d^{\prime}/d+\epsilon
d^{\prime}/4d}+\max_{1\leq
d^{\prime\prime}<d^{\prime}}\Big{\\{}N^{-d^{\prime\prime}/d+241d^{\prime}/89d+58d^{\prime\prime}/89d+5\epsilon}\Big{\\}}N^{52/89}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}.$

Since the coefficient of $d^{\prime\prime}/d$ in the exponent of $N$ is
negative, the reasonable choice is $d^{\prime\prime}=1$. Thus,

(4.18) $\displaystyle I_{\phi}\ll N^{4+d^{\prime}/d+\epsilon
d^{\prime}/4d}+N^{52/89+241d^{\prime}/89d-31/89d+5\epsilon}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}.$

The remaining case in (4.14), when $\mathcal{D}$ is a non trivial proper
subset of $D^{\prime}$. Assume that $|\mathcal{D}|=r$. Then by using (4.12),
it is easy to see that

$\displaystyle I_{\mathcal{D}}\ll
N^{(1/d+\delta/d)r}\int_{\begin{subarray}{c}|y_{l}|\leq
1\>:\>l\in\mathcal{D}\\\ |y_{l}|>1\>:\>l\in
D^{\prime}\setminus\mathcal{D}\end{subarray}}|P(\boldsymbol{y})|^{2}\Phi\left(\boldsymbol{y}_{\mathbb{T}}\right)\prod_{l\in
D^{\prime}\setminus\mathcal{D}}G(y_{l})d\boldsymbol{y}.$

Then we see that the integral in the right hand side can be obtain in a
similar way as the estimate of $I_{\phi}$ in (4.18). In particular, we get
such bound with the replacement $d^{\prime}$ by $d^{\prime}-r$ in (4.18).
Essentially the bound for $I_{\mathcal{D}}$, is given by

$\displaystyle I_{\mathcal{D}}\ll
N^{(1/d+\epsilon/4d)r}\Big{(}N^{4+(d^{\prime}-r)/d+\epsilon(d^{\prime}-r)/4d}+N^{52/89+241(d^{\prime}-r)/89d-31/89d+5\epsilon}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}\Big{)}\ll
I_{\phi}.$

Above estimate and (4.18) combining with (4.14) we obtain

(4.19) $\displaystyle I\ll N^{4+d^{\prime}/d+\epsilon
d^{\prime}/4d}+N^{52/89+241d^{\prime}/89d-31/89d+5\epsilon}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}.$

Therefore, (4.19) and the bound $2^{u_{1}}\cdots
2^{u_{d^{\prime}}}/T=1/N^{d^{\prime}/d+d^{\prime}\epsilon/2d}$ prove Lemma
4.4. ∎

For $d\geq 3,$ combining the bounds for (3.9) from (4.2) in Case 1 and (4.8)
in Case 2, we get

$\displaystyle\sum_{\begin{subarray}{c}\boldsymbol{j},\boldsymbol{t}\in\mathbb{N}^{d^{\prime}}\\\
2^{u_{l}-1}\leq j_{l},t_{l}<2^{u_{l}}\end{subarray}}\sum_{1\leq m,n\leq
M}\mathds{1}_{\left\|\boldsymbol{j}\boldsymbol{z}_{m}^{D^{\prime}}-\boldsymbol{t}\boldsymbol{z}_{n}^{D^{\prime}}\right\|_{\infty}<1}\ll
N^{3+2/d+2\epsilon}+N^{4-\epsilon
d^{\prime}/4d}+N^{\frac{152d^{\prime}-31}{89d}+\frac{52}{89}+5\epsilon}E(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}$

Replacing above estimate in (3.2) we deduce that

$\displaystyle\text{Var}(H_{N},\boldsymbol{\mu})$ $\displaystyle\ll(\log
N)^{2d}\left(N^{-\epsilon d^{\prime}/4d}+\max_{D^{\prime}\subseteq
D}\Big{\\{}N^{\frac{152d^{\prime}-31}{89d}+\frac{52}{89}-4+5\epsilon}E^{*}(\boldsymbol{X}_{N}^{D^{\prime}})^{76/89}\Big{\\}}\right)\ll
N^{-\kappa},$

for some $\kappa>0,$ provided that the hypothesis on the joint additive energy
(1.4) is true.

To get an upper bound estimate of (3.9) for d=2, breaking into two cases is
not enough, we need to break in few more cases. We already addressed the case
$\min(z_{m}^{(l)},z_{n}^{(l)})<N^{1/d-\epsilon}$ for some $l=1,2$ in Case 1.
The complement situation of this case can be break in four parts depending on
the range of $\min(z_{m}^{(l)},z_{n}^{(l)})$.

  * (i)

$\min(z_{m}^{(l)},z_{n}^{(l)})\in[N^{1/2-\epsilon},N^{1/2+2\epsilon})$ for
$l=1,2$,

  * (ii)

$\min(z_{m}^{(1)},z_{n}^{(1)})\in[N^{1/2-\epsilon},N^{1/2+2\epsilon})$ and
$\min(z_{m}^{(2)},z_{n}^{(2)})\geq N^{1/2+2\epsilon}$,

  * (iii)

$\min(z_{m}^{(2)},z_{n}^{(2)})\in[N^{1/2-\epsilon},N^{1/2+2\epsilon})$ and
$\min(z_{m}^{(1)},z_{n}^{(1)})\geq N^{1/2+2\epsilon}$,

  * (iv)

$z_{m}^{(l)},z_{n}^{(l)}\geq N^{1/2+2\epsilon}$ for $l=1,2.$

Observe that $2^{u_{l}-1}\leq j_{l},t_{l}<2^{u_{l}}$ and
$|j_{l}z_{m}^{(l)}-j_{l}z_{m}^{(l)}|<1$ together imply
$z_{m}^{(l)}/z_{n}^{(l)}\in[1/4,4].$ So,
$\min(z_{m}^{(l)},z_{n}^{(l)})\in[C,D]$ implies
$z_{m}^{(l)},z_{n}^{(l)}\in[C/4,4D].$

For the case (i), it is enough to consider
$z_{m}^{(l)},z_{n}^{(l)}\in[N^{\beta_{l}}/4,8N^{\beta_{l}})$ as we can cover
$[N^{1/2-\epsilon},N^{1/2+2\epsilon})$ by $\ll\log N$ many intervals of the
form $[N^{\beta}_{l},2N^{\beta}_{l})$, where $1/2-\epsilon\leq\beta_{l}\leq
1/2+2\epsilon.$ Then we can make the setup similar to Case 2 but here we
define $T_{l}=2^{u_{l}-1}N^{\beta_{l}},\>l=1,2$ and $b_{\boldsymbol{k}}$ as
follows

$b_{\boldsymbol{k}}=\sum_{\begin{subarray}{c}1\leq m\leq M\\\
z_{m}^{(l)}\in[N^{\beta_{l}}/4,8N^{\beta_{l}}),\>l=1,2\end{subarray}}\mathds{1}(\boldsymbol{z}_{m}\in[\boldsymbol{k},\boldsymbol{k}+1)).$

By Cauchy-Schwarz inequality we get $|P(0)|^{2}\ll
E(\boldsymbol{X}_{N})N^{\beta_{1}+\beta_{2}}$ , which can use in the place of
trivial upper bound $|P(0)|^{2}\ll N^{4}$. Then we follow the Case 2 to bound
(3.9).

For case (ii), we consider
$z_{n}^{(1)},z_{m}^{(1)}\in[N^{\beta}/4,8N^{\beta})$ and
$z_{m}^{(2)},z_{n}^{(2)}>N^{1/2+2\epsilon}$ where $1/2-\epsilon\leq\beta\leq
1/2+2\epsilon.$ In this case we define $T_{1}=2^{u_{1}-1}N^{\beta}$ and
$T_{2}=2^{u_{2}-1}N^{1/2+2\epsilon}$ and $b_{\boldsymbol{k}}$ as follows

$b_{\boldsymbol{k}}=\sum_{\begin{subarray}{c}1\leq m\leq M\\\
z_{m}^{(1)}\in[N^{\beta}/4,8N^{\beta})\\\
z_{m}^{(2)}>N^{1/2+2\epsilon}\end{subarray}}\mathds{1}(\boldsymbol{z}_{m}\in[\boldsymbol{k},\boldsymbol{k}+1)).$

Here we have to work with the trivial bound $|P(0)|^{2}\ll N^{4}.$ Then
following Case 2, we obtain bound of (3.9). The case (iii) is same with case
(ii), only we have to interchange the role of the components. The case (iv)
essentially follow the Case 2 with using the definition
$T_{l}=2^{u_{l}}N^{1/2+2\epsilon}$ for $l=1,2$. In all first three cases, we
can choose $\delta<\epsilon$ and for case (iv) take $\delta=\epsilon/2$ in the
definition of function $K$ in (4.6). Finally, combining all the estimates in
these cases for (3.9) and then replacing in (3.2), under the hypothesis on
joint additive energy in the theorem we get
$\text{Var}(H_{N},\boldsymbol{\mu})\ll N^{-\kappa}$, for some $\kappa>0$.

This is the main part of the proof. The rest of the arguments follow a
standard method of applying Chebyshev’s inequality and Borel-cantelli lemma
(see the proof of Theorem 1 in [3] and also [9, 17]).

##  5\. Proof of Theorem 1.1

When $x_{n}^{(1)}=\cdots=x_{n}^{(d)}:=x_{n}$ (say), the
$\text{Var}(H_{N},\boldsymbol{\mu})$(see subsection 3.2) is bounded above by

(5.1) $\displaystyle\frac{(\log
N)^{d}}{N^{4}}\sum_{\begin{subarray}{c}\|\boldsymbol{u}\|_{\infty}\leq U\\\
\boldsymbol{u}\in\mathbb{N}^{d}\end{subarray}}\sum_{\begin{subarray}{c}\boldsymbol{j},\boldsymbol{t}\in\mathbb{N}^{d}\\\
2^{\boldsymbol{u}-1}\leq\boldsymbol{j},\boldsymbol{t}<2^{\boldsymbol{u}}\end{subarray}}\sum_{1\leq
m,n\leq
M}\mathds{1}_{\left\|\boldsymbol{j}z_{m}-\boldsymbol{t}z_{n}\right\|_{\infty}<1},$

where the multi-set $\\{z_{1},\dots,z_{M}\\}=\\{|x_{m}-x_{n}|:1\leq n\neq
m\leq N\\}.$ We have $d\geq 3$ and $\boldsymbol{u}\in\mathbb{N}^{d}$ be fixed
in (5.1). We divide the sum over $m,n$ into three cases depending on the size
of $\min(z_{m},z_{n}).$
Case 1: $\min(z_{m},z_{n})<N^{1/d+\epsilon}$.
This case is same as to the Case 1 of section 4.
Case 2: $N^{1/d+\epsilon}\leq\min(z_{m},z_{n})<N^{1+\epsilon}$
Case 3: $\min(z_{m},z_{n})\geq N^{1+\epsilon}$
In Case 2 as well as in Case 3 we observe that for all $1\leq l,\nu\leq d$,

$\displaystyle\left|\frac{j_{l}}{t_{l}}-\frac{z_{n}}{z_{m}}\right|<\frac{1}{z_{m}t_{l}}\implies|j_{l}t_{\nu}-j_{\nu}t_{l}|<\frac{t_{l}+t_{\nu}}{z_{m}}\leq\frac{4(rN)^{1/d}}{N^{1/d+\epsilon}}.$

As a conclusion we have for sufficiently large $N$ and for all $1\leq
l,\nu\leq d$, the equality ${j_{l}}/{t_{l}}={j_{\nu}}/{t_{\nu}}$. So in Case 2
and Case 3, the estimate for the sums over $m,n$ and
$\boldsymbol{j},\boldsymbol{t}$ in (5.1) is bounded above by

(5.2)
$\displaystyle\sum_{\begin{subarray}{c}\boldsymbol{j},\boldsymbol{t}\in\mathbb{N}^{d}\\\
2^{\boldsymbol{u}-1}\leq\boldsymbol{j},\boldsymbol{t}<2^{\boldsymbol{u}}\\\
j_{1}/t_{1}=\cdots=j_{d}/t_{d}\end{subarray}}\sum_{1\leq m,n\leq
M}\mathds{1}_{\left\|\boldsymbol{j}z_{m}-\boldsymbol{t}z_{n}\right\|_{\infty}<1}.$

Similar to the estimate of (3.9) in the proof of Theorem 1.3 we have to bound
(5.2) for Case 2 and Case 3 here.

For Case 3, let $u=\max(u_{1}\dots,u_{d})=u_{\nu}$ (say), set
$T=2^{u-1}N^{1+\epsilon/2}$ and recall $b_{k},a_{h}$ from (4.3) and (4.4) for
dimension one. Then (5.2) is bounded above by

(5.3) $\displaystyle\sum_{2^{u-1}\leq j_{\nu},t_{\nu}<2^{u}}\sum_{1\leq
m,n\leq
M}\mathds{1}_{\left|j_{\nu}z_{m}-t_{\nu}z_{n}\right|<1}\sum_{\begin{subarray}{c}\boldsymbol{j},\boldsymbol{t}\in\mathbb{N}^{d-1}\\\
2^{\boldsymbol{u}-1}\leq\boldsymbol{j},\boldsymbol{t}<2^{\boldsymbol{u}}\\\
j_{1}/t_{1}=\cdots=j_{d}/t_{d}\end{subarray}}1.$

The condition $j_{1}/t_{1}=\cdots=j_{d}/t_{d}$ gives estimate for the sum over
$\boldsymbol{j},\boldsymbol{t}\in\mathbb{N}^{d-1}$ is $\ll(N^{1/d})^{d-1}$ as
$2^{u}\ll N^{1/d}$. The remaining sums have been estimated in Lemma 3, Lemma 4
and Lemma 5 of [1], where they choose $2^{u}\ll N$. But here $2^{u}\ll
N^{1/d}$ and thus we use the function $K$ and $G$ defined in (4.6) and (4.9),
which depend on $d$, in their Lemma 4 and Lemma 5. Specifically, we use the
estimate from the first passage of page no. 499 in [1] to achieve the upper
bound estimate for (5.2) as

$\displaystyle\ll
N^{(1-1/d)}\frac{2^{u}}{T}\left(N^{4+1/d+\epsilon/4}+T^{105/89+\epsilon}N^{52/89}E(X_{N})^{76/89}\right)$
$\displaystyle\ll
N^{4-\epsilon/4}+N^{157/89+16/89d+2\epsilon}E(X_{N})^{76/89},$

since $T=2^{u-1}N^{1+\epsilon/2}\ll N^{1+1/d+\epsilon/2}.$

Now note that $2^{u-1}\leq j_{\nu},t_{\nu}<2^{u}$ and
$|j_{\nu}z_{m}-t_{\nu}z_{n}|<1$ together imply $z_{m}/z_{n}\in[1/4,4].$
Therefore, if $\min(z_{m},z_{n})\in[C,D)$ then $z_{m},z_{n}\in[C/4,4D].$

By the above observation, in Case 2, it is enough to consider
$z_{m},z_{n}\in[N^{\beta}/4,8N^{\beta})$ for some
$\beta\in[1/d+\epsilon,1+\epsilon]$. As we can cover the interval
$[N^{1/d+\epsilon},N^{1+\epsilon})$ by $\ll\log N$ many sub-intervals of the
form $[N^{\beta},2N^{\beta})$. Let $\beta$ be fixed and $T=2^{u-1}N^{\beta}.$
Then similar to (5.3), the estimate for (5.2) is given by

$\displaystyle\ll(\log{N})(N^{1/d})^{d-1}\sum_{2^{u-1}\leq
j_{\nu},t_{\nu}<2^{u}}\sum_{\begin{subarray}{c}1\leq m,n\leq M\\\
z_{m},z_{n}\in[N^{\beta}/4,8N^{\beta}]\end{subarray}}\mathds{1}_{\left|j_{\nu}z_{m}-t_{\nu}z_{n}\right|<1}.$

Then following case 2 in page no. 500 of [1] we get

$\displaystyle\sum_{2^{u-1}\leq
j_{\nu},t_{\nu}<2^{u}}\sum_{\begin{subarray}{c}1\leq m,n\leq M\\\
z_{m},z_{n}\in[N^{\beta}/4,8N^{\beta}]\end{subarray}}\mathds{1}_{\left|j_{\nu}z_{m}-t_{\nu}z_{n}\right|<1}\ll\frac{2^{u}}{T}\left(E(X_{N})N^{1/d+\beta+\epsilon/4}+E(X_{N})N^{13\beta/89}T^{105/89+\epsilon}\right).$

Since $T=2^{u-1}N^{\beta}\ll N^{1/d+\beta}$, in Case 2, the upper bound
estimate of (5.2) is

$\displaystyle\ll\log{N}N^{1-1/d-\beta}\left(E(X_{N})N^{1/d+\beta+\epsilon/4}+E(X_{N})N^{118\beta/89+105/89d+(1+\beta)\epsilon}\right)$
$\displaystyle\ll E(X_{N})N^{118/89+16/89d+3\epsilon}.$

Combining the estimates in all three cases, the variance i.e., the upper bound
of (5.1) is

$\displaystyle\ll
N^{-1+2/d+3\epsilon}+E(X_{N})N^{-238/89+16/89d+4\epsilon}+N^{-\epsilon/8}+N^{-199/89+16/89d+2\epsilon}E(X_{N})^{76/89}\ll
N^{-\epsilon^{\prime}},$

provided that the hypothesis on the additive energy bound is true. The rest of
the arguments follow a standard method of applying Chebyshev’s inequality and
Borel-cantelli lemma (see the proof of Theorem 1 in [3] and also [9, 17]).

## Acknowledgement

The authors would like to express sincere appreciation to Professor Christoph
Aistleitner for several insightful suggestions. Also, they thank Professor
Olivier Ramaré for some helpful suggestions.

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