#  Central limit theorems for lattice point counting on tessellated domains

Sourav Das  Department of Mathematics and Statistics, Indian Institute of
Technology Kanpur, Kanpur, 208016, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Following the approach of Bj$\ddot{\text{o}}$rklund and Gorodnik, we have
considered the discrepancy function for lattice point counting on domains that
can be nicely tessellated by the action of a diagonal semigroup. We have shown
that suitably normalized discrepancy functions for lattice point counting on
certain tessellated domains satisfy a non-degenerate central limit theorem.
Furthermore, we have also addressed the same problem for affine and congruence
lattice point counting, proving analogous non-degenerate central limit
theorems for them. The main ingredients of the proofs are the method of
cumulants and quantitative multiple mixing estimates.

###### Key words and phrases:

Lattice point counting, Central limit theorem, affine lattice, congruence
lattice

###### 2020 Mathematics Subject Classification:

11P61, 11J83, 60F05

##  1\. Introduction

One of the central problems in the geometry of numbers is the lattice point
counting problem in certain domains of the Euclidean space. Minkowski’s first
theorem is one of the fundamental results in the geometry of numbers, which
says that a centrally symmetric convex set $C$ of $\mathbb{R}^{k}$ with
$\operatorname{vol}(C)>2^{k}$ contains a non-zero integer point. This theorem
can be extended for a general lattice $g\mathbb{Z}^{k}$ for some
$g\in\operatorname{GL}_{k}(\mathbb{R}).$ Before proceeding further, let us set
a few notations. The space of all unimodular lattices of $\mathbb{R}^{k}$ is
denoted by $X_{k},$ which is naturally identified with
$\operatorname{SL}_{k}(\mathbb{R})/\operatorname{SL}_{k}(\mathbb{Z}),$ which
supports a natural $\operatorname{SL}_{k}(\mathbb{R})$-invariant measure
$\mu_{k}$ coming from the Haar measure on $\operatorname{SL}_{k}(\mathbb{R})$
such that $\mu_{k}(X_{k})=1.$ Throughout this article, given a finite set $B$,
we denote the number of elements of $B$ by $|B|$. In $1945,$ Siegel ([20])
proved a probabilistic result of the first of its kind in the geometry of
numbers. Suppose that $\Omega$ is a measurable subset of $\mathbb{R}^{k},$ not
containing zero. Then Siegel’s mean value theorem ([20]) states that if we
choose an unimodular lattice $\Lambda$ at random, on average $\Lambda$
contains $\operatorname{vol}(\Omega)$ many points of $\Omega.$ Along the same
lines, Roger ([17]) gives formulas for higher moments for the counting
function $|\Lambda\cap\Omega|.$ To date, Siegel’s mean value theorem and
Roger’s formulas have applications in numerous number theoretic and ergodic
problems, such as lattice point counting, Oppenheim-type problems related to
quadratic forms, effective ergodic theorems, and more.

There has been extensive study to understand the behavior of the error term
$|\Lambda\cap\Omega|-\operatorname{vol}(\Omega)$ of lattice point counting
problems for a varying, sufficiently nice family of domains. Let
$\\{\Omega_{T}\\}$ be an increasing family of finite volume Borel subsets of
$\mathbb{R}^{k}$ such that $\operatorname{vol}(\Omega_{T})\rightarrow\infty$
as $T\rightarrow\infty.$ In [18], Schmidt prove that for $\mu_{k}$ almost
every unimodular lattice $\Lambda$,

$|\Lambda\cap\Omega_{T}|=\operatorname{vol}(\Omega_{T})+O\left(\operatorname{vol}(\Omega_{T})^{1/2}\log^{2}(\operatorname{vol}(\Omega_{T}))\right).$
(1.1)

Note that Schmidt’s result was much more general than this; for simplicity, we
are considering it in this form. The key ingredient in proving (1.1) is
Roger’s second moment formula (in particular the discrepancy bound). One must
note that we can view Schmidt’s result as an analog of the Law of Large
Numbers. This motivates the question of whether other probabilistic limit
laws, such as the central limit theorem, the law of iterated logarithm, etc.,
also hold for the above kind of lattice point counting problems. In the paper
[9], Bj$\ddot{\text{o}}$rklund and Gorodinik studied the discrepancy functions
given by

$D_{T}(\Lambda)=|\Lambda\cap\Omega_{T}|-\operatorname{vol}(\Omega_{T}).$ (1.2)

and affirmatively answered the above question by proving that suitably
normalized discrepancy functions satisfy a non-degenerate central limit
theorem for domains defined by products of linear forms. Our current project
is devoted to proving certain central limit theorems for discrepancy functions
of unimodular lattices, affine unimodular lattices, and congruence unimodular
lattices for domains, which can be nicely tessellated by means of the action
of a diagonal subgroup. The domains we have considered appear naturally in
lattice point counting problems and Diophantine approximations, and have
applications in spiraling of Diophantine approximates (see [2, 5, 7] and
Remark 1.4). It is worth mentioning a few recent works in this context.
Recently K. Holm proved a central limit theorem for symplectic lattice point
counting in [16]. Using the method of cumulants and quantitative multiple
mixing estimate, Bj$\ddot{\text{o}}$rklund and Gorodnik ([7]) proved a central
limit theorem for Diophantine approximations. Prior to this, the results of
this paper for unweighted cases were proved by Dolgopyat, Fayad, and
Vinogradov in [10], although their approach was completely different from [7].
In a very recent paper [1], Aggarwal and Ghosh extended the results of [7] for
inhomogeneous Diophantine approximation and Diophantine approximation with
congruence conditions.

###  1.1. The space of affine unimodular lattices

We denote the space of affine unimodular lattices of $\mathbb{R}^{k}$ by
$X_{k,a}.$

$X_{k,a}\cong\operatorname{SL}_{k}(\mathbb{R})\ltimes\mathbb{R}^{k}/\operatorname{SL}_{k}(\mathbb{Z})\ltimes\mathbb{Z}^{k}.$

$\operatorname{SL}_{k}(\mathbb{R})\ltimes\mathbb{R}^{k}$ acts naturally on
$X_{k,a}$, which is given by

$((g,\mathbf{v}),\Lambda)\mapsto
g\Lambda+\mathbf{v},\,\,\,\,\text{for}\,\,(g,\mathbf{v})\in\operatorname{SL}_{k}(\mathbb{R})\ltimes\mathbb{R}^{k}\,\,\text{and}\,\,\Lambda\in
X_{k,a}.$

Note that $X_{k,a}$ supports a natural
$\operatorname{SL}_{k}(\mathbb{R})\ltimes\mathbb{R}^{k}$-invariant probability
measure $\mu_{k,a}$, coming from the left-invariant Haar measure on
$\operatorname{SL}_{k}(\mathbb{R})\ltimes\mathbb{R}^{k}.$

Let $G_{k}=\operatorname{SL}_{k}(\mathbb{R})$ and
$G_{k,a}=\operatorname{SL}_{k}(\mathbb{R})\ltimes\mathbb{R}^{k}$. Then
$G_{k,a}$ can be identified as a subgroup of $G_{k+1}$ in view of the map
$h:G_{k,a}\rightarrow G_{k+1}$ defined by

$(g,\mathbf{v})\mapsto\begin{bmatrix}g&\mathbf{v}\\\
0&1\end{bmatrix}\quad\text{for}\,\,(g,\mathbf{v})\in G_{k,a}.$ (1.3)

Clearly $h$ induces an injective map $\tilde{h}:X_{k,a}\rightarrow X_{k+1}.$
Furthermore there is surjection $\pi:G_{k,a}\rightarrow G_{k}$ given by

$\pi(g,\mathbf{v})=g\quad\text{for all}\,\,(g,\mathbf{v})\in G_{k,a}.$ (1.4)

$\pi$ also induces a surjection $\tilde{\pi}:X_{k,a}\rightarrow X_{k}.$ It is
easy to observe that $\tilde{\pi}$ is measure preserving.

###  1.2. The space of congruence lattices

Let $(\mathbf{v},N)\in\mathbb{Z}^{k}\times\mathbb{N}$ be such that
$\gcd(\mathbf{v},N)=1.$ Denote the space of all affine unimodular lattices of
$\mathbb{R}^{k}$ of the form $g(\mathbb{Z}^{k}+\frac{\mathbf{v}}{N})$ by
$X_{k,c}.$ We can identify $X_{k,c}$ with the homogeneous space
$\operatorname{SL}_{k}(\mathbb{R})/\Gamma_{\mathbf{v},N}$ via

$g(\mathbb{Z}^{k}+\frac{\mathbf{v}}{N})\mapsto g\Gamma_{\mathbf{v},N},$

where
$\Gamma_{\mathbf{v},N}=\\{g\in\operatorname{SL}_{k}(\mathbb{Z}):g\mathbf{v}\equiv\mathbf{v}\,\,(\text{mod
N})\\}.$ Since $\Gamma_{\mathbf{v},N}$ is a lattice in
$\operatorname{SL}_{k}(\mathbb{R}),$ there exists a
$\operatorname{SL}_{k}(\mathbb{R})$-invariant measure $\mu_{k,c}$ on $X_{k,c}$
coming from the Haar measure on $\operatorname{SL}_{k}(\mathbb{R})$ such that
$\mu_{k,c}(X_{k,c})=1.$

###  1.3. Main results

First, we note that (1.1) is also true for almost every affine and congruence
unimodular lattices of $\mathbb{R}^{k}.$ The proof of these results follows
the same streamline as of [18, Theorem 1] just replace the use of discrepancy
bounds for unimodular lattices with the analogous bounds for the affine and
congruence unimodular lattices. In this article, we are interested in studying
the discrepancy functions on $X_{k}$, $X_{k,a}$, and $X_{k,c}$ for the domains
of the following type.
Let $c_{1},\dots,c_{m}\in\mathbb{R}_{>0}$ and
$u_{1},\dots,u_{m}\in\mathbb{R}_{>0}$ be such that $u_{1}+\dots+u_{m}=n.$ Now
for $T>0$, we consider the domains of the form

$\Omega_{T}=\left\\{(\mathbf{x},\mathbf{y})\in\mathbb{R}^{m}\times\mathbb{R}^{n}:1\leq\|\mathbf{y}\|\leq
T,\,\,|x_{i}|\|\mathbf{y}\|^{u_{i}}<c_{i}\,\,\text{for}\,\,i=1,\dots,m\right\\},$
(1.5)

where $\mathbf{x}=(x_{1},\dots,x_{m}),$ $\|\cdot\|$ be a norm on
$\mathbb{R}^{n}$ and $|\cdot|$ is the usual absolute value on $\mathbb{R}.$
Let

$N_{\xi}(0,\sigma^{2}):=\frac{1}{\sqrt{2\pi\sigma}}\int_{-\infty}^{\xi}e^{\frac{-x^{2}}{2\sigma}}\,\,dx,$
(1.6)

denotes the normal distribution with mean $0$ and variance $\sigma^{2}$,
$\zeta(x)=\sum_{m=1}^{\infty}\frac{1}{m^{x}}$ denotes the Riemann zeta
function and

$\omega_{n}=\int_{0}^{2\pi}\int_{0}^{\pi}\dots\int_{0}^{\pi}S(\theta_{1},\dots,\theta_{n-1})\,\,d\theta_{1}\dots
d\theta_{n-1},$ (1.7)

where $r^{n-1}S(\theta_{1},\dots,\theta_{n-1})drd\theta_{1}\dots
d\theta_{n-1}$ is the spherical volume element on $\mathbb{R}^{n}.$

######  Theorem 1.1.

Let $\Omega_{T}$ be as in (1.5) and $m+n\geq 5$. Then $\exists$ a
$\sigma_{u}>0$ such that

$\mu_{m+n}\left\\{\Lambda\in
X_{m+n}:\frac{|\Lambda\cap\Omega_{T}|-\operatorname{vol}(\Omega_{T})}{\operatorname{vol}(\Omega_{T})^{1/2}}<\xi\right\\}\longrightarrow
N_{\xi}(0,\sigma_{u}^{2})\,\,\,\,\text{as}\,\,T\rightarrow\infty,$ (1.8)

where

$\sigma_{u}^{2}:=2\left(\frac{2\zeta(m+n-1)}{\zeta(m+n)}-1\right).$ (1.9)

######  Theorem 1.2.

Let $\Omega_{T}$ be as in (1.5) and $m+n\geq 5$. Then

$\mu_{m+n,a}\left\\{\Lambda\in
X_{m+n,a}:\frac{|\Lambda\cap\Omega_{T}|-\operatorname{vol}(\Omega_{T})}{\operatorname{vol}(\Omega_{T})^{1/2}}<\xi\right\\}\longrightarrow
N_{\xi}(0,1)\,\,\,\,\text{as}\,\,T\rightarrow\infty.$ (1.10)

######  Theorem 1.3.

Let $\Omega_{T}$ be as in (1.5) and $m+n\geq 5$. Then $\exists$ a
$\sigma_{c}>0$ such that

$\mu_{m+n,c}\left\\{\Lambda\in
X_{m+n,c}:\frac{|\Lambda\cap\Omega_{T}|-\operatorname{vol}(\Omega_{T})}{\operatorname{vol}(\Omega_{T})^{1/2}}<\xi\right\\}\longrightarrow
N_{\xi}(0,\sigma_{c}^{2})\,\,\,\,\text{as}\,\,T\rightarrow\infty,$ (1.11)

where

$\sigma_{c}^{2}:=\frac{2}{\zeta_{N}(m+n)}\left(1+\frac{2}{\zeta_{N}(m+n)}\sum_{s\in
C_{N}}\sum_{s_{2}\geq 1}\frac{s_{2}-1}{(Ns_{2}+s)^{m+n}}\right),$ (1.12)
$C_{N}=\\{s\in\mathbb{Z}:0\leq s<N\,\,\text{and}\,\,\gcd(s,N)=1\\},$

and

$\zeta_{N}(x)=\sum_{\tiny\begin{array}[]{c}s_{1}\geq 1\\\
\gcd(s_{1},N)=1\end{array}}s^{-x}.$

######  Remark 1.4.

  1. (1)

With minor modifications, we can easily prove versions of Theorem 1.1, 1.2 and
1.3 for domains of the form $\Omega^{\prime}_{T}$. Here, for $T,b>0$ and a
Borel measurable subset $A\subseteq\mathbb{S}^{m-1},$ we define
$\Omega^{\prime}_{T}$ as:

$\Omega^{\prime}_{T}=\left\\{(\mathbf{x},y)\in\mathbb{R}^{m}\times\mathbb{R}:\frac{\mathbf{x}}{\|\mathbf{x}\|}\in
A,1\leq|y|\leq
T,\,\,\|\mathbf{x}\|^{m}|y|<b\,\,\text{for}\,\,i=1,\dots,m\right\\}.$ (1.13)

Note that domains of this form naturally appear in problems of Diophantine
approximation. More specifically, these domains are closely related to the
spiraling of approximations. For more details see the interesting paper by
Athreya, Ghosh, and Tseng [5]. This theme was further explored over number
fields by Alam and Ghosh in [2]. It is worth mentioning that the study of
spiraling of Diophantine approximants in [5] is related to counting lattice
points in the domains of the form $\Omega^{\prime}_{T}.$ The crucial thing is
to notice that the domains $\Omega^{\prime}_{T}$ admit an approximate
tessellation by means of an action by a diagonal element of
$\operatorname{SL}_{m+1}(\mathbb{R})$ (and for $T=2^{M}$, $M\in\mathbb{N}$,
$\Omega^{\prime}_{T}$ possesses an exact tessellation). Thus, one can apply
the methods of this article to easily derive analogous central limit theorems
for domains $\Omega^{\prime}_{T}.$ This can be viewed as a central limit
theorem for spiraling.

  2. (2)

We give a detailed proof of Theorem 1.2. For the proof of Theorem 1.1 and 1.3
we omit some details, as they are analogous to the proof of Theorem 1.2. We
indicate the necessary changes required in the proofs of Theorem 1.1 and 1.3.

###  1.4. Outline of the proof of the above Theorems

To prove the above theorems, we closely follow the strategy of
Bj$\ddot{\text{o}}$rklund and Gorodnik ([7]). We provide a brief outline of
the proof for Theorem 1.2, noting that the proofs of the other two theorems
follow a similar strategy. By an easy reduction, we can show that it is enough
to prove Theorem 1.2 for $T$ of the form $2^{M},$ with $M\in\mathbb{N}$. If
$T=2^{M},$ $\Omega_{T}$ can be nicely tessellated by means of the action of a
diagonal subgroup of $SL_{m+n}(\mathbb{R}).$ To handle the case of $T=2^{M},$
we use the method of cumulants and a powerful theorem of Fréchet and Shohat
(Theorem 2.8), which gives a criterion under which a sequence of bounded
measurable functions on a probability space converge in distribution to the
normal distribution. However, we cannot directly apply Theorem 2.8 to the
functions

$\Lambda\longmapsto\frac{|\Lambda\cap\Omega_{T}|-\operatorname{vol}(\Omega_{T})}{\operatorname{vol}(\Omega_{T})^{1/2}},\,\,\text{where}\,\,T=2^{M}\,\,\text{with}\,\,M\in\mathbb{N},$

because these functions are typically unbounded. To address this issue, we
exploit the tessellation property of our domain to observe that, for
$T=2^{M}$, the conclusion of Theorem 1.2 can be reformulated (see Theorem 2.6)
in terms of
$\operatorname{SL}_{m+n}(\mathbb{R})\ltimes\mathbb{R}^{m+n}$-translates of the
Siegel transform $\hat{\chi}$ of the indicator function $\chi$ of
$\Omega_{2}$. Furthermore, we can approximate $\hat{\chi}$ by a family of
smooth and bounded $C_{c}^{\infty}$ functions on the space of affine
unimodular lattices of $\mathbb{R}^{m+n}$. We show that it is sufficient to
prove the central limit theorem for
$\operatorname{SL}_{m+n}(\mathbb{R})\ltimes\mathbb{R}^{m+n}$-translates of the
smooth approximation to $\hat{\chi}$ (see Proposition 2.17). We then verify
the CLT criteria of Fréchet and Shohat for these bounded functions. This
amounts to showing that the variance is finite and that the cumulant of order
$r$ ($r\geq 3$) for these functions tends to $0$ as $M\rightarrow\infty.$ In
the computation of cumulants, the main tools are the quantitative multiple
mixing estimate of [6, Theorem 1.1] and a combinatorial tool developed by
Bj$\ddot{\text{o}}$rklund and Gorodnik in [8, Proposition 6.2 ] to analyze the
cumulants.

Aknowledgement: The author is grateful to his advisor, Prof. Arijit Ganguly,
for helpful discussions and continuous encouragement. Part of this work was
done when the author was visiting the University of Z$\ddot{\text{u}}$rich to
attend the Z$\ddot{\text{u}}$rich Dynamics School and the
Z$\ddot{\text{u}}$rich dynamics conference in June $2023$. The author thanks
the University of Z$\ddot{\text{u}}$rich for their generous hospitality.
Thanks are also due to Prof. Micheal Bj$\ddot{\text{o}}$rklund and Prof.
Alexander Gorodnik for helpful discussions during the Z$\ddot{\text{u}}$rich
dynamics conference.

##  2\. Proof of the CLT for affine unimodular lattices

To begin with, we show that it is enough to prove Theorem 1.2 for the case
$T=2^{M}$ with $M\in\mathbb{N}$.

######  Proposition 2.1.

Suppose that Theorem 1.2 holds true for $T$ of the form $2^{M}$ with
$M\in\mathbb{N}$. Then Theorem 1.2 holds true in its full generality.

###### Proof.

First, let us compute the volume of $\Omega_{T}.$ Let $d\mathbf{x}$ and
$d\mathbf{y}$ be the usual Lebesgue measure on $\mathbb{R}^{m}$ and
$\mathbb{R}^{n}$ respectively.

$\displaystyle\operatorname{vol}(\Omega_{T})$ $\displaystyle=$
$\displaystyle{\int\int}_{\Omega_{T}}d\mathbf{x}\,\,d\mathbf{y}$ (2.1)
$\displaystyle=$ $\displaystyle\int_{1\leq\|\mathbf{y}\|\leq
T}\left(\int_{-\frac{c_{1}}{{\|\mathbf{y}\|}^{u_{1}}}}^{\frac{c_{1}}{{\|\mathbf{y}\|}^{u_{1}}}}\dots\int_{-\frac{c_{m}}{{\|\mathbf{y}\|}^{u_{m}}}}^{\frac{c_{m}}{{\|\mathbf{y}\|}^{u_{m}}}}dx_{1}\dots
dx_{m}\right)d\mathbf{y}$ $\displaystyle=$
$\displaystyle\int_{1\leq\|\mathbf{y}\|\leq T}\frac{2^{m}c_{1}\dots
c_{m}}{\|\mathbf{y}\|^{n}}d\mathbf{y}$

Now we change the Cartesian coordinates $\mathbf{y}=(y_{1},\dots,y_{n})$ of
$\mathbb{R}^{n}$ to spherical coordinates $(r,\theta_{1},\dots,\theta_{n-1}).$
Let $r^{n-1}S(\theta_{1},\dots,\theta_{n-1})drd\theta_{1}\dots d\theta_{n-1}$
be the spherical volume element on $\mathbb{R}^{n}$ and

$\omega_{n}=\int_{0}^{2\pi}\int_{0}^{\pi}\dots\int_{0}^{\pi}S(\theta_{1},\dots,\theta_{n-1})\,\,d\theta_{1}\dots
d\theta_{n-1}.$ (2.2)

Then in view of the above coordinate change, we have

$\displaystyle\operatorname{vol}(\Omega_{T})$ $\displaystyle=$ $\displaystyle
2^{m}c_{1}\dots c_{m}\omega_{n}\int_{1}^{T}\frac{1}{r^{n}}r^{n-1}\,\,dr$ (2.3)
$\displaystyle=$ $\displaystyle 2^{m}c_{1}\dots c_{m}\omega_{n}\log T.$

Given any real parameter $T>0$ there exists $M=M(T)\in\mathbb{Z}_{\geq 0}$
such that $\frac{T}{2}<2^{M}\leq T\leq 2^{M+1}.$ Now for any affine lattice
$\Lambda\in X_{m+n,a},$ we have

$\displaystyle\frac{|\Lambda\cap\Omega_{T}|-\operatorname{vol}(\Omega_{T})}{\operatorname{vol}(\Omega_{T})^{1/2}}$
$\displaystyle=$
$\displaystyle\frac{|\Lambda\cap\Omega_{2^{M}}|-\operatorname{vol}(\Omega_{2^{M}})}{\operatorname{vol}(\Omega_{T})^{1/2}}+\frac{|\Lambda\cap(\Omega_{T}\setminus\Omega_{2^{M}})|}{\operatorname{vol}(\Omega_{T})^{1/2}}-\frac{\operatorname{vol}(\Omega_{T}\setminus\Omega_{2^{M}})}{\operatorname{vol}(\Omega_{T})^{1/2}}$
$\displaystyle=$ $\displaystyle a_{T}X_{T}+Y_{T}-Z_{T},$

where

$a_{T}=\left(\frac{\operatorname{vol}(\Omega_{2^{M}})}{\operatorname{vol}(\Omega_{T})}\right)^{1/2},X_{T}=\frac{|\Lambda\cap\Omega_{2^{M}}|-\operatorname{vol}(\Omega_{2^{M}})}{\operatorname{vol}(\Omega_{2^{M}})^{1/2}},Y_{T}=\frac{|\Lambda\cap(\Omega_{T}\setminus\Omega_{2^{M}})|}{\operatorname{vol}(\Omega_{T})^{1/2}},Z_{T}=\frac{\operatorname{vol}(\Omega_{T}\setminus\Omega_{2^{M}})}{\operatorname{vol}(\Omega_{T})^{1/2}}.$
(2.5)

By assumption $X_{T}$ converges in distribution to $N(0,\sigma^{2})$ as
$T\rightarrow\infty.$ Hence in order to show (1.10), it is enough to show that
$a_{T}\rightarrow 1,$ $Y_{T}\rightarrow 0$ and $Z_{T}\rightarrow 0$ as
$T\rightarrow\infty.$

Note that $a_{T}=\left(\frac{M\log 2}{\log T}\right)^{1/2}$ and also we have

$\begin{array}[]{lcl}\displaystyle\quad\quad\frac{T}{2}<2^{M}\leq T\\\
\displaystyle\implies\log T-\log 2<M\log 2\leq\log T\\\
\displaystyle\implies\left(1-\frac{\log 2}{\log T}\right)^{1/2}<a_{T}\leq 1\\\
\displaystyle\implies a_{T}\rightarrow
1\,\,\text{as}\,\,T\rightarrow\infty.\end{array}$

$\displaystyle Z_{T}$ $\displaystyle=$ $\displaystyle\gamma_{m,n}\frac{\log
T-\log 2^{M}}{(\log T)^{1/2}}=\gamma_{m,n}\frac{\log\frac{T}{2^{M}}}{(\log
T)^{1/2}}\leq\gamma_{m,n}\frac{\log 4}{(\log T)^{1/2}}\rightarrow
0\,\,\text{as}\,\,T\rightarrow\infty,$ (2.6)

where $\gamma_{m,n}$ is a constant depending upon $m$ and $n.$

Since $\operatorname{vol}(\Omega_{T}\setminus\Omega_{2^{M}})\leq\log 2$ for
all $T,$ we must have $Y_{T}\rightarrow 0.$ Hence we are done.

∎

Siegel transform is one the main tools in the study of lattice point counting
problems. Throughout this section we denote the set of all affine unimodular
lattices on $\mathbb{R}^{m+n}$ by $X_{a}$, $\mu_{a}$ denotes the measure
$\mu_{m+n,a}$ defined in the last section, and $l:=m+n$. Let
$f:\mathbb{R}^{l}\rightarrow\mathbb{R}$ be a Borel measurable function with
compact support. The Siegel transform of $f$ is
$\hat{f}:X_{a}\rightarrow\mathbb{R}$ defined by

$\hat{f}(\Lambda)=\sum_{\lambda\in\Lambda\setminus\\{0\\}}f(\lambda)\quad\text{for}\,\,\Lambda\in
X_{a}.$ (2.7)

Let us record the analog of Siegel’s mean value theorem and Roger’s second
order moment formula for affine lattices. We refer to [4], [11], and [14] for
the proof of the following proposition.

######  Proposition 2.2.

For $f\in L^{1}(\mathbb{R}^{l})\cap L^{2}(\mathbb{R}^{l})$, we have the
following:

$\int_{X_{a}}\hat{f}(\Lambda)\,\,d\mu_{a}(\Lambda)=\int_{\mathbb{R}^{l}}f(\mathbf{x})\,\,d\mathbf{x}$
(2.8)
$\int_{X_{a}}{\hat{f}(\Lambda)}^{2}\,\,d\mu_{a}(\Lambda)=\left(\int_{\mathbb{R}^{l}}f(\mathbf{x})\,\,d\mathbf{x}\right)^{2}+\int_{\mathbb{R}^{l}}{f(\mathbf{x})}^{2}\,\,d\mathbf{x},$
(2.9)

where $d\mathbf{x}$ is the usual Lebesgue measure on $\mathbb{R}^{l}$

Siegel transform of bounded functions are typically unbounded and the
following function has a close relationship with the growth of the Siegel
transform.

######  Definition 2.3.

Let $\Lambda$ be any lattice of $\mathbb{R}^{l}$ and covolume of $\Lambda$ is
$d(\Lambda)<\infty.$ We define

$\alpha_{l}(\Lambda):=\sup_{V}\left\\{d(V\cap\Lambda)^{-1}:V\cap\Lambda\,\,\text{is
a lattice in}\,\,V\right\\},$ (2.10)

where the supremum is taken over all non-zero subspaces $V$ of
$\mathbb{R}^{l}.$

An immediate consequence of Mahler’s compactness criterion shows that
$\alpha_{l}$ is a proper map on $X_{l}.$ We define
$\alpha_{l,a}:X_{a}\rightarrow\mathbb{R}$ by

$\alpha_{l,a}(\Lambda)=\alpha_{l+1}(\tilde{h}(\Lambda))=\alpha_{l}(\tilde{\pi}(\Lambda)),$
(2.11)

where $\alpha_{l}$ is given by (2.10) and $\tilde{h}$, $\tilde{\pi}$ are
defined in Subsection 1.1. A routine verification shows that the above
definition indeed makes sense. Let us recall the following propositions, which
will be useful later.

######  Proposition 2.4.

[1, Proposition 3.14] Let $g:\mathbb{R}^{l}\rightarrow\mathbb{R}$ be a bounded
function with compact support. Then

$|\hat{g}(\Lambda)|\ll_{\operatorname{supp}(g)}\|g\|_{\infty}\cdot\alpha_{l,a}(\Lambda)\quad\forall\Lambda\in
X_{a},$

where $\hat{g}$ denotes the Siegel transform of $g$ on the space of all affine
unimodular lattices of $\mathbb{R}^{l}.$

######  Proposition 2.5.

[1, Proposition 3.17] Let $\alpha_{l,a}$ be the function defined on $X_{a}$ by
(2.10). Then $\alpha_{l,a}\in L^{p}(X_{a})$ for $1\leq p<l$ and

$\mu_{a}(\\{\alpha_{l,a}\geq L\\})\ll_{p}L^{-p}\quad\forall\,\,p<l.$ (2.12)

Now we reformulate our problem using the Siegel transform. Let
$\chi_{\Omega_{T}}$ denotes the indicator function of $\Omega_{T}.$ For
$T=2^{M}$ and $\Lambda\in X_{a}$, we have

$|\Lambda\cap\Omega_{2^{M}}|=\hat{\chi}_{\Omega_{2^{M}}}(\Lambda)\quad\text{and
}\quad\operatorname{vol}(\Omega_{2^{M}})=\int_{X_{a}}\hat{\chi}_{\Omega_{2^{M}}}(\Lambda)\,\,d\mu_{a}(\Lambda)$
(2.13)

Note that our domain $\Omega_{2^{M}}$ can be tessellated by the action of a
diagonal element of $\operatorname{SL}_{l}(\mathbb{R}).$ Let

${c}_{0}=\operatorname{diag}(2^{u_{1}},\dots,2^{u_{m}},2^{-1},\dots,2^{-1})$
(2.14)

and

$c=(\operatorname{diag}(2^{u_{1}},\dots,2^{u_{m}},2^{-1},\dots,2^{-1}),0)\in\operatorname{SL}_{l}(\mathbb{R})\ltimes\mathbb{R}^{n}.$
(2.15)

Then

$\Omega_{2^{M}}=\bigsqcup_{k=0}^{M-1}{c}_{0}^{-k}\Omega_{2}.$ (2.16)

Hence

$|\Lambda\cap\Omega_{2^{M}}|=\sum_{k=0}^{M-1}\hat{\chi}_{\Omega_{2}}(c^{k}\Lambda)$
(2.17)

and

$\operatorname{vol}(\Omega_{2^{M}})=\int_{X_{a}}\sum_{k=0}^{M-1}\hat{\chi}_{\Omega_{2}}(c^{k}\Lambda)\,\,d\mu_{a}(\Lambda)=\sum_{k=0}^{M-1}\operatorname{vol}(\Omega_{2})=M\cdot\operatorname{vol}(\Omega_{2})$
(2.18)

From now on we will denote $\chi_{\Omega_{2}}$ by $\chi$. From the above
discussion, it is clear that the following Theorem is equivalent to Theorem
1.2.

######  Theorem 2.6.

Under the hypothesis of Theorem 1.2,

$H_{M}:=\frac{1}{\sqrt{M}}\left(\sum_{k=0}^{M-1}\hat{\chi}\circ
c^{k}-M\cdot\operatorname{vol}(\Omega_{2})\right)\longrightarrow
N(0,\eta^{2})\,\,\text{in distribution as}\,\,M\rightarrow\infty,$ (2.19)

where $\eta=\operatorname{vol}(\Omega_{2})^{1/2}$ and $N(0,\eta^{2})$ denotes
the normal distribution with mean $0$ and variance $\eta^{2}.$

We prove Theorem 2.6 using the CLT criteria of Fréchet and Shohat. Let us
first recall the notion of a _cumulant_.

######  Definition 2.7.

Let $(Y,\nu)$ be a probability space. For $\psi_{1},\dots,\psi_{r}\in
L^{\infty}(Y),$ their _joint cumulant_ of order $r$ is defined by

$\text{Cum}_{[r]}(\psi_{1},\dots,\psi_{r})=\sum_{\mathcal{P}}(-1)^{|\mathcal{P}|-1}\left(|\mathcal{P}|-1\right)!\prod_{I\in\mathcal{P}}\int_{Y}\prod_{i\in
I}\psi_{i}\,\,d\nu,$

where $\mathcal{P}$ is the set of all partition of the set $\\{1,\dots,r\\}.$

For a partition $\mathcal{Q}$ of $\\{1,\dots,r\\}$, the _conditional joint
cumulant_ of $\psi_{1},\dots,\psi_{r}\in L^{\infty}(Y)$ with respect to
$\mathcal{Q}$ is defined as

$\text{Cum}_{[r]}(\psi_{1},\dots,\psi_{r}|\mathcal{Q})=\sum_{\mathcal{P}}(-1)^{|\mathcal{P}|-1}\left(|\mathcal{P}|-1\right)!\prod_{I\in\mathcal{P}}\prod_{J\in\mathcal{Q}}\int_{Y}\prod_{i\in
I\cap J}\psi_{i}\,\,d\nu.$

For $\psi\in L^{\infty}(Y),$ we define

$\text{Cum}_{[r]}(\psi):=\text{Cum}_{[r]}(\psi,\dots,\psi).$

Note that $\text{Cum}_{[r]}$ is multi linear in the functions
$\psi_{1},\dots,\psi_{r}.$ We now state the CLT criteria of Fréchet and Shohat
from [13].

######  Theorem 2.8.

Let $(Y,\nu)$ be a probability space and $\\{\Psi_{M}\\}$ be a sequence of
real-valued measurable bounded functions on $Y$ satisfying

$\int_{Y}\Psi_{M}\,\,d\nu=0\quad\text{and}\quad\sigma^{2}:=\lim_{M\rightarrow\infty}\int_{Y}\Psi_{M}^{2}<\infty$
(2.20)

and

$\lim_{M\rightarrow\infty}\text{Cum}_{[r]}(\Psi_{M})=0\quad\text{for all
}\,\,r\geq 3.$ (2.21)

Then for every $\xi\in\mathbb{R}$

$\nu(\\{\Psi_{M}<\xi\\})\rightarrow{N}_{\xi}(0,\sigma^{2})\quad\text{as}\quad
M\rightarrow\infty.$

######  Proposition 2.9.

Let $\psi_{1},\dots,\psi_{r}\in L^{\infty}(Y,\mu)$ and $\mathcal{Q}$ be any
partition of $\\{1,\dots,r\\}$ with $|\mathcal{Q}|\geq 2.$ Then we have

$\text{Cum}_{[r]}(\psi_{1},\dots,\psi_{r}|\mathcal{Q})=0.$

Before proceeding further, we recall the notion of operator norm on
$G_{a}:=\operatorname{SL}_{l}(\mathbb{R})\ltimes\mathbb{R}^{l}$ and the
Sobolev norms on the space $C_{c}^{\infty}(X_{a}).$ $G_{a}$ is a connected
semisimple Lie group with trivial centre. $G_{a}$ carries a right-invariant
Riemannian metric which decends to the quotient $G_{a}/\Gamma_{a},$ where
$\Gamma_{a}:=\operatorname{SL}_{l}(\mathbb{Z})\ltimes\mathbb{Z}^{l}.$ Let
$\rho_{G_{a}}$ be the right-invariant distance function on $G_{a}$ induced by
the right-invariant Riemannian metric and $\rho_{X_{a}}$ be the corresponding
distance function on $G_{a}/\Gamma_{a}.$

######  Definition 2.10.

Let $\|\cdot\|$ be the norm on the Lie algebra
$\mathfrak{g}=\operatorname{Lie}(G_{a})$ coming from the Riemannian metric on
$G_{a}.$ We define the submultiplicative operator norm $\|\cdot\|_{\text{op}}$
on $G_{a}$ by

$\|g\|_{\text{op}}:=\max\\{\|\operatorname{Ad}(g)Y\|:Y\in\mathfrak{g},\,\,\|Y\|=1\\}\quad\forall
g\in G_{a}.$ (2.22)

In other words, the operator norm of $g$ is just the operator norm of the
adjoint map $\operatorname{Ad}(g):\mathfrak{g}\rightarrow\mathfrak{g}.$ Now we
record the following lemma which will be useful later. The proof of the
following lemma can be found in [6, Lemma 2.1].

######  Lemma 2.11.

  1. (i)

For all $g\in G_{a},$ $\exists$ constants $c_{1}\geq 1$ and $c_{2}>0$ such
that

$c_{1}^{-1}\log\|g\|_{\text{op}}-c_{2}\leq\rho_{G_{a}}(g,e)\leq
c_{1}\log\|g\|_{\text{op}}+c_{2},$ (2.23)

where $e$ is the identity element of $G_{a}.$

  2. (ii)

Let
$c=(\operatorname{diag}(2^{u_{1}},\dots,2^{u_{m}},2^{-1},\dots,2^{-1}),0)\in
G_{a}.$ Then $\exists$ $\lambda>1$ such that for any $q\in\mathbb{N},$

$\|c^{q}\|_{\text{op}}\geq\lambda^{q}.$

Every $Z\in\operatorname{Lie}(G_{a})$ gives rise to a first order differential
operator $\mathcal{D}_{Z}$ on $C_{c}^{\infty}(X_{a})$ defined by

$\mathcal{D}_{Z}(\psi)(x):=\lim_{t\rightarrow
0}\frac{\psi(\exp(tZ)x)-\psi(x)}{t}\quad\forall\psi\in C_{c}^{\infty}(X_{a}).$

If $\\{Z_{1},\dots,Z_{r}\\}$ be a fix ordered basis of
$\operatorname{Lie}(G_{a}),$ then every element of the universal enveloping
algebra $\mathcal{U}(\operatorname{Lie}(G_{a}))$ of
$\operatorname{Lie}(G_{a})$ can be written as a linear combination of
monomials $Z_{1}^{n_{1}}\dots Z_{r}^{n_{r}}$ in the basis elements. Hence we
extend the differential operator to $\mathcal{U}(\operatorname{Lie}(G_{a}))$
by defining it for monomials $Y=Z_{1}^{n_{1}}\dots Z_{r}^{n_{r}}$ as

$\mathcal{D}_{Y}=\mathcal{D}_{Z_{1}}^{n_{1}}\dots\mathcal{D}_{Z_{r}}^{n_{r}}.$

The degree of $\mathcal{D}_{Y}$ is defined as $\deg(Y):=n_{1}+\dots+n_{r}.$
Now we define a family of Sobolev norms on $C_{c}^{\infty}(X_{a}).$

######  Definition 2.12.

Let $d\in\mathbb{N}$ and $\psi\in C_{c}^{\infty}(X_{a}).$ We define the
following

$\|\psi\|_{C^{d}}:=\max\\{\|\mathcal{D}_{Y}\psi\|_{\infty}:Y\in\mathcal{U}(\operatorname{Lie}(G_{a}))\,\,\text{is
a monomial,}\,\,\deg(Y)\leq d\\},$
$S_{k}(\psi):=\max\\{\|\psi\|_{\infty},\|\psi\|_{C^{d}}\\}.$

######  Lemma 2.13.

The family of Sobolev norms on $C_{c}^{\infty}(X_{a})$ satisfies the following
properties.

  1. (1)

For any $\psi\in C_{c}^{\infty}(X_{a})$ and $g\in G_{a}$,

$S_{k}(\psi\circ g)\ll_{k}\|g\|_{\text{op}}^{k}S_{k}(\psi).$

  2. (2)

For any $\psi_{1},\psi_{2}\in C_{c}^{\infty}(X_{a}),$

$S_{k}(\psi_{1}\psi_{2})\ll_{k}S_{k}(\psi_{1})S_{k}(\psi_{2}).$

###### Proof.

The proof of these properties of Sobolev norms is very standard. Hence we
leave the details to interested readers. ∎

Next, we recall the quantitative exponential mixing of all orders for Lie
groups from [6], which will be very instrumental in the computation of
cumulants later. The following is a particular case of [6, Theorem 1.1].

######  Theorem 2.14.

For all integer $r\geq 2$ and all sufficiently large $q,$ there exists
$\delta=\delta(r,q)>0$ such that for all $\psi_{1},\dots,\psi_{r}\in
C_{c}^{\infty}(X_{a})$ and $g_{1},\dots,g_{r}\in G_{a}$

$\left|\int_{X_{a}}(\psi_{1}\circ g_{1})\dots(\psi_{r}\circ
g_{r})\,\,d\mu_{a}-\prod_{i=1}^{r}\left(\int_{X_{a}}\psi_{i}\,\,d\mu_{a}\right)\right|\ll_{r,q}e^{-\delta\min_{i\neq
j}\rho_{G_{a}}(g_{i},g_{j})}S_{q}(\psi_{1})\dots S_{q}(\psi_{r}).$ (2.24)

To apply Theorem 2.14 in the computation of cumulants, we need to approximate
$\hat{\chi}$ by smooth functions. For that first we approximate $\chi$
(following [7, Section 6])by a family of non-negative smooth functions
$f_{\varepsilon}\in C_{c}^{\infty}(\mathbb{R}^{m+n})$ such that supp
$f_{\varepsilon}\subseteq(\Omega_{2})_{\varepsilon}$, and

$\chi\leq f_{\varepsilon}\leq
1,\,\,\|f_{\varepsilon}\|_{C^{k}}\ll\varepsilon^{-k},\,\,\|\chi-
f_{\varepsilon}\|_{1}\ll\varepsilon,\,\,\|\chi-
f_{\varepsilon}\|_{2}\ll\sqrt{\varepsilon},$ (2.25)

where $(\Omega_{2})_{\varepsilon}$ is an $\varepsilon$-neighbourhood of the
set $\Omega_{2}.$

For simplicity from now on by $\alpha$ we denote the function $\alpha_{a,l}$
on $X_{a}$ defined by (2.11).

######  Lemma 2.15.

For every $b>0$, there is a family of smooth functions $\\{\eta_{L}\\}\in
C_{c}^{\infty}(X_{a})$ such that, for every $L>0$

$0\leq\eta_{L}\leq 1,\,\,\eta_{L}=1\,\,\text{on}\,\,\\{\alpha\leq
b^{-1}L\\},\,\,\eta_{L}=0\,\,\text{on}\,\,\\{\alpha>bL\\},\,\,\|\eta_{L}\|_{C^{k}}\ll
1.$ (2.26)

###### Proof.

The proof of this Lemma is analogous to [7, Lemma 4.11]. ∎

The truncated Siegel transform of a bounded function
$f:\mathbb{R}^{m+n}\rightarrow\mathbb{R}$ with compact support is defined as

$\hat{f}^{(L)}:=\hat{f}\cdot\eta_{L}.$

######  Lemma 2.16.

If $f\in C^{\infty}_{c}(X_{a}),$ then the truncated Siegel transform
$\hat{f}^{(L)}\in C^{\infty}_{c}(X_{a})$ and also we have

$\left\|\hat{f}^{(L)}\right\|_{\infty}\ll_{\operatorname{supp}(f)}L\|f\|_{\infty},$
(2.27)
$\left\|\hat{f}^{(L)}\right\|_{C^{k}}\ll_{\operatorname{supp}(f)}L\|f\|_{C^{k}},$
(2.28)
$\left\|\hat{f}-\hat{f}^{(L)}\right\|_{1}\ll_{\operatorname{supp}(f),p}L^{-p}\|f\|_{\infty}\quad\forall\,\,p<m+n-1,$
(2.29)
$\left\|\hat{f}-\hat{f}^{(L)}\right\|_{2}\ll_{\operatorname{supp}(f),p}L^{-(p-1)/2}\|f\|_{\infty}\quad\forall\,\,p<m+n-1$
(2.30)
$S_{k}\left(\hat{f_{\varepsilon}}^{(L)}\right)\ll_{k}\varepsilon^{-k}L,$
(2.31)

where $f_{\varepsilon}$ is given by (2.25).

###### Proof.

The proof of the first four inequalities follows from [7, Lemma 4.12]. For the
last inequality, consider

$\displaystyle S_{k}\left(\hat{f_{\varepsilon}}^{(L)}\right)$ $\displaystyle=$
$\displaystyle\max\left\\{\left\|\hat{f_{\varepsilon}}^{(L)}\right\|_{\infty},\left\|\hat{f_{\varepsilon}}^{(L)}\right\|_{C^{k}}\right\\}$
(2.32) $\displaystyle\ll_{\operatorname{supp}(f_{\varepsilon})}$
$\displaystyle\max\left\\{L\|f_{\varepsilon}\|_{\infty},L\|f_{\varepsilon}\|_{C^{k}}\right\\}$
(2.33) $\displaystyle\ll$
$\displaystyle\max\\{L,L\varepsilon^{-k}\\},\,\,\text{using}\,\,\eqref{equ:chi_f_epsilon}$
(2.34) $\displaystyle=$ $\displaystyle\varepsilon^{-k}L.$ (2.35)

∎

Now we are going to approximate $\hat{\chi}$ by means of truncated Siegel
transforms of $f_{\varepsilon},$ where $f_{\varepsilon}$ are given by (2.25).
Next we approximate $H_{M}$ by $H_{M}^{(\varepsilon,L)}$ such that
$\|H_{M}-H_{M}^{(\varepsilon,L)}\|_{1}\rightarrow 0.$ Then $H_{M}$ and
$H_{M}^{(\varepsilon,L)}$ converge to the same limit in distribution. Due to
the approximation, we get two sequences $\varepsilon(M)$ and $L(M)$ which
depends on $M.$ At the end we give explicit choices of these sequences.

######  Proposition 2.17.

Define

$H_{M}^{(\varepsilon,L)}:=\frac{1}{\sqrt{M}}\left(\sum_{k=0}^{M-1}\hat{f_{\varepsilon}}^{(L)}\circ
c^{k}-M\int_{X_{a}}\hat{f_{\varepsilon}}^{(L)}\,\,d\mu_{a}\right),$ (2.36)

where $\hat{f_{\varepsilon}}^{(L)}=\hat{f_{\varepsilon}}\eta_{L}\in
C_{c}^{\infty}(X_{a})$ are smooth approximations to $\hat{\chi}.$ Then to
prove Theorem 2.6 it is enough to show that

$H_{M}^{(\varepsilon,L)}\rightarrow N(0,\eta^{2})\,\,\text{in distribution
as}\,\,M\rightarrow\infty,$

where $\eta=\operatorname{vol}(\Omega_{2})^{1/2}$.

###### Proof.

First note that to prove this proposition it is enough to show that

$\|H_{M}^{(\varepsilon,L)}-H_{M}\|_{1}\rightarrow
0\quad\text{as}\,\,M\rightarrow\infty.$
$\displaystyle\|H_{M}^{(\varepsilon,L)}-H_{M}\|_{1}$ $\displaystyle=$
$\displaystyle\left\|\frac{1}{\sqrt{M}}\left(\sum_{k=0}^{M-1}\left(\hat{f_{\varepsilon}}^{(L)}-\hat{\chi}\right)\circ
c^{k}-M\int_{X_{a}}\left(\hat{f_{\varepsilon}}^{(L)}-\hat{\chi}\right)\,\,d\mu_{a}\right)\right\|_{1}$
(2.37) $\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{M}}\sum_{k=0}^{M-1}\left\|\left(\hat{f_{\varepsilon}}^{(L)}-\hat{\chi}\right)\circ
c^{k}\right\|_{1}+\sqrt{M}\left\|\hat{f_{\varepsilon}}^{(L)}-\hat{\chi}\right\|_{1}$
$\displaystyle=$ $\displaystyle
2\sqrt{M}\left\|\hat{f_{\varepsilon}}^{(L)}-\hat{\chi}\right\|_{1},\,\,\text{by
using}\,\,G_{a}\text{-invariance of}\,\,\mu$ $\displaystyle\leq$
$\displaystyle
2\sqrt{M}\left(\left\|\hat{f_{\varepsilon}}-\hat{f_{\varepsilon}}^{(L)}\right\|_{1}+\left\|\hat{f_{\varepsilon}}-\hat{\chi}\right\|_{1}\right)$
$\displaystyle\ll$ $\displaystyle\sqrt{M}(L^{-(l-2)/2}+\varepsilon),$

by using (2.25), Lemma 2.16, the fact that $f_{\varepsilon}\leq 1$ and the
family $\\{\operatorname{supp}\,\,f_{\varepsilon}\\}_{\varepsilon}$ is
uniformly bounded.

To get our desired result we choose the parameters $\varepsilon$ and $L$ as
functions of $M$ such that as $M\rightarrow\infty$,

$\varepsilon=o(M^{-1/2})\quad\text{and}\quad M=o(L^{l-2}).$ (2.38)

∎

######  Lemma 2.18.

$\|H_{M}^{(\varepsilon,L)}-H_{M}\|_{2}\rightarrow 0$ as $M\rightarrow\infty.$

###### Proof.

By computations analogous to the proof of Proposition 2.17, we get

$\displaystyle\|H_{M}^{(\varepsilon,L)}-H_{M}\|_{2}$ $\displaystyle\leq$
$\displaystyle\sqrt{M}\left(\left\|\hat{f_{\varepsilon}}^{(L)}-\hat{\chi}\right\|_{2}+\left\|\hat{f_{\varepsilon}}^{(L)}-\hat{\chi}\right\|_{1}\right)$
$\displaystyle\leq$
$\displaystyle\sqrt{M}\left(\left\|\hat{f_{\varepsilon}}-\hat{f_{\varepsilon}}^{(L)}\right\|_{2}+\left\|\hat{f_{\varepsilon}}-\hat{\chi}\right\|_{2}+\left\|\hat{f_{\varepsilon}}-\hat{f_{\varepsilon}}^{(L)}\right\|_{1}+\left\|\hat{f_{\varepsilon}}-\hat{\chi}\right\|_{1}\right)$
$\displaystyle\ll$
$\displaystyle\sqrt{M}\left(L^{-(l-3)/2}+\left({\left\|f_{\varepsilon}-\chi\right\|_{1}}^{2}+{\|f_{\varepsilon}-\chi\|_{2}}^{2}\right)^{1/2}+L^{-(l-2)}+\varepsilon\right)$
$\displaystyle\ll$
$\displaystyle\sqrt{M}\left(L^{-(l-3)/2}+\left(\varepsilon^{2}+{\sqrt{\varepsilon}}^{2}\right)^{1/2}+L^{-(l-2)}+\varepsilon\right)$
$\displaystyle\ll$
$\displaystyle\sqrt{M}\left(L^{-(l-3)/2}+\varepsilon+\sqrt{\varepsilon}+L^{-(l-2)}+\varepsilon\right),$

where in the third line we have used Roger’s second moment formula, Lemma 2.16
and (2.25). Beyond that repeated use of Lemma 2.16 and (2.25) gives us the
above inequality.

Given the above inequality if we choose the parameters $\varepsilon$ and $L$
such that

$\varepsilon=o(M^{-1})\quad and\quad M=o(L^{l-3}),$ (2.39)

then $\|H_{M}^{(\varepsilon,L)}-H_{M}\|_{2}\rightarrow 0$ as
$M\rightarrow\infty.$ Note that (2.39) imply (2.38). Hence We choose the
parameters $\varepsilon$ and $L$ satisfying (2.39) at the end of this section
after considering all other things. ∎

In view of the Proposition (2.17), Theorem 2.6 (hence Theorem 1.2) is
equivalent to showing central limit theorem for $H_{M}^{(\varepsilon,L)}$.

######  Theorem 2.19.

Under the hypothesis of Theorem 1.2,

$H_{M}^{(\varepsilon,L)}\rightarrow N(0,\eta^{2})\,\,\text{in distribution
as}\,\,M\rightarrow\infty,$

where $\eta=\operatorname{vol}(\Omega_{2})^{1/2}$.

To prove Theorem 2.19 we use the CLT criteria of Fréchet and Shohat. By using
Theorem 2.8, we immediately get Theorem 2.19 if we can prove that for some
choice of parameters $\varepsilon$ and $L$

$\sigma^{2}:=\lim_{M\rightarrow\infty}\|H_{M}^{(\varepsilon,L)}\|_{2}^{2}<\infty$
(2.40)

$\lim_{M\rightarrow\infty}\text{Cum}_{[r]}\left(H_{M}^{(\varepsilon,L)}\right)=0\quad\text{for
all}\,\,r\geq 3.$ (2.41)

Computation of variance: First note that in view of the triangle inequality we
have

$\|H_{M}\|_{2}-\|H_{M}-H_{M}^{(\varepsilon,L)}\|_{2}\leq\|H_{M}^{(\varepsilon,L)}\|_{2}\leq\|H_{M}\|_{2}+\|H_{M}-H_{M}^{(\varepsilon,L)}\|_{2},$
(2.42)

where $H_{M}$ is given by (2.19). Hence in order to show (2.40) it is enough
to show that

$\lim_{M\rightarrow\infty}\|H_{M}\|_{2}^{2}<\infty$ (2.43)

and

$\|H_{M}-H_{M}^{(\varepsilon,L)}\|_{2}\rightarrow 0.$ (2.44)

Also if we can show (2.43) and (2.44), we get the expression of variance as

$\sigma^{2}=\lim_{M\rightarrow\infty}\|H_{M}^{(\varepsilon,L)}\|_{2}^{2}=\lim_{M\rightarrow\infty}\|H_{M}\|_{2}^{2}.$
(2.45)

Recall that

$H_{M}=\frac{1}{\sqrt{M}}\sum_{k=0}^{M-1}(\hat{\chi}\circ
c^{k}-\operatorname{vol}(\Omega_{2}))=\frac{1}{\sqrt{M}}\sum_{k=0}^{M-1}\phi_{k},$
(2.46)

where $\phi_{k}=\hat{\chi}\circ c^{k}-\operatorname{vol}(\Omega_{2}).$

$\displaystyle\|H_{M}\|_{2}^{2}=\frac{1}{M}\sum_{k_{1},k_{2}=0}^{M-1}\int_{X_{a}}\phi_{k_{1}}\phi_{k_{2}}\,\,d\mu_{a}$
$\displaystyle=$
$\displaystyle\frac{1}{M}\sum_{k_{1},k_{2}=0}^{M-1}\int_{X_{a}}\phi_{k_{1}-k_{2}}\phi_{0}\,\,d\mu_{a}$
(2.47) $\displaystyle=$ $\displaystyle\frac{1}{M}\sum_{\pm
k=0}^{M-1}(M-|k|)\int_{X_{a}}\phi_{k}\phi_{0}\,\,d\mu_{a}$ $\displaystyle=$
$\displaystyle\sum_{k=-\infty}^{\infty}\chi_{B_{M}}\left(1-\frac{|k|}{M}\right)\int_{X_{a}}\phi_{k}\phi_{0}\,\,d\mu_{a},$

where $B_{M}:=\\{k\in\mathbb{Z}:|k|\leq M-1\\}.$ Now by using the dominated
convergence theorem, we get that that

$\lim_{M\rightarrow\infty}\|H_{M}\|_{2}^{2}=\sum_{k=-\infty}^{\infty}\int_{X_{a}}\phi_{k}\phi_{0}\,\,d\mu_{a}$
(2.48)

Roger’s second order moment formula gives

$\displaystyle\int_{X_{a}}\phi_{k}\phi_{0}\,\,d\mu_{a}$ $\displaystyle=$
$\displaystyle\int_{X_{a}}\hat{\chi}\cdot(\hat{\chi}\circ
c^{k})\,\,d\mu_{a}-\operatorname{vol}(\Omega_{2})^{2}$ (2.49) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\int_{X_{a}}(\hat{\chi}+\hat{\chi}\circ
c^{k})^{2}\,\,d\mu_{a}-2\int_{X_{a}}\hat{\chi}^{2}\,\,d\mu_{a}-2\operatorname{vol}(\Omega_{2})^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\left(\int_{\mathbb{R}^{l}}(\chi+\chi\circ
c^{k})\,\,\right)^{2}+\int_{\mathbb{R}^{l}}(\chi+\chi\circ
c^{k})^{2}\,\,-2\left(\left(\int_{\mathbb{R}^{l}}\chi\right)^{2}+\int_{\mathbb{R}^{l}}\chi^{2}\,\,\right)-2\operatorname{vol}(\Omega_{2})^{2}\right)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{l}}\chi\cdot(\chi\circ
c^{k})\,\,d\mathbf{x}$ $\displaystyle=$
$\displaystyle\operatorname{vol}(\Omega_{2}\cap c^{-k}\Omega_{2}).$

Hence

$\lim_{M\rightarrow\infty}\|H_{M}\|_{2}^{2}=\sum_{k=-\infty}^{\infty}\operatorname{vol}(\Omega_{2}\cap
c^{-k}\Omega_{2})=\operatorname{vol}(\Omega_{2})<\infty,$ (2.50)

since $\operatorname{vol}(\Omega_{2}\cap c^{-k}\Omega_{2})=0$ for all nonzero
integer $k.$

Computation of cumulants:

First, let us write

$H_{M}^{(\varepsilon,L)}:=\frac{1}{\sqrt{M}}\sum_{k=0}^{M-1}\psi_{k},$ (2.51)

where $\psi_{k}=\hat{f_{\varepsilon}}^{(L)}\circ
c^{k}-\mu_{a}\left(\hat{f_{\varepsilon}}^{(L)}\right)$ and
$\mu_{a}\left(\hat{f_{\varepsilon}}^{(L)}\right)=\displaystyle\int_{X_{a}}\hat{f_{\varepsilon}}^{(L)}\,\,d\mu_{a}.$

By the multilinearity of $\text{cum}_{[r]}$ we get that

$\text{cum}_{[r]}\left(H_{M}^{(\varepsilon,L)}\right)=\frac{1}{M^{r/2}}\sum_{k_{1},\dots,k_{r}=0}^{M-1}\text{cum}_{[r]}(\psi_{k_{1}},\dots,\psi_{k_{r}})$
(2.52)

Now we want to decompose the sum of (2.52) into sub-sums according to our
convenience following [8, Proposition 6.2 ] and [7, Eq. (3.8)].

######  Proposition 2.20.

Let $r\in\mathbb{N}$ with $r\geq 3.$ Let $0\leq\alpha<\beta$ and a partition
$\mathcal{Q}$ of $\\{1,\dots,r\\}$ be given, and
$\mathbf{k}=(k_{1},\dots,k_{r})\in\mathbb{R}^{r}_{+}$. We define

$\Delta(\alpha):=\\{\mathbf{k}\in\mathbb{R}^{r}_{+}:|k_{i}-k_{j}|\leq\alpha\,\,\forall
i,j\\}$

and

$\Delta_{\mathcal{Q}}(\alpha,\beta)=\left\\{\mathbf{k}\in\mathbb{R}^{r}_{+}:\max_{I\in\mathcal{Q}}\max_{i,j\in
I}\\{|k_{i}-k_{j}|\\}\leq\alpha\,\,\text{and}\,\,\min_{I,J\in\mathcal{Q},I\neq
J}\min_{i\in I,j\in J}\\{|k_{i}-k_{j}|\\}>\beta\right\\}.$

Then, given
$0=\alpha_{0}<\beta_{1}<\alpha_{1}=(3+r)\beta_{1}<\beta_{2}<\dots<\beta_{r}<\alpha_{r-1}=(3+r)\beta_{r-1}<\beta_{r}$,
we have

$\mathbb{R}^{r}_{+}=\Delta(\beta_{r})\cup\left(\bigcup_{j=0}^{r-1}\bigcup_{|\mathcal{Q}|\geq
2}\Delta_{\mathcal{Q}}(\alpha_{j},\beta_{j+1})\right).$

Intersecting with $\\{0,1,\dots,M-1\\}^{r},$ we get that

$\\{0,1,\dots,M-1\\}^{r}=\Omega(\beta_{r},M)\cup\left(\bigcup_{j=0}^{r-1}\bigcup_{|\mathcal{Q}|\geq
2}\Omega_{\mathcal{Q}}(\alpha_{j},\beta_{j+1},M)\right),$

where

$\Omega(\beta_{r},M):=\\{0,1,\dots,M-1\\}^{r}\cap\Delta(\beta_{r}),$

and

$\Omega_{\mathcal{Q}}(\alpha_{j},\beta_{j+1},M):=\\{0,1,\dots,M-1\\}^{r}\cap\Delta_{\mathcal{Q}}(\alpha_{j},\beta_{j+1}).$

To estimate the cumulant (2.52), the strategy involves separately estimating
the sums over $\Omega(\beta_{r},M)$ and
$\Omega_{\mathcal{Q}}(\alpha_{j},\beta_{j+1},M)$. After accounting for all
other factors, the sequences $\\{\alpha_{j}\\}$ and $\\{\beta_{j}\\}$ are
chosen at the final step.

Case: 1 Summing over $\mathbf{\Omega(\beta_{r},M)}.$

Suppose the index $\mathbf{k}=(k_{1},\dots,k_{r})$ runs over
${\Omega(\beta_{r},M)}.$ Then

$|k_{i}-k_{j}|\leq\beta_{r}\,\,\text{for all}\,\,i,j.$

Given this, we have

$|\Omega(\beta_{r},M)|\ll M\beta_{r}^{r-1}$

We claim that

$\frac{1}{M^{r/2}}\sum_{(k_{1},\dots,k_{r})\in\Omega(\beta_{r},M)}|\text{cum}_{[r]}(\psi_{k_{1}},\dots,\psi_{k_{r}})|\ll_{r}M^{1-r/2}\beta_{r}^{r-1}\left\|\hat{f_{\varepsilon}}^{(L)}\right\|^{(r-l)^{+}}_{\infty}$
(2.53)

where $(r-l)^{+}=\max\\{0,r-l\\}.$

To prove (2.53), it is enough to show that

$\int_{X_{a}}|\psi_{k_{1}}\dots\psi_{k_{r}}|\,\,d\mu_{a}\ll_{r}\left\|\hat{f_{\varepsilon}}^{(L)}\right\|^{(r-l)^{+}}_{\infty}$
(2.54)

First, consider the case $r\leq l$. In this case we apply the generalized
H$\ddot{\text{o}}$lder inequality to the $l$ functions
$\psi_{k_{1}},\dots,\psi_{k_{r}},1,\dots,1$ to get

$\int_{X_{a}}|\psi_{k_{1}}\dots\psi_{k_{r}}|\,\,d\mu_{a}\ll\|\psi_{k_{1}}\|_{l}\dots\|\psi_{k_{r}}\|_{l}$
(2.55)

Now for any $k$

$\displaystyle\|\psi_{k}\|_{l}$ $\displaystyle\leq$
$\displaystyle\left\|\hat{f_{\varepsilon}}^{(L)}\right\|_{l}+\mu_{a}\left(\hat{f_{\varepsilon}}^{(L)}\right),\,\,\text{by
using $G_{a}$-invariance of $\mu_{a}$}$ $\displaystyle\leq$
$\displaystyle\|\alpha_{l,a}\|_{l}+\|f_{\varepsilon}\|_{1},\,\,\text{by
Proposition \ref{prop:Siegel_alpha_bound_affine}}$ $\displaystyle\leq$
$\displaystyle\|\alpha_{l,a}\|_{l}+\|f_{\varepsilon}-\chi\|_{1}+\|\chi\|_{1}$
$\displaystyle\ll$
$\displaystyle\|\alpha_{l,a}\|_{l}+\varepsilon+\operatorname{vol}(\Omega_{2})$

By (2.55) and (2), we get

$\int_{X_{a}}|\psi_{k_{1}}\dots\psi_{k_{r}}|\,\,d\mu_{a}\ll 1$ (2.56)

This shows that (2.54) holds if $r\leq l.$ Next we consider the case when
$r>l.$ In this case

$\displaystyle\int_{X_{a}}|\psi_{k_{1}}\dots\psi_{k_{r}}\psi_{k_{r+1}}\dots\psi_{k_{l}}|\,\,d\mu_{a}$
$\displaystyle\leq$
$\displaystyle\|\psi_{k_{r+1}}\dots\psi_{k_{l}}\|_{\infty}\int_{X_{a}}|\psi_{k_{1}}\dots\psi_{k_{r}}|\,\,d\mu_{a}$
$\displaystyle\leq$ $\displaystyle
2^{r-l}\left\|\hat{f_{\varepsilon}}^{(L)}\right\|_{\infty}^{r-l}\|\psi_{k_{1}}\|_{l}\dots\|\psi_{k_{r}}\|_{l}$
$\displaystyle\ll$ $\displaystyle
2^{r-l}\left\|\hat{f_{\varepsilon}}^{(L)}\right\|_{\infty}^{r-l},\,\,\text{using
above estimate for $\|\psi_{k}\|_{l}$}$ $\displaystyle\ll$
$\displaystyle\left\|\hat{f_{\varepsilon}}^{(L)}\right\|_{\infty}^{r-l}.$

Therefore (2.54) holds and hence we get our desired result.

Case: 2 Summing over $\mathbf{\Omega_{\mathcal{Q}}(\alpha_{j},\beta_{j+1},M)}$
with $\mathbf{|\mathcal{Q}|\geq 2}.$

Suppose that $\mathcal{Q}=\\{J_{1},\dots,J_{d}\\}$ with $d\geq 2.$ Given an
arbitrary subset $I$ of $\\{1,\dots,r\\}$ first we want to show that

$\int_{X_{a}}\prod_{i\in
I}\psi_{k_{i}}\,\,d\mu_{a}\approx\prod_{t=0}^{d}\int_{X_{a}}\prod_{i\in I\cap
J_{t}}\psi_{k_{i}}\,\,d\mu_{a}.$ (2.57)

If $I\subseteq J_{t}$ $(\text{hence}\,\,I\cap J_{t}\,\,\text{is empty for all
indices}\,\,t\,\,\text{except one})$ for some $t=1,\dots,d$, then it is easy
to observe that equality holds in (2.57). In other cases, we will show that
equality in (2.57) holds with an error term.

If $I=I_{1}\sqcup\dots\sqcup I_{s}$, where each $I_{t}$ is nonempty and
$I_{t}\in\\{I\cap J:J\in\mathcal{Q}\\}$ for $t=1,\dots,s.$ Let
$g=\hat{f_{\varepsilon}}^{(L)}-\mu_{a}\left(\hat{f_{\varepsilon}}^{(L)}\right)$,
$\mathbf{f}=\hat{f_{\varepsilon}}^{(L)}$ and $k_{I_{j}}:=\max\\{k_{i}:i\in
I_{j}\\}.$ Then

$\displaystyle\int_{X_{a}}\prod_{i\in I}\psi_{k_{i}}\,\,d\mu_{a}$
$\displaystyle=$ $\displaystyle\int_{X_{a}}\prod_{i\in I}g\circ
c^{k_{i}}\,\,d\mu_{a}$ $\displaystyle=$
$\displaystyle\int_{X_{a}}\prod_{j=1}^{s}\left(\prod_{i\in I_{j}}g\circ
c^{k_{i}}\right)\,\,d\mu_{a}$ $\displaystyle=$
$\displaystyle\int_{X_{a}}\prod_{j=1}^{s}\left(\prod_{i\in I_{j}}g\circ
c^{k_{i}-k_{I_{j}}}\right)\circ c^{k_{I_{j}}}\,\,d\mu_{a}$ $\displaystyle=$
$\displaystyle\int_{X_{a}}\prod_{j=1}^{s}\left(\sum_{K_{j}\subseteq
I_{j}}(-\mu_{a}\left(\mathbf{f}\right))^{|K_{j}|}\prod_{i\in I_{j}\setminus
K_{j}}\mathbf{f}\circ c^{k_{i}-k_{I_{j}}}\right)\circ
c^{k_{I_{j}}}\,\,d\mu_{a}$ $\displaystyle=$
$\displaystyle\int_{X_{a}}\sum_{K_{j}\subseteq
I_{j},j=1,\dots,s}(-\mu_{a}\left(\mathbf{f}\right))^{\sum_{d=1}^{s}|K_{d}|}\prod_{j=1}^{s}\left(\prod_{i\in
I_{j}\setminus K_{j}}\mathbf{f}\circ c^{k_{i}-k_{I_{j}}}\right)\circ
c^{k_{I_{j}}}\,\,d\mu_{a}$ $\displaystyle=$ $\displaystyle\sum_{K_{j}\subseteq
I_{j},j=1,\dots,s}(-\mu_{a}\left(\mathbf{f}\right))^{\sum_{d=1}^{s}|K_{d}|}\int_{X_{a}}\prod_{j=1}^{s}\left(\prod_{i\in
I_{j}\setminus K_{j}}\mathbf{f}\circ c^{k_{i}-k_{I_{j}}}\right)\circ
c^{k_{I_{j}}}\,\,d\mu_{a}$

Let $\mathbf{f}_{K_{j}}:=\prod_{i\in I_{j}\setminus K_{j}}\mathbf{f}\circ
c^{k_{i}-k_{I_{j}}}$ for $j=1,\dots,s.$ Then from above we have

$\int_{X_{a}}\prod_{i\in I}\psi_{k_{i}}\,\,d\mu_{a}=\sum_{K_{j}\subseteq
I_{j},j=1,\dots,s}(-\mu_{a}\left(\mathbf{f}\right))^{\sum_{d=1}^{s}|K_{d}|}\int_{X_{a}}\prod_{j=1}^{s}\mathbf{f}_{K_{j}}\circ
c^{k_{I_{j}}}\,\,d\mu_{a}.$ (2.58)

We want to use the quantitative estimate for higher order correlations, i.e.,
Theorem 2.14 to compute the above integral. Before that let us compute a few
other things. Using Lemma 2.13, we get

$\displaystyle\prod_{j=1}^{s}S_{q}(\mathbf{f}_{K_{j}})$ $\displaystyle=$
$\displaystyle\prod_{j=1}^{s}S_{q}\left(\prod_{i\in I_{j}\setminus
K_{j}}\mathbf{f}\circ c^{k_{i}-k_{I_{j}}}\right)$ $\displaystyle\ll$
$\displaystyle\prod_{j=1}^{s}\left(\prod_{i\in I_{j}\setminus
K_{j}}S_{q}\left(\mathbf{f}\circ c^{k_{i}-k_{I_{j}}}\right)\right)$
$\displaystyle\ll$
$\displaystyle\prod_{j=1}^{s}S_{q}(\mathbf{f})^{|I_{j}\setminus
K_{j}|}\prod_{i\in I_{j}\setminus
K_{j}}\left\|(c^{-1})^{k_{I_{j}}-k_{i}}\right\|_{\text{op}}^{q}$
$\displaystyle\ll$
$\displaystyle\prod_{j=1}^{s}S_{q}(\mathbf{f})^{|I_{j}\setminus
K_{j}|}\left\|c^{-1}\right\|_{\text{op}}^{q|I_{j}\setminus
K_{j}|\alpha_{j}},\,\,\text{since}\,\,k_{I_{j}}-k_{i}\leq\alpha_{j}\,\,\forall
i\in I_{j}\setminus K_{j}$ $\displaystyle\ll$ $\displaystyle
S_{q}(\mathbf{f})^{r}\|c^{-1}\|^{qr\alpha_{j}}_{\text{op}}.$

Hence

$\prod_{j=1}^{s}S_{q}(\mathbf{f}_{K_{j}})\ll
S_{q}(\mathbf{f})^{r}e^{qr\alpha_{j}\tau},\,\,\text{where}\,\,\tau=\log\|c^{-1}\|_{\text{op}}>0.$
(2.59)

Using (2.58), (2.59) and Theorem 2.14, we get

$\int_{X_{a}}\prod_{i\in I}\psi_{k_{i}}\,\,d\mu_{a}=\sum_{K_{j}\subseteq
I_{j},j=1,\dots,s}(-\mu_{a}\left(\mathbf{f}\right))^{\sum_{d=1}^{s}|K_{d}|}\left(\prod_{j=1}^{s}\int_{X_{a}}\mathbf{f}_{K_{j}}\,\,d\mu_{a}+O_{q,r}(E^{\prime})\right)$
(2.60)

where

$E^{\prime}=e^{-\delta\min_{i\neq
j}\rho_{G_{a}}\left(c^{k_{I_{i}}},c^{k_{I_{j}}}\right)}\prod_{j=1}^{s}S_{q}(\mathbf{f}_{K_{j}})\ll
e^{-\delta\min_{i\neq
j}\rho_{G_{a}}\left(1,c^{k_{I_{j}}-k_{I_{i}}}\right)}S_{q}(\mathbf{f})^{r}e^{qr\alpha_{j}\tau}$
(2.61)

First note that for any $i\neq j,$ $|k_{I_{j}}-k_{I_{i}}|>\beta_{j+1}.$
Suppose that $k_{I_{j}}-k_{I_{i}}>0.$ Then using Lemma 2.11, we get
$0<D_{1}\leq 1,D_{2}>0$ such that

$\rho_{G_{a}}\left(1,c^{k_{I_{j}}-k_{I_{i}}}\right)\geq
D_{1}\log\|c^{k_{I_{j}}-k_{I_{i}}}\|_{\text{op}}-D_{2}\geq
D_{1}\beta_{j+1}\log\lambda-D_{2}.$ (2.62)

Let $\delta^{\prime}=\delta D_{1}\log\lambda$. Combining (2.61) and (2.62) we
get that

$E^{\prime}\ll
e^{-\delta^{\prime}\beta_{j+1}}S_{q}(\mathbf{f})^{r}e^{qr\alpha_{j}\tau}=e^{-(\delta^{\prime}\beta_{j+1}-qr\alpha_{j}\tau)}S_{q}(\mathbf{f})^{r}$
(2.63)

Now if we let
$E=e^{-(\delta^{\prime}\beta_{j+1}-qr\alpha_{j}\tau)}S_{q}(\mathbf{f})^{r},$
then from (2.60) we get

$\displaystyle\int_{X_{a}}\prod_{i\in I}\psi_{k_{i}}\,\,d\mu_{a}$
$\displaystyle=$ $\displaystyle\sum_{K_{j}\subseteq
I_{j},j=1,\dots,s}(-\mu_{a}\left(\mathbf{f}\right))^{\sum_{d=1}^{s}|K_{d}|}\prod_{j=1}^{s}\int_{X_{a}}\mathbf{f}_{K_{j}}\,\,d\mu_{a}+O_{q,r}(E)$
$\displaystyle=$ $\displaystyle\prod_{j=1}^{s}\sum_{K_{j}\subseteq
I_{j}}(-\mu_{a}\left(\mathbf{f}\right))^{|K_{j}|}\int_{X_{a}}\mathbf{f}_{K_{j}}\,\,d\mu_{a}+O_{q,r}(E)$
$\displaystyle=$ $\displaystyle\prod_{j=1}^{s}\int_{X_{a}}\prod_{i\in
I_{j}}\left(g\circ c^{k_{i}}\right)\,\,d\mu_{a}+O_{q,r}(E)$ $\displaystyle=$
$\displaystyle\prod_{j=1}^{s}\int_{X_{a}}\prod_{i\in
I_{j}}\psi_{k_{i}}\,\,d\mu_{a}+O_{q,r}(E)$

Hence finally for any
$\mathbf{k}=(k_{1},\dots,k_{r})\in\Omega_{\mathcal{Q}}(\alpha_{j},\beta_{j+1},M)$
with $|\mathcal{Q}|\geq 2,$ we have

$\int_{X_{a}}\prod_{i\in
I}\psi_{k_{i}}\,\,d\mu_{a}=\prod_{t=0}^{d}\int_{X_{a}}\prod_{i\in I\cap
J_{t}}\psi_{k_{i}}\,\,d\mu_{a}+O_{q,r}(E),$ (2.64)

where
$E=e^{-(\delta^{\prime}\beta_{j+1}-qr\alpha_{j}\tau)}S_{q}(\mathbf{f})^{r}.$
Now summing the above estimate over all partitions $\mathcal{P}$ of
$\\{1,\dots,r\\}$ and denoting an element of $\mathcal{P}$ by $I,$ we get

$\displaystyle\text{cum}_{[r]}(\psi_{k_{1}},\dots,\psi_{k_{r}})=\text{cum}_{[r]}(\psi_{k_{1}},\dots,\psi_{k_{r}}|\mathcal{Q})+O_{q,r}(E)$
(2.65)

Now by using Proposition 2.9, we get

$|\text{cum}_{[r]}(\psi_{k_{1}},\dots,\psi_{k_{r}})|\ll_{q,r}E$ (2.66)

Final estimate on the cumulants:

Combining all the estimates, we get

$\text{cum}_{[r]}\left(H_{M}^{(\varepsilon,L)}\right)\ll
M^{1-r/2}\beta_{r}^{r-1}L^{(r-l)^{+}}+\varepsilon^{-qr}L^{r}M^{r/2}\max_{j}\left\\{e^{-(\delta^{\prime}\beta_{j+1}-qr\alpha_{j}\tau)}\right\\}$
(2.67)

Now we choose the parameters $\alpha_{j}$ and $\beta_{j}$ such that the right
hand side of (2.67) goes to zero as $M\rightarrow\infty.$ We do this by
choosing a single parameter $\eta>0.$ We define the parameters inductively by

$\beta_{1}=\eta\quad\text{and}\quad\beta_{j+1}=\max\\{\eta+(3+r)\beta_{j},\eta+(\delta^{\prime})^{-1}r(3+r)q\tau\\}$
(2.68)

for $j=1,\dots,r-1.$

The above choice of $\beta_{j+1}$ fulfills the requirement, i.e.,
$\alpha_{j}=(3+r)\beta_{j}<\beta_{j+1}$ of Proposition 2.20. Also we have

$\delta^{\prime}\beta_{j+1}-qr\tau\alpha_{j}\geq\delta^{\prime}\eta>0.$ (2.69)

By induction, it easily follows that $\beta_{r}\ll_{r}\eta.$ Hence from (2.67)
we get

$\text{cum}_{[r]}\left(H_{M}^{(\varepsilon,L)}\right)\ll
M^{1-r/2}{\eta}^{r-1}L^{(r-l)^{+}}+\varepsilon^{-qr}L^{r}M^{r/2}e^{-\delta^{\prime}\eta}.$
(2.70)

Further we want to choose parameters $\varepsilon$ and $L$ such that

$M^{1-r/2}{\eta}^{r-1}L^{(r-l)^{+}}\rightarrow 0$ (2.71)

and

$\varepsilon^{-qr}L^{r}M^{r/2}e^{-\delta^{\prime}\eta}\rightarrow 0$ (2.72)

as $M\rightarrow\infty.$

We take the parameter $\eta=C_{r}\log M$ for some constant $C_{r}>0.$

If $r-l<0,$ then $L^{(r-l)^{+}}=0.$ Hence in this case (2.71) follows by our
choice of the parameter $\eta$ as above, since $r\geq 3$, $1-r/2\leq-1/2$ and
$\eta^{r-1}=o(M^{1/2}).$

Now assume that $r-l>0.$ Then if we choose the parameter $L=M^{d}$ for some
real number $d$ (we will make the choice of $d$ later), (2.71) holds provided
$1-r/2+d(r-l)>0.$ Hence $d$ must satisfy

$d<\frac{r-2}{2(r-l)}.$ (2.73)

We also want (2.39) to hold and this forces $d(l-3)>1.$ Therefore we can find
the suitable choice of $d$ if and only if $\frac{1}{l-3}<\frac{r-2}{2(r-l)},$
i.e., $r(l-5)+6>0.$ This holds since by assumption $l\geq 5.$

Finally, the only things remaining to choose are $C_{r}$ and $\varepsilon.$
Choose $\varepsilon=N^{-2}.$ Then (2.72) holds if
$r(2q+d+1/2)-\delta^{\prime}C_{r}<0$ and this holds if
$C_{r}>\frac{r}{\delta^{\prime}}(2q+d+1/2).$ Hence for such a choice of
$C_{r}$, $\varepsilon$ and $L$, (2.71) and (2.72) holds. This completes the
proof of (2.41) for $r\geq 3.$ Now as an immediate application of Theorem 2.8,
Theorem 2.19 follows.

##  3\. Proof of the CLT for congruence unimodular lattices

Throughout this section, we will denote the set of all congruence unimodular
lattices on $\mathbb{R}^{l}$ by $X_{c},$ where $l=m+n.$ Let
$g:\mathbb{R}^{l}\rightarrow\mathbb{R}$ be a Borel measurable function with
compact support. The Siegel transform of $g$ is
$\hat{g}:X_{c}\rightarrow\mathbb{R}$ defined by

$\hat{g}(\Lambda)=\sum_{\lambda\in\Lambda\setminus\\{0\\}}g(\lambda)\quad\text{for}\,\,\Lambda\in
X_{c}.$ (3.1)

Now let us record the analogs of Siegel’s transform and Roger’s second moment
formula for congruence lattice from [15]. The proof of the same can also be
found in [3].

######  Proposition 3.1.

[15, Theorem 3.2]. Let $g:\mathbb{R}^{l}\rightarrow\mathbb{R}$ be a bounded
Borel measurable function with compact support. Then

$\int_{X_{c}}\hat{g}(\Lambda)\,\,d\mu_{c}(\Lambda)=\int_{\mathbb{R}^{l}}g(\mathbf{x})\,\,d\mathbf{x}$
(3.2)
$\int_{X_{c}}{\hat{g}(\Lambda)}^{2}\,\,d\mu_{c}(\Lambda)=\left(\int_{\mathbb{R}^{l}}g(\mathbf{x})\,\,d\mathbf{x}\right)^{2}+\frac{1}{\zeta_{N}(m+n)}\sum_{\tiny\begin{array}[]{c}s_{1}\geq
1\\\
\gcd(s_{1},N)=1\end{array}}\sum_{\tiny\begin{array}[]{c}s_{2}\in\mathbb{Z}\setminus\\{0\\}\\\
s_{2}\equiv s_{1}(\mod
N)\end{array}}\int_{\mathbb{R}^{l}}{g(s_{1}\mathbf{x})}g(s_{2}\mathbf{x})\,\,d\mathbf{x},$
(3.3)

where $\zeta_{N}$ is defined in Theorem 1.3.

By arguments similar to the previous section it is easy to see that Theorem
1.3 is equivalent to the following theorem.

######  Theorem 3.2.

Under the hypothesis of Theorem 1.3,

$R_{M}:=\frac{1}{\sqrt{M}}\sum_{k=0}^{M-1}\left(\hat{\chi}\circ
c_{0}^{k}-M\cdot\operatorname{vol}(\Omega_{2})\right)\rightarrow
N(0,\eta^{2}\sigma_{c}^{2})\,\,\text{in distribution
as}\,\,M\rightarrow\infty,$ (3.4)

where $\sigma_{c}^{2}$ is given by Theorem 1.3,
$\eta=\operatorname{vol}(\Omega_{2})^{1/2}$ and $\hat{\chi}$ is the Siegel
transform of $\chi_{\Omega_{2}}$ on $X_{c}.$

Again as we have seen in the case of affine unimodular lattices, here also
Theorem 3.2 is equivalent to proving the following theorem.

######  Theorem 3.3.

Under the hypothesis of Theorem 1.3,

$R_{M}^{(\varepsilon,L)}:=\frac{1}{\sqrt{M}}\left(\sum_{k=0}^{M-1}\hat{g_{\varepsilon}}^{(L)}\circ
c_{0}^{k}-M\int_{X_{c}}\hat{g_{\varepsilon}}^{(L)}\,\,d\mu_{c}\right)\rightarrow
N(0,\eta^{2}\sigma_{c}^{2})\,\,\text{in distribution
as}\,\,M\rightarrow\infty,$ (3.5)

where $\hat{g_{\varepsilon}}^{(L)}=\hat{g_{\varepsilon}}\eta_{L}\in
C_{c}^{\infty}(X_{c})$ are smooth approximations to $\hat{\chi}$,
$\sigma_{c}^{2}$ is given by Theorem 1.3 and
$\eta=\operatorname{vol}(\Omega_{2})^{1/2}$.

Again we use the CLT criteria of Fréchet and Shohat to prove Theorem 3.3.
Proving Theorem 3.3 amounts to show that for some choice of parameters
$\varepsilon$ and $L$

$\sigma^{2}:=\lim_{M\rightarrow\infty}\|R_{M}^{(\varepsilon,L)}\|_{2}^{2}<\infty$
(3.6)

$\lim_{M\rightarrow\infty}\text{Cum}_{[r]}\left(R_{M}^{(\varepsilon,L)}\right)=0\quad\text{for
all}\,\,r\geq 3.$ (3.7)

The computation of cumulant (i.e., Equation (3.7)) is analogous to the affine
case (with appropriate modifications) discussed in detail in the previous
section. Hence we omit the details here. Rather we do the computation of
variance for the congruence case.

Computation of variance:

First note that given the triangle inequality, we have

$\|R_{M}\|_{2}-\|R_{M}-R_{M}^{(\varepsilon,L)}\|_{2}\leq\|R_{M}^{(\varepsilon,L)}\|_{2}\leq\|R_{M}\|_{2}+\|R_{M}-R_{M}^{(\varepsilon,L)}\|_{2},$
(3.8)

where $R_{M}$ is given by (3.4). Hence in order to show (3.6) it is enough to
show that

$\lim_{M\rightarrow\infty}\|R_{M}\|_{2}^{2}<\infty$ (3.9)

and

$\|R_{M}-R_{M}^{(\varepsilon,L)}\|_{2}\rightarrow 0.$ (3.10)

Also if we can show (3.9) and (3.10), we get the expression of variance as

$\sigma^{2}=\lim_{M\rightarrow\infty}\|R_{M}^{(\varepsilon,L)}\|_{2}^{2}=\lim_{M\rightarrow\infty}\|R_{M}\|_{2}^{2}.$
(3.11)

Recall that

$R_{M}=\frac{1}{\sqrt{M}}\sum_{k=0}^{M-1}(\hat{\chi}\circ
c_{0}^{k}-\operatorname{vol}(\Omega_{2}))=\frac{1}{\sqrt{M}}\sum_{k=0}^{M-1}\phi_{k},$
(3.12)

where $\phi_{k}=\hat{\chi}\circ c_{0}^{k}-\operatorname{vol}(\Omega_{2}).$

$\displaystyle\|R_{M}\|_{2}^{2}=\frac{1}{M}\sum_{k_{1},k_{2}=0}^{M-1}\int_{X_{c}}\phi_{k_{1}}\phi_{k_{2}}\,\,d\mu_{c}$
$\displaystyle=$
$\displaystyle\frac{1}{M}\sum_{k_{1},k_{2}=0}^{M-1}\int_{X_{c}}\phi_{k_{1}-k_{2}}\phi_{0}\,\,d\mu_{c}$
(3.13) $\displaystyle=$ $\displaystyle\frac{1}{M}\sum_{\pm
k=0}^{M-1}(M-|k|)\int_{X_{c}}\phi_{k}\phi_{0}\,\,d\mu_{c}$ $\displaystyle=$
$\displaystyle\sum_{k=-\infty}^{\infty}\chi_{B_{M}}\left(1-\frac{|k|}{M}\right)\int_{X_{c}}\phi_{k}\phi_{0}\,\,d\mu_{c},$

where $B_{M}:=\\{k\in\mathbb{Z}:|k|\leq M-1\\}.$ Again using the dominated
convergence theorem, we get

$\lim_{M\rightarrow\infty}\|R_{M}\|_{2}^{2}=\sum_{k=-\infty}^{\infty}\int_{X_{c}}\phi_{k}\phi_{0}\,\,d\mu_{c}.$
(3.14)

Now Roger’s second order moment formula gives

$\displaystyle\int_{X_{c}}\phi_{k}\phi_{0}\,\,d\mu_{c}$ $\displaystyle=$
$\displaystyle\int_{X_{c}}\hat{\chi}\cdot(\hat{\chi}\circ
c_{0}^{k})\,\,d\mu_{c}-\operatorname{vol}(\Omega_{2})^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\int_{X_{c}}(\hat{\chi}+\hat{\chi}\circ
c_{0}^{k})^{2}\,\,d\mu_{c}-2\int_{X_{c}}\hat{\chi}^{2}\,\,d\mu_{c}-2\operatorname{vol}(\Omega_{2})^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\left(\int_{\mathbb{R}^{l}}(\chi+\chi\circ
c_{0}^{k})\,\,d\mathbf{x}\right)^{2}-2\left(\int_{\mathbb{R}^{l}}\chi\,\,d\mathbf{x}\right)^{2}-2\operatorname{vol}(\Omega_{2})^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2\zeta_{N}(m+n)}\sum_{\tiny\begin{array}[]{c}s_{1}\geq
1\\\
\gcd(s_{1},N)=1\end{array}}\sum_{\tiny\begin{array}[]{c}s_{2}\in\mathbb{Z}\setminus\\{0\\}\\\
s_{2}\equiv s_{1}(\mod N)\end{array}}\int_{\mathbb{R}^{l}}((\chi+\chi\circ
c_{0}^{k})(s_{1}\mathbf{x})(\chi+\chi\circ c_{0}^{k})(s_{2}\mathbf{x})$
$\displaystyle-$ $\displaystyle
2\chi(s_{1}\mathbf{x})\chi(s_{2}\mathbf{x}))\,\,d\mathbf{x}$ $\displaystyle=$
$\displaystyle\frac{1}{\zeta_{N}(m+n)}\left(\sum_{\tiny\begin{array}[]{c}s_{1}\geq
1\\\
\gcd(s_{1},N)=1\end{array}}\sum_{\tiny\begin{array}[]{c}s_{2}\in\mathbb{Z}\setminus\\{0\\}\\\
s_{2}\equiv s_{1}(\mod N)\end{array}}\int_{\mathbb{R}^{l}}(\chi\circ
c_{0}^{k})(s_{1}\mathbf{x})\chi(s_{2}\mathbf{x})\,\,d\mathbf{x}\right).$
(3.24)

Hence

$\displaystyle\lim_{M\rightarrow\infty}\|R_{M}\|_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\zeta_{N}(m+n)}\sum_{k=-\infty}^{\infty}\sum_{\tiny\begin{array}[]{c}s_{1}\geq
1\\\ \gcd(s_{1},N)=1\end{array}}\sum_{\tiny\begin{array}[]{c}s_{2}\geq 1\\\
s_{2}\equiv s_{1}(\mod N)\end{array}}\int_{\mathbb{R}^{l}}(\chi\circ
c_{0}^{k})(s_{1}\mathbf{x})\chi(s_{2}\mathbf{x})\,\,d\mathbf{x}$ (3.29)
$\displaystyle=$
$\displaystyle\frac{2}{\zeta_{N}(m+n)}\sum_{\tiny\begin{array}[]{c}s_{1}\geq
1\\\ \gcd(s_{1},N)=1\end{array}}\sum_{\tiny\begin{array}[]{c}s_{2}\geq 1\\\
s_{2}\equiv s_{1}(\mod
N)\end{array}}\int_{\mathbb{R}^{l}}\left(\sum_{k=-\infty}^{\infty}\chi\circ
c_{0}^{k}\right)(s_{1}\mathbf{x})\chi(s_{2}\mathbf{x})\,\,d\mathbf{x}.$ (3.34)

Now let

$\Omega=\left\\{(\mathbf{x},\mathbf{y})\in\mathbb{R}^{m}\times\mathbb{R}^{n}:\,\,|x_{i}|\|\mathbf{y}\|^{u_{i}}<c_{i}\,\,\text{for}\,\,i=1,\dots,m\right\\}.$
(3.35)

Then by changing the coordinates to spherical coordinates, we get

$\displaystyle\int_{\mathbb{R}^{l}}\left(\sum_{k=-\infty}^{\infty}\chi\circ
c_{0}^{k}\right)(s_{1}\mathbf{x})\chi(s_{2}\mathbf{x})\,\,d\mathbf{x}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{l}}\chi_{\Omega}(s_{1}\mathbf{x})\chi(s_{2}\mathbf{x})\,\,d\mathbf{x}$
$\displaystyle=$ $\displaystyle\operatorname{vol}\left(s_{1}^{-1}\Omega\cap
s_{2}^{-1}\Omega_{2}\right)$ $\displaystyle=$
$\displaystyle\int_{1/s_{2}\leq\|\mathbf{y}\|\leq
2/s_{2}}\prod_{i=1}^{m}\left(\frac{2c_{i}}{\max\\{s_{1},s_{2}\\}^{1+u_{i}}\|\mathbf{y}\|^{u_{i}}}\right)d\mathbf{y}$
$\displaystyle=$ $\displaystyle\frac{2^{m}c_{1}\dots
c_{m}\omega_{n}}{\max\\{s_{1},s_{2}\\}^{m+n}}\int_{1/s_{2}}^{2/s_{2}}\frac{1}{r^{n}}r^{n-1}dr$

Hence

$\int_{\mathbb{R}^{l}}\left(\sum_{k=-\infty}^{\infty}\chi\circ
c_{0}^{k}\right)(s_{1}\mathbf{x})\chi(s_{2}\mathbf{x})\,\,d\mathbf{x}=2^{m}(\log
2)\max\\{s_{1},s_{2}\\}^{-m-n}c_{1}\dots c_{n}\omega_{n},$ (3.37)

where $\omega_{n}$ is given by (1.7).

Let $C_{N}=\\{s\in\mathbb{Z}:0\leq s<N\,\,\text{and}\,\,\gcd(s,N)=1\\}.$ In
view of (3.37), we finally get that

$\displaystyle\sigma^{2}$ $\displaystyle=$ $\displaystyle\frac{2^{m+1}(\log
2)c_{1}\dots
c_{n}\omega_{n}}{\zeta_{N}(m+n)}\sum_{\tiny\begin{array}[]{c}s_{1}\geq 1\\\
\gcd(s_{1},N)=1\end{array}}\sum_{\tiny\begin{array}[]{c}s_{2}\geq 1\\\
s_{2}\equiv s_{1}(\mod N)\end{array}}\max\\{s_{1},s_{2}\\}^{-m-n}$
$\displaystyle=$ $\displaystyle\frac{2^{m+1}(\log 2)c_{1}\dots
c_{n}\omega_{n}}{\zeta_{N}(m+n)}\sum_{s\in C_{N}}\sum_{s_{1},s_{2}\geq
1}\max\\{Ns_{1}+s,Ns_{2}+s\\}^{-m-n}$ $\displaystyle=$
$\displaystyle\frac{2^{m+1}(\log 2)c_{1}\dots
c_{n}\omega_{n}}{\zeta_{N}(m+n)}\sum_{s\in C_{N}}\left(\sum_{s_{1}\geq
1}(Ns_{1}+s)^{-m-n}+2\sum_{1\leq s_{1}<s_{2}}(Ns_{2}+s)^{-m-n}\right)$
$\displaystyle=$ $\displaystyle\frac{2^{m+1}(\log 2)c_{1}\dots
c_{n}\omega_{n}}{\zeta_{N}(m+n)}\left(\zeta_{N}(m+n)+2\sum_{s\in
C_{N}}\sum_{s_{2}\geq 1}\frac{s_{2}-1}{(Ns_{2}+s)^{m+n}}\right)$ (3.43)
$\displaystyle=$ $\displaystyle\frac{2^{m+1}(\log 2)c_{1}\dots
c_{n}\omega_{n}}{\zeta_{N}(m+n)}\left(1+\frac{2}{\zeta_{N}(m+n)}\sum_{s\in
C_{N}}\sum_{s_{2}\geq 1}\frac{s_{2}-1}{(Ns_{2}+s)^{m+n}}\right)$ (3.44)

##  4\. Proof of the CLT for unimodular lattices

As mentioned earlier we denote the set of all unimodular lattices of
$\mathbb{R}^{l}$ by $X_{l},$ where $l=m+n.$ Let
$h:\mathbb{R}^{l}\rightarrow\mathbb{R}$ be a Borel measurable function with
compact support. The Siegel transform of $h$ is
$\hat{h}:X_{l}\rightarrow\mathbb{R}$ defined by

$\hat{h}(\Lambda)=\sum_{\lambda\in\Lambda\setminus\\{0\\}}h(\lambda)\quad\text{for}\,\,\Lambda\in
X_{l}.$ (4.1)

We recall Siegel’s integral formula and Roger’s second moment formula for the
space of unimodular lattices from [20, 17].

######  Proposition 4.1.

Let $h:\mathbb{R}^{l}\rightarrow\mathbb{R}$ be a bounded Riemann integrable
function with compact support and
$F:\mathbb{R}^{l}\times\mathbb{R}^{l}\rightarrow\mathbb{R}$ be a non-negative
measurable function. Then

$\int_{X_{l}}\hat{h}(\Lambda)\,\,d\mu_{l}(\Lambda)=\int_{\mathbb{R}^{l}}h(\vec{\mathbf{z}})\,\,d\vec{\mathbf{z}}$
(4.2)

and

$\int_{X_{l}}\sum_{\vec{\mathbf{z_{1}}},\vec{\mathbf{z_{2}}}\in
P(\mathbb{Z}^{l})}F(g\vec{\mathbf{z_{1}}},g\vec{\mathbf{z_{2}}})\,\,d\mu_{l}(g\mathbb{Z}^{l})=\zeta(l)^{-2}\int_{\mathbb{R}^{l}\times\mathbb{R}^{l}}F(\vec{\mathbf{z_{1}}},\vec{\mathbf{z_{2}}})d\vec{\mathbf{z_{1}}}d\vec{\mathbf{z_{2}}}+\zeta(l)^{-1}\int_{\mathbb{R}^{l}}\left(F(\vec{\mathbf{z}},\vec{\mathbf{z}})+F(\vec{\mathbf{z}},\vec{\mathbf{-z}})\right)d\vec{\mathbf{z}},$
(4.3)

where $\zeta$ denotes the Riemann’s $\zeta$-function, $P(\mathbb{Z}^{l})$
denotes the the set of all primitive integral vectors in $\mathbb{Z}^{l}$ and
$d\vec{\mathbf{z}}$ denotes the Lebesgue measure on $\mathbb{R}^{l}$

Again by arguments similar to Section 2, it is easy to see that Theorem 1.1 is
equivalent to the following theorem.

######  Theorem 4.2.

Under the hypothesis of Theorem 1.1,

$G_{M}^{(\varepsilon,L)}:=\frac{1}{\sqrt{M}}\left(\sum_{k=0}^{M-1}\hat{h_{\varepsilon}}^{(L)}\circ
c_{0}^{k}-M\int_{X_{l}}\hat{h_{\varepsilon}}^{(L)}\,\,d\mu_{l}\right)\rightarrow
N(0,\eta^{2}\sigma_{u}^{2})\,\,\text{in distribution
as}\,\,M\rightarrow\infty,$ (4.4)

where $\hat{h_{\varepsilon}}^{(L)}=\hat{h_{\varepsilon}}\eta_{L}\in
C_{c}^{\infty}(X_{l})$ are smooth approximations to $\hat{\chi}$,
$\sigma_{u}^{2}$ is given by Theorem 1.1 and
$\eta=\operatorname{vol}(\Omega_{2})^{1/2}$.

By the CLT criteria of Fréchet and Shohat, proving Theorem 4.2 amounts to show
that for some choice of parameters $\varepsilon$ and $L$

$\sigma^{2}:=\lim_{M\rightarrow\infty}\|G_{M}^{(\varepsilon,L)}\|_{2}^{2}<\infty$
(4.5)

$\lim_{M\rightarrow\infty}\text{Cum}_{[r]}\left(G_{M}^{(\varepsilon,L)}\right)=0\quad\text{for
all}\,\,r\geq 3.$ (4.6)

The computation of cumulant (i.e., Equation (4.6)) is analogous to the affine
case (with appropriate modifications) discussed in detail in Section 2. Hence
we skip the details here. Rather we do the computation of the variance.

Computation of variance: By an easy reduction we have that

$\sigma^{2}=\lim_{M\rightarrow\infty}\|G_{M}^{(\varepsilon,L)}\|_{2}^{2}=\lim_{M\rightarrow\infty}\|G_{M}\|_{2}^{2}<\infty,$
(4.7)

where

$G_{M}=\frac{1}{\sqrt{M}}\sum_{k=0}^{M-1}\phi_{k}\quad\text{and}\quad\phi_{k}=\hat{\chi}\circ
c_{0}^{k}-\operatorname{vol}(\Omega_{2}).$ (4.8)

Also arguments similar to Section 2 give us

$\sigma^{2}=\lim_{M\rightarrow\infty}\|H_{M}\|_{2}^{2}=\sum_{k=-\infty}^{\infty}\int_{X_{l}}\phi_{k}\phi_{0}\,\,d\mu_{l}=\sum_{k=-\infty}^{\infty}\left(\int_{X_{l}}\hat{\chi}\cdot(\hat{\chi}\circ
c_{0}^{k})\,\,d\mu_{l}-\operatorname{vol}(\Omega_{2})^{2}\right).$ (4.9)

Now by applying Roger’s second order moment formula to the function

$F_{k}(\vec{\mathbf{z_{1}}},\vec{\mathbf{z_{2}}}):=\sum_{s_{1},s_{2}\in\mathbb{N}}\chi(s_{1}c_{0}^{k}\vec{\mathbf{z_{1}}})\chi(s_{2}\vec{\mathbf{z_{2}}}),\,\,\,\,(\vec{\mathbf{z_{1}}},\vec{\mathbf{z_{2}}})\in\mathbb{R}^{l}\times\mathbb{R}^{l},$
(4.10)

we get

$\displaystyle\int_{X_{l}}\hat{\chi}\cdot(\hat{\chi}\circ
c_{0}^{k})\,\,d\mu_{l}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\mathbf{z_{1}}},\vec{\mathbf{z_{2}}}\in
P(\mathbb{Z}^{l})}F_{k}(g\vec{\mathbf{z_{1}}},g\vec{\mathbf{z_{2}}})\,\,d\mu_{l}(g\mathbb{Z}^{l})$
$\displaystyle=$
$\displaystyle\operatorname{vol}(\Omega_{2})^{2}+\zeta(l)^{-1}\sum_{s_{1},s_{2}\in\mathbb{N}}\left(\int_{\mathbb{R}^{l}}\chi(s_{1}c_{0}^{k}\vec{\mathbf{z}})\chi(s_{2}\vec{\mathbf{z}})\,\,d\vec{\mathbf{z}}+\int_{\mathbb{R}^{l}}\chi(s_{1}c_{0}^{k}\vec{\mathbf{z}})\chi(-s_{2}\vec{\mathbf{z}})\,\,d\vec{\mathbf{z}}\right)$
$\displaystyle=$
$\displaystyle\operatorname{vol}(\Omega_{2})^{2}+2\zeta(l)^{-1}\sum_{s_{1},s_{2}\in\mathbb{N}}\left(\int_{\mathbb{R}^{l}}\chi(s_{1}c_{0}^{k}\vec{\mathbf{z}})\chi(s_{2}\vec{\mathbf{z}})\,\,d\vec{\mathbf{z}}\right),$
(4.12)

since

$\displaystyle\zeta(l)^{-2}\int_{\mathbb{R}^{l}\times\mathbb{R}^{l}}F_{k}(\vec{\mathbf{z_{1}}},\vec{\mathbf{z_{2}}})\,\,d\vec{\mathbf{z_{1}}}\,\,d\vec{\mathbf{z_{2}}}$
$\displaystyle=$
$\displaystyle\zeta(l)^{-2}\sum_{s_{1},s_{2}\in\mathbb{N}}\operatorname{vol}\left(\frac{1}{s_{1}}c_{0}^{-k}\Omega_{2}\right)\operatorname{vol}\left(\frac{1}{s_{2}}\Omega_{2}\right)$
(4.13) $\displaystyle=$
$\displaystyle\zeta(l)^{-2}\operatorname{vol}(\Omega_{2})^{2}\sum_{s_{1},s_{2}\in\mathbb{N}}\frac{1}{s_{1}^{l}}\frac{1}{s_{2}^{l}}$
(4.14) $\displaystyle=$ $\displaystyle\operatorname{vol}(\Omega_{2})^{2}.$
(4.15)

Therefore, we have

$\displaystyle\sigma^{2}$ $\displaystyle=$ $\displaystyle
2\zeta(l)^{-1}\sum_{k=-\infty}^{\infty}\sum_{s_{1},s_{2}\in\mathbb{N}}\left(\int_{\mathbb{R}^{l}}\chi(s_{1}c_{0}^{k}\vec{\mathbf{z}})\chi(s_{2}\vec{\mathbf{z}})\,\,d\vec{\mathbf{z}}\right)$
(4.16) $\displaystyle=$ $\displaystyle
2\zeta(l)^{-1}\sum_{s_{1},s_{2}\in\mathbb{N}}\int_{\mathbb{R}^{l}}\left(\sum_{k=-\infty}^{\infty}\chi\circ
c_{0}^{k}\right)(s_{1}\vec{\mathbf{z}})\,\,\chi(s_{2}\vec{\mathbf{z}})\,\,d\vec{\mathbf{z}}$
(4.17) $\displaystyle=$ $\displaystyle 2^{m+1}\zeta(l)^{-1}(\log 2)c_{1}\dots
c_{n}\omega_{n}\sum_{s_{1},s_{2}\in\mathbb{N}}\max\\{s_{1},s_{2}\\}^{-l},\,\,\text{by
using}\,\,\eqref{equ:xi_Omega_vol}$ (4.18) $\displaystyle=$ $\displaystyle
2^{m+1}\zeta(l)^{-1}(\log 2)c_{1}\dots
c_{n}\omega_{n}\left(\sum_{s=1}^{\infty}s_{1}^{-l}+2\sum_{1\leq
s_{1}<q}s_{2}^{-l}\right)$ (4.19) $\displaystyle=$ $\displaystyle
2^{m+1}\zeta(l)^{-1}(\log 2)c_{1}\dots
c_{n}\omega_{n}\left(\zeta(l)+2\sum_{s_{2}\geq
1}\frac{s_{2}-1}{s_{2}^{l}}\right)$ (4.20) $\displaystyle=$ $\displaystyle
2^{m+1}\zeta(l)^{-1}(\log 2)c_{1}\dots c_{n}\omega_{n}(2\zeta(l-1)-\zeta(l))$
(4.21) $\displaystyle=$ $\displaystyle 2^{m+1}(\log 2)c_{1}\dots
c_{n}\omega_{n}\left(\frac{2\zeta(l-1)}{\zeta(l)}-1\right).$ (4.22)

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