# A central limit theorem for Hilbert Modular Forms

Jishu Das  Indian Institute of Science Education and Research Pune, Dr. Homi
Bhabha Road, Pune 411008<EMAIL_ADDRESS>and  Neha Prabhu
Sai University, One Hub Road, Paiyanur, Tamil Nadu 603104
<EMAIL_ADDRESS>
###### Abstract.

For a prime ideal $\mathfrak{p}$ in a totally real number field $L$ with the
adele ring $\mathbb{A}$, we study the distribution of angles
$\theta_{\pi}(\mathfrak{p})$ coming from Satake parameters corresponding to
unramified $\pi_{\mathfrak{p}}$ where $\pi_{\mathfrak{p}}$ comes from a global
$\pi$ ranging over a certain finite set $\Pi_{\underline{k}}(\mathfrak{n})$ of
cuspidal automorphic representations of GL${}_{2}(\mathbb{A})$ with trivial
central character. For such a representation $\pi$, it is known that the
angles $\theta_{\pi}(\mathfrak{p})$ follow the Sato-Tate distribution. Fixing
an interval $I\subseteq[0,\pi]$, we prove a central limit theorem for the
number of angles $\theta_{\pi}(\mathfrak{p})$ that lie in $I$, as
$\mathrm{N}(\mathfrak{p})\to\infty$. The result assumes $\mathfrak{n}$ to be a
squarefree integral ideal, and that the components in the weight vector
$\underline{k}$ grow suitably fast as a function of $x$.

###### 2020 Mathematics Subject Classification:

Primary: 11F41, 11F72, Secondary: 11F30

##  1\. Introduction

The statistics of eigenvalues of Hecke operators have been a topic of interest
for a few decades. More recently, following the series of papers which settled
the Sato-Tate conjecture in various settings, such as [Tay08], [BLGG11]
[BLGG11], the study of error terms in these theorems has received significant
attention, see [PS19, BP19, Tho21, HIJTS22] for example. This article
investigates the statistics of the error term in the Sato-Tate theorem for
Hilbert Modular Forms, building on a similar study conducted in [PS19], which
we briefly describe. Let $\mathcal{F}_{N,k}$ be the set of normalized non-CM
cusp forms of weight $k$ and level $N$ that are also eigenforms for Hecke
operators $T_{n}$ acting on spaces of cusp forms $S(N,k)$. In particular, any
$f\in\mathcal{F}_{N,k}$ has a Fourier expansion

$f(z)=\sum\limits_{n=1}^{\infty}n^{\frac{k-1}{2}}a_{f}(n)q^{n}$

with $a_{f}(1)=1$ and $q=e^{2\pi iz}$. By the work of Deligne, it is known
that for $p$ prime with $\gcd(p,N)=1$, the sequence $a_{f}(p)$ lies in
$[-2,2]$ and the Sato-Tate theorem reveals the distribution of this sequence.
That is, for a fixed $f\in\mathcal{F}_{N,k}$, if we set
$a_{f}(p)=2\cos\theta_{f}(p)$, and fix an interval $I\subseteq[0,\pi]$ and let

$N_{I}(f,x):=\\#\\{p\leq x:\gcd(p,N)=1,\theta_{f}(p)\in I\\},$

then

$\lim\limits_{x\to\infty}\frac{N_{I}(f,x)}{\pi(x)}=\int_{I}\mu_{\infty}(t)~{}dt.$

Here, $\pi(x)$ denotes the number of primes not exceeding $x$, and
$\mu_{\infty}(t)=\frac{2}{\pi}\sin^{2}t$ is the function associated with the
Sato-Tate measure of the interval $I$. In [PS19], it was shown that under
suitable growth conditions of the weight $k=k(x)$, the error term in the above
asymptotic statement exhibits a Gaussian distribution when one averages over
$f$ in $\mathcal{F}_{N,k}$. In this article, we prove the analogous result in
the Hilbert modular setting.

Let $L$ denote a totally real field of degree $d$ over $\mathbb{Q}$ and
$\mathcal{O}$ be its ring of integers. Let $\nu=\nu_{\mathfrak{p}}$ be the
discrete valuation associated with the prime ideal $\mathfrak{p}$ in
$\mathcal{O}.$ The local field $L_{\nu}$ is the completion of $L$ with respect
to the topology induced by $\nu$, and $\mathcal{O}_{\nu}$ denotes the ring of
integers in the local field $L_{\nu}.$ Let $\mathbb{A}$ denote the Adele ring
of $\mathbb{Q}$ and $\mathbb{A}_{f}$ denote the finite adeles. Let $G$ denote
the algebraic group that is the Weil restriction of scalars of
$\text{GL}_{2/L}$ from $L$ to $\mathbb{Q}$. Let $G_{f}=G(\mathbb{A}_{f})$ and
$G_{\infty}=G(\mathbb{R})$. For an integral ideal $\mathfrak{n}$, let
$K_{0}(\mathfrak{n})$ be the congruence subgroup of level $\mathfrak{n}.$ If
the prime factorisation of $\mathfrak{n}$ is given by
$\mathfrak{n}=\mathfrak{q}_{1}^{a_{1}}\,\dots\,\mathfrak{q}_{r}^{a_{r}},$ then
the congruence subgroup $K_{0}(\mathfrak{n})$ is defined as

$K_{0}(\mathfrak{n})=\Bigg{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\prod_{\nu}GL_{2}(\mathcal{O}_{\nu})\,\Big{|}\,c_{\mathfrak{q}_{i}}\equiv
0\,\text{mod}\,\mathfrak{q}_{i}^{a_{i}}\,\text{for
all}\,i=1,\dots,r\Bigg{\\}}.$

Let the weight be given by the $d$-tuple $\underline{k}=(k_{1},\dots,k_{d})$
where $k_{i}$ is an even integer and $k_{i}\geq 4$ for all $i=1,\dots,d.$ Let
$\mathcal{A}^{\text{cusp}}_{\underline{k}}(G(\mathbb{Q})\backslash
G(\mathbb{A}))$ denote the space of cuspidal automorphic forms on
$G(\mathbb{A})$ with trivial central character (see for instance [BJ79,
Section 4] for definitions). The group $G(\mathbb{A})$ acts on
$\mathcal{A}^{\text{cusp}}_{\underline{k}}(G(\mathbb{Q})\backslash
G(\mathbb{A}))$ by right translations. An irreducible representation $\pi$ of
$G(\mathbb{A})$ is called a cuspidal automorphic representation if it is
isomorphic to a subrepresentation of
$\mathcal{A}^{\text{cusp}}_{\underline{k}}(G(\mathbb{Q})\backslash
G(\mathbb{A})).$ We have the following decomposition of
$\pi=\pi_{f}\otimes\pi_{\infty}$ where $\pi_{f}$ and $\pi_{\infty}$ are
representations of $G_{f}$ and $G_{\infty}$ respectively.

Let $\Pi_{\underline{k}}(\mathfrak{n})$ be the set of cuspidal unitary
automorphic representations $\pi$ with respect to $G$ such that

  * •

$\pi_{f}$ has a $K_{0}(\mathfrak{n})$ fixed vector,

  * •

$\pi_{\infty}=\otimes_{i=1}^{d}D_{k_{i}-1},$ where $D_{k}$ is the discrete
series representation of $\text{GL}_{2}(\mathbb{R})$ with minimal $K$-type of
weight $k+1.$

The set $\Pi_{\underline{k}}(\mathfrak{n})$ is finite (see (2.4)). Let
$\pi\in\Pi_{\underline{k}}(\mathfrak{n})$ and consider $\mathfrak{p}$, a prime
ideal in $L$ for which $\pi_{\mathfrak{p}}$ is unramified. The Satake
parameter associated with $\pi_{\mathfrak{p}}$ is a conjugacy class

$\begin{pmatrix}e^{i\pi\theta_{\pi}(\mathfrak{p})}&\\\
&e^{-i\pi\theta_{\pi}(\mathfrak{p})}\end{pmatrix}\in\text{SU(2)}/\sim$

with $\theta_{\pi}(\mathfrak{p})\in[0,1].$ For the classical setting of
$L=\mathbb{Q}$ and a classical eigenform $f,$ let $\pi(f)$ denote the
automorphic representation associated with $f$. The angles $\theta_{f}(p)$
considered in [PS19] are exactly the $\theta_{\pi}(\mathfrak{p})$ with
$\mathfrak{p}=\langle p\rangle.$

We set

$\pi_{L}(x)=\\#\\{\mathfrak{p}:\mathfrak{p}\text{ prime ideal in $L$ with
}\mathfrak{p}\nmid\mathfrak{n},\mathrm{N}(\mathfrak{p})\leq x\\}.$

For an interval $I=[\alpha,\beta]\subseteq[0,\pi]$, the Sato-Tate theorem
studies the distribution of

$\displaystyle
N_{I}(\pi,x):=\sum\limits_{\begin{subarray}{c}\mathrm{N}(\mathfrak{p})\leq
x\\\
\mathfrak{p}\nmid\mathfrak{n}\end{subarray}}\chi_{I}\left(\theta_{\pi}(\mathfrak{p})\right)$

and from the work of Barnet-Lamb, Gee and Geraghty [BLGG11], it is known that

(1.1) $N_{I}(\pi,x)\sim\pi_{L}(x)\mu_{\infty}(I).$

We are interested in studying the statistics of the error term in this
theorem. To be precise, let $\phi$ be a complex-valued function defined on
$\Pi_{\underline{k}}(\mathfrak{n})$. We denote the average

$\langle\phi(\pi)\rangle:=\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}\phi(\pi).$

The task then, is to study the behaviour of the moments

(1.2) $\langle(N_{I}(\pi,x)-\pi_{L}(x)\mu_{\infty}(I))^{r}\rangle$

for each $r\in\mathbb{N}$. We prove

######  Theorem 1.1.

Consider a family $\Pi_{\underline{k}}(\mathfrak{n})$ for fixed squarefree
level $\mathfrak{n}$ and even weights $\underline{k}=\underline{k}(x)$ such
that $\frac{\sum_{i=1}^{d}\log k_{i}}{\sqrt{x}\log x}\to\infty$ as
$x\to\infty$. Fix an interval $I\subseteq[0,\pi]$. Then for any continuous
real-valued function $g$ on $\mathbb{R}$, we have

$\lim\limits_{x\to\infty}g\left(\frac{N_{I}(\pi,x)-\pi_{L}(x)\mu_{\infty}(I)}{\sqrt{\pi_{L}(x)(\mu_{\infty}(I)-\mu_{\infty}(I)^{2})}}\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}g(t)e^{-\frac{t^{2}}{2}}~{}dt.$

The motivation for the above result comes from ideas in probability theory.
For a given interval $I\subseteq[0,\pi]$ and a representation $\pi$ in
$\Pi_{\underline{k}}(\mathfrak{n})$, consider a prime ideal $\mathfrak{p}$
such that $\pi_{\mathfrak{p}}$ is unramified. Note that since
$\mathfrak{p}\nmid\mathfrak{n}$, choosing $\pi$ with a $K_{0}(\mathfrak{n})$
fixed vector gives us that the representations $\pi_{\mathfrak{p}}$ are
unramified. Let

$X_{\mathfrak{p},\pi}=\chi_{I}\left(\theta_{\pi}(\mathfrak{p})\right).$

Consider a non-archimedean valuation $\nu$ corresponding to $\mathfrak{p}$ and
set

$d\mu_{\nu}(\theta)=\frac{\mathrm{N}(\mathfrak{p})+1}{\left(\mathrm{N}(\mathfrak{p})^{\frac{1}{2}}+\mathrm{N}(\mathfrak{p})^{-\frac{1}{2}}\right)^{2}-4\cos^{2}\theta}d\mu_{\infty}(\theta),$

where $d\mu_{\infty}(\theta)=\frac{2}{\pi}\sin^{2}\theta~{}d\theta.$ From the
work of Li [Li09], we know that

$\langle X_{\mathfrak{p},\pi}\rangle\sim\mu_{\nu}(I)$

if we assume appropriate growth conditions on $\underline{k}$ or
$\mathfrak{n}$. More generally, Lau-Li-Wang [LLW14, Theorem 1.1] prove an
effective joint distribution result for these angles. Using their result, we
infer that if $I_{1},\ldots,I_{h}$ are each intervals in $[0,\pi]$ and
$\mathfrak{p}_{1},\ldots,\mathfrak{p}_{h}$ are distinct prime ideals that are
relatively prime to $\mathfrak{n},$ then

$\langle X_{\mathfrak{p}_{1},\pi}\cdots
X_{\mathfrak{p}_{h},\pi}\rangle\sim\prod_{j=1}^{T}\mu_{\nu_{j}}(I_{j})$

under suitable growth conditions on $\underline{k}$. Here, $\nu_{j}$ is the
valuation corresponding to $\mathfrak{p}_{j}$. Thus, viewed as random
variables, the $X_{\mathfrak{p},\pi}$ have distributions that depend on
$\mathfrak{p}$, but behave like independent random variables for large enough
$\underline{k}$.

It is then natural to investigate the distribution of the sum of random
variables $\sum_{\mathrm{N}(\mathfrak{p})\leq x}X_{\mathfrak{p},\pi}$.
Firstly, we view $\mu_{\nu}(I)$ to be $\mbox{E}[X_{\mathfrak{p},\pi}]$. Then a
guess for the variance would be

$\text{Var}[X_{\mathfrak{p},\pi}]=\mbox{E}[X_{\mathfrak{p},\pi}^{2}]-\mbox{E}[X_{\mathfrak{p},\pi}]^{2}=\mu_{\nu}(I)-\mu_{\nu}(I)^{2}.$

However, keeping (1.1) in mind, and observing that $d\mu_{\nu}(\theta)\to
d\mu_{\infty}(\theta)$ as $\mathrm{N}(\mathfrak{p})\to\infty$, the quantity
$\pi_{L}(x)\mu_{\infty}(I)$ is a reasonable expression for the expectation of
the sum of random variables $\sum_{\mathrm{N}(\mathfrak{p})\leq
x}X_{\mathfrak{p},\pi}$ for $x$ large enough. Similarly, the variance of the
sum of random variables is heuristically
$\pi_{L}(x)(\mu_{\infty}(I)-\mu_{\infty}(I)^{2})$ as $x\to\infty$. With these
ideas in mind, we are motivated to find necessary conditions for a central
limit theorem to hold. This also adds to the literature on central limit
theorems for Hecke eigenvalues in various settings, for example [Nag06],
[MP20], [LNW19] and [FW21].

The proof of Theorem 1.1 adapts the underlying technique used in the proof of
[PS19, Theorem 1.1], namely the method of moments applied to quantities
approximating the moments (1.2), and the Eichler-Selberg trace formula.
However, the proof of Theorem 1.1 is cleaner, and is achieved by two
processes. First, we expand the approximating functions using a basis of
Chebyshev polynomials. Secondly, we simplify the proof of the theorem
pertaining to the calculation of higher moments. This proof is adapted from
[SWZ], who worked out a simplification of the theorem on higher moments in
[PS19]. This is achieved by expressing the main term of the analogous trace
formula, given in (2.3), as an integral involving Chebyshev polynomials (see
Lemma 5.2).

The structure of the paper is as follows. Section 2 covers the properties of
Chebyshev polynomials and Beurling-Selberg polynomials needed to follow the
proof of the theorem. Section 3 gives the details of the first-moment
calculation, culminating in the average Sato-Tate result for Hilbert Modular
Forms. After explaining the outline of the proof of Theorem 1.1 in Section 4,
we give the details of the simplified proof of higher moments in Section 5. We
conclude the article by stating a smooth version of Theorem 4.1 that should
hold, resulting in a smooth version of the central limit theorem with weaker
growth conditions on $\underline{k}=\underline{k}(x)$.

Acknowledgements The first named author is supported by a Ph.D. fellowship
from CSIR. The second named author’s research is supported by a DST-INSPIRE
Faculty Fellowship from the Government of India and acknowledges Savitribai
Phule Pune University, where she was working when this work began. The authors
are grateful to Baskar Balasubramanyam and Kaneenika Sinha for their valuable
comments on a previous version of this article.

##  2\. Preliminaries

###  2.1. Chebyshev polynomials

The Chebyshev polynomials of the second kind, denoted by $U_{n}$, are defined
by the recurrence relations

$\displaystyle U_{0}(x)$ $\displaystyle=1$ $\displaystyle U_{1}(x)$
$\displaystyle=2x$ $\displaystyle U_{n}(x)$
$\displaystyle=2x\,U_{n-1}(x)-U_{n-2}(x)\qquad\text{for }n\geq 2.$

These polynomials form an orthonormal basis with respect to the Sato-Tate
measure. More explicitly, the following holds. For non-negative integers $m,n$

(2.1)
$\displaystyle\frac{2}{\pi}\int_{0}^{\pi}U_{m}(\cos\theta)U_{n}(\cos\theta)\sin^{2}\theta~{}d\theta=\begin{cases}1&\text{
if }m=n\\\ 0&\text{ otherwise.}\end{cases}$

The following recursive relations also hold. For non-negative integers $m$ and
$n$ with $m\geq n$,

(2.2) $U_{m}(x)U_{n}(x)=\sum\limits_{k=0}^{n}U_{m+n-2k}(x).$

###  2.2. A trace formula

We state the version of Arthur’s trace formula proved in [LLW14] that we will
frequently use. This also appears in [BS19].

######  Proposition 2.1 ([BS19], Proposition 18).

Let $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{h}$ be distinct primes coprime to
squarefree $\mathfrak{n}$. Let $\underline{m}=(m_{1},\ldots,m_{h})$ be a tuple
of non-negative integers, and
$\mathfrak{a}=\mathfrak{p}_{1}^{m_{1}}\cdots\mathfrak{p}_{h}^{m_{h}}$. Then

(2.3)
$\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}\prod_{i=1}^{h}U_{m_{i}}(\cos\theta_{\pi}(\mathfrak{p}_{i}))=C_{L}\mathrm{N}(\mathfrak{n})\delta_{2\mid\underline{m}}\prod_{i=1}^{d}\frac{k_{i}-1}{4\pi}\mathrm{N}(\mathfrak{a})^{-\frac{1}{2}}+O\left(\mathrm{N}(\mathfrak{n})^{\epsilon}\mathrm{N}(\mathfrak{a})^{\frac{3}{2}}\right).$

Here,

  * •

$C_{L}$ is a constant that depends only on the number field $L$.

  * •

The quantity $\delta_{2\mid\underline{m}}$ is equal to one if all the $m_{i}$
are even, and zero otherwise.

As an immediate consequence, we have

(2.4)
$\\#\Pi_{\underline{k}}(\mathfrak{n})=C_{L}\mathrm{N}(\mathfrak{n})\prod_{i=1}^{d}\frac{k_{i}-1}{4\pi}+O\left(\mathrm{N}(\mathfrak{n})^{\epsilon}\right).$

######  Remark.

For simplicity, we stick to squarefree level $\mathfrak{n}$ keeping
Proposition 2.1 in mind. A similar result may be obtained by taking
$\mathfrak{n}$ not necessarily squarefree, using [[LLW14], Theorem 6.3].

We also state well-known results on sums of powers of prime ideal norms that
will be useful.

######  Lemma 2.1.

We have

(2.5) $\sum_{\mathrm{N}(\mathfrak{p})\leq
x}\frac{1}{\mathrm{N}(\mathfrak{p})}=\log\log x+O_{L}(1)$

and

(2.6) $\sum_{r\geq 2}\sum_{\mathrm{N}(\mathfrak{p})\leq
x}\frac{1}{\mathrm{N}(\mathfrak{p})^{r}}=O_{L}(1).$

###### Proof.

Equation (2.5) follows from Lemma 2.4 of [Ros99]. For (2.6), recall that for a
prime ideal $\mathfrak{p},$ the value of $\mathrm{N}(\mathfrak{p})=p^{t}$
where $\langle p\rangle=\mathfrak{p}\cap\mathbb{Z}$ and $1\leq t\leq d.$ Let
the ideal counting function be given by

$a_{l}=|\\{\mathfrak{m}\subset\mathcal{O}:\mathfrak{m}\,\text{ideal with
}\mathrm{N}(\mathfrak{m})=l.\\}|.$

Then, it is known that $a_{l}\leq\tau(l)^{d-1}$ (see [CN63, equation (68)]),
where $\tau$ is the usual divisor function.

Therefore,

$\displaystyle\sum_{r\geq 2}\sum_{\mathrm{N}(\mathfrak{p})\leq
x}\frac{1}{\mathrm{N}(\mathfrak{p})^{r}}$ $\displaystyle=\sum_{r\geq
2}\sum_{p\leq x}\sum_{t=1}^{d}\frac{a_{p^{t}}}{p^{tr}}\leq\sum_{r\geq
2}\sum_{p\leq x}\sum_{t=1}^{d}\frac{(t+1)^{d-1}}{p^{tr}}$
$\displaystyle\leq\sum_{r\geq 2}\sum_{p\leq
x}\frac{d(d+1)^{d-1}}{p^{r}}\ll_{L}\sum_{r\geq 2}\sum_{p\leq
x}\frac{1}{p^{r}}=O_{L}(1).$

∎

###  2.3. Beurling-Selberg polynomials

The main underlying technique used in this article is to approximate the
characteristic function by appropriate trigonometric polynomials of finite
degree. This technique has been used widely to obtain fluctuations in error
terms in equidistribution theorems, following the work of [FR10]. Here, we
state the properties that we need, and refer the reader to a detailed
exposition in [Mon94, Chapter 1]. Let
$J=[\alpha,\beta]\subseteq[-\frac{1}{2},\frac{1}{2}]$ and $M\geq 1$ be an
integer. There exist trigonometric polynomials $S_{J,M}^{+}(x)$ and
$S_{J,M}^{-}(x)$ of degree not exceeding $M$ such that for all
$x\in\mathbb{R}$,

$S_{J,M}^{-}(x)\leq\chi_{J}(x)\leq S_{J,M}^{+}(x),$

where $\chi_{J}$ is the usual indicator function of the interval $J$. While
the explicit definition of these polynomials can be found in [Mon94], we will
heavily use properties of the coefficients in their Fourier expansions for our
calculations. Before we describe these properties, we fix some notation. Let
$e(t):=e^{2\pi it}.$ Then, the Fourier expansion of $\chi_{I}$ is given by

$\chi_{J}(x)=\sum_{n\in\mathbb{Z}}\hat{\chi}_{J}(n)e(nx),\qquad\qquad\hat{\chi}_{J}(n)=\int_{J}e(-nt)~{}dt.$

Note that

(2.7)
$\hat{\chi}_{J}(0)=\beta-\alpha,\qquad\hat{\chi}_{J}(n)=\frac{e(-n\alpha)-e(-n\beta)}{2\pi
in}\quad\text{ for }|n|\geq 1.$

The Beurling-Selberg polynomials $S_{J,M}^{+}$ and $S_{J,M}^{-}$ are good
approximations of the indicator function and this is evident from the
properties of its Fourier coefficients. For $0\leq|m|\leq M$,

(2.8) $\hat{S}_{J,M}^{\pm}(m)=\hat{\chi}_{J}(m)+O\left(\frac{1}{M+1}\right)$

and $\hat{S}_{J,M}^{\pm}(m)=0$ for $|m|>M$. Since we are interested in finer
statistics of the sequence of angles $\\{\theta_{\pi}(\mathfrak{p})\\}$, which
are equidistributed with respect to the measure $\frac{2}{\pi}\sin^{2}\theta$,
it is useful to consider Fourier expansions of the polynomials using the
orthonormal basis of Chebyshev polynomials $U_{n}(\cos\theta)$. The following
lemma accomplishes this.

######  Lemma 2.2 ([RT17, Lemma 1.3]).

Let $I=[a,b]\subseteq[0,\pi]$, and let $M$ be a positive integer. There exist
trigonometric polynomials

$F^{\pm}_{I,M}(\theta)=\sum\limits_{m=0}^{M}\hat{F}^{\pm}_{I,M}(n)U_{m}(\cos\theta)$

such that for $0\leq\theta\leq\pi$, we have

$F^{-}_{I,M}(\theta)\leq\chi_{I}(\theta)\leq F^{+}_{I,M}(\theta).$

The key idea in the proof is to take $\alpha=\frac{a}{2\pi}$ and
$\beta=\frac{b}{2\pi}$ so that we now have an interval in
$[-\frac{1}{2},\frac{1}{2}]$, and study the properties of

$F^{\pm}_{I,M}(\theta)=S^{\pm}_{J,M}\left(\frac{\theta}{2\pi}\right)+S^{\pm}_{J,M}\left(-\frac{\theta}{2\pi}\right).$

The relation between the Fourier coefficients $\hat{S}_{J,M}^{\pm}(m)$ and
$\hat{F}_{I,M}^{\pm}(m)$ is the following. For $0\leq m\leq M$, let

(2.9)
$\hat{\mathcal{S}}_{J,M}^{\pm}(m)=\hat{S}_{J,M}^{\pm}(m)+\hat{S}_{J,M}^{\pm}(-m).$

Then

(2.10) $\displaystyle\hat{F}_{I,M}^{\pm}(m)$
$\displaystyle=\hat{\mathcal{S}}_{J,M}^{\pm}(m)-\hat{\mathcal{S}}_{J,M}^{\pm}(m+2).$

Moreover, using (2.7) and (2.8) in (2.9), we get that for $0<m\leq M$,

(2.11) $\displaystyle\hat{\mathcal{S}}_{J,M}^{\pm}(m)$
$\displaystyle=\frac{\sin(2\pi m\beta)-\sin(2\pi
m\alpha)}{m\pi}+O\left(\frac{1}{M+1}\right)$
$\displaystyle=\frac{\sin(mb)-\sin(ma)}{m\pi}+O\left(\frac{1}{M+1}\right),\quad\text{
and }$ (2.12) $\displaystyle\hat{\mathcal{S}}_{J,M}^{\pm}(0)$
$\displaystyle=2(\beta-\alpha)=\frac{a}{\pi}-\frac{b}{\pi}.$

Henceforth, to simplify notation we will fix the interval $I=[a,b]$ throughout
the rest of the article, and write $F_{M}^{\pm}$ and $\hat{F}_{M}^{\pm}$
without mentioning $I$ in the subscript.

We now record an important result about the Fourier coefficients
$\hat{F}_{M}^{\pm}(m)$.

######  Proposition 2.2.

(2.13) $\displaystyle\hat{F}_{M}^{\pm}(0)$
$\displaystyle=\mu_{\infty}(I)+O\left(\frac{1}{M+1}\right).$ (2.14)
$\displaystyle\sum_{m=1}^{M}\hat{F}_{M}^{\pm}(m)^{2}$
$\displaystyle=\mu_{\infty}(I)-\mu_{\infty}(I)^{2}+O\left(\frac{\log
M}{M}\right).$

###### Proof.

Equation (2.13) follows easily from noting that

$\displaystyle\hat{F}_{M}^{\pm}(0)$
$\displaystyle=\hat{\mathcal{S}}_{J,M}^{\pm}(0)-\hat{\mathcal{S}}_{J,M}^{\pm}(2)$
$\displaystyle=\frac{a}{\pi}-\frac{b}{\pi}-\frac{\sin(2b)-\sin(2a)}{2\pi}+O\left(\frac{1}{M+1}\right)$
$\displaystyle=\frac{2}{\pi}\int_{a}^{b}\sin^{2}\theta~{}d\theta+O\left(\frac{1}{M+1}\right)$

Equation (2.14) is less straightforward, and has been worked out in [Pra17,
Proposition 3.6.1]. ∎

##  3\. First Moment

Throughout the rest of this article, whenever we write
$\mathrm{N}(\mathfrak{p})\leq x$, in a sum, we will assume that the sum runs
over prime ideals $\mathfrak{p}\nmid\mathfrak{n}$.

Approximating the quantity $N_{I}(\pi,x)$ above and below by the Beurling-
Selberg polynomials, we have

$\sum\limits_{\mathrm{N}(\mathfrak{p})\leq
x}F^{-}_{M}(\theta_{\pi}(\mathfrak{p}))\leq
N_{I}(\pi,x)\leq\sum\limits_{\mathrm{N}(\mathfrak{p})\leq
x}F^{+}_{M}(\theta_{\pi}(\mathfrak{p})).$

Writing

$\sum\limits_{\mathrm{N}(\mathfrak{p})\leq
x}F^{\pm}_{M}(\theta_{\pi}(\mathfrak{p}))=\pi_{L}(x)\hat{F}^{\pm}_{M}(0)+\sum_{m=1}^{M}\hat{F}^{\pm}_{M}(m)\sum_{\mathrm{N}(\mathfrak{p})\leq
x}U_{m}(\cos(\theta_{\pi}(\mathfrak{p}))),$

and using (2.13), there exist constants $C$ and $D$ such that

$\displaystyle D\frac{\pi_{L}(x)}{M+1}+F^{-}(M,\pi)(x)\leq
N_{I}(\pi,x)-\pi_{L}(x)\mu_{\infty}(I)\leq
F^{+}(M,\pi)(x)+C\frac{\pi_{L}(x)}{M+1}$

where

(3.1)
$F^{\pm}(M,\pi)(x):=\sum_{m=1}^{M}\hat{F}_{M}^{\pm}(m)\sum_{\mathrm{N}(\mathfrak{p})\leq
x}U_{m}(\cos\theta_{\pi}(\mathfrak{p})).$

We compute the first moment by estimating

$\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}F^{\pm}(M,\pi)(x).$

We first focus on the upper bound:

$\displaystyle
N_{I}(\pi,x)\leq\pi_{L}(x)\hat{F}^{+}_{M}(0)+\sum_{m=1}^{M}\hat{F}^{+}_{M}(m)\sum_{\mathrm{N}(\mathfrak{p})\leq
x}U_{m}(\cos(\theta_{\pi}(\mathfrak{p}))).$

Since the first summand on the left-hand side is independent of $\pi$, in
order to compute the first moment we need to estimate

$\displaystyle\sum_{m=1}^{M}\hat{F}^{+}_{M}(m)\sum_{\mathrm{N}(\mathfrak{p})\leq
x}\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}U_{m}(\cos(\theta_{\pi}(\mathfrak{p})).$

Using (2.3) we see that this is

$\displaystyle=\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\sum_{\begin{subarray}{c}m=2\\\
m\,\textrm{even}\end{subarray}}^{M}\hat{F}_{M}^{+}(m)\sum_{\mathrm{N}(\mathfrak{p})\leq
x}\left(C_{L}\mathrm{N}(\mathfrak{n})\prod_{i=1}^{d-1}\frac{k_{i}-1}{4\pi}\mathrm{N}(\mathfrak{p})^{-\frac{m}{2}}+O(\mathrm{N}(\mathfrak{p})^{\frac{3m}{2}}\mathrm{N}(\mathfrak{n})^{\epsilon})\right)$
$\displaystyle\quad+O_{L}\left(\sum_{m=1}^{M}\left|\hat{F}_{M}^{+}(m)\right|\sum_{\mathrm{N}(\mathfrak{p})\leq
x}\frac{\mathrm{N}(\mathfrak{p})^{\frac{3m}{2}}\mathrm{N}(\mathfrak{n})^{\epsilon}}{\prod_{i=1}^{d-1}(k_{i}-1)}\right).$

Using Proposition 2.1 with $h=1$, the estimate
$\hat{F}^{+}_{M}(m)\ll\frac{1}{m}$ and Lemma 2.1, we see that

$\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\sum_{m=1}^{M}\hat{F}_{M}^{+}(m)\sum_{\mathrm{N}(\mathfrak{p})\leq
x}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}U_{m}(\cos(\theta_{\pi}(\mathfrak{p}))\ll_{L}\log\log
x+\frac{\pi_{L}(x)x^{\frac{3}{2}M}}{\prod_{i=1}^{d-1}k_{i}}.$

Keeping (2.13) in mind we therefore get,

$\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}N_{I}(\pi,x)-\pi_{L}(x)\mu_{\infty}(I)\ll_{L}\log\log
x+\frac{\pi_{L}(x)x^{\frac{3}{2}M}}{\prod_{i=1}^{d-1}k_{i}}+\frac{\pi_{L}(x)}{M+1}.$

On choosing $M=\left[\frac{2d\sum_{i=1}^{d-1}\log k_{i}}{3\log x}\right]$ for
some $0<d<1$, we have proved the following proposition.

######  Proposition 3.1.

Let $\sum_{i=1}^{d-1}\log k_{i}$ be a function of $x$. Then, for any
$I\subseteq[0,\pi]$ we have

$\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}N_{I}(\pi,x)=\pi_{L}(x)\mu_{\infty}(I)+O_{L}\left(\frac{\pi_{L}(x)\log
x}{\sum_{i=1}^{d-1}\log k_{i}}+\log\log x\right).$

Remark: The above proposition is the analogue of the effective average Sato-
Tate theorem for holomorphic cusp forms [PS19, Proposition 4.1] and Maass
forms [Wan14, Theorem 1.1].

##  4\. Outline of the proof of the main theorem.

We compute all higher moments by proving the following result.

######  Theorem 4.1.

Let $M=\lfloor\sqrt{\pi_{L}(x)}\log\log x\rfloor$ and $F^{\pm}(M,\pi)(x)$ be
as defined in (3.1). Suppose $\frac{\sum_{i=1}^{d}\log k_{i}}{\sqrt{x}\log
x}\to\infty$ as $x\to\infty$. Then,

(4.1)
$\lim_{x\to\infty}\left\langle\left(\frac{F^{\pm}(M,\pi)(x))}{\sqrt{\pi_{L}(x)}}\right)^{n}\right\rangle=\begin{cases}0&\text{
if }n\text{ is odd,}\\\
\frac{n!}{2^{\frac{n}{2}}\frac{n}{2}!}\left(\mu_{\infty}(I)-\mu_{\infty}(I)^{2}\right)^{\frac{n}{2}}&\text{
if }n\text{ is even.}\end{cases}$

Following the strategy in [PS19, Section 6], and making the necessary
modifications i.e., using the trace formula in Proposition 2.3 instead of the
Eichler-Selberg trace formula, one can prove the following analogue of [PS19,
Proposition 6.2]. The proof is very similar, so we omit details.

######  Proposition 4.1.

Let $I=[a,b]\subseteq[0,\pi]$ and $M=\lfloor\sqrt{\pi_{L}(x)}\log\log
x\rfloor$. Suppose $\frac{\sum_{i=1}^{d}\log k_{i}}{\sqrt{x}\log x}\to\infty$
as $x\to\infty$,

$\lim\limits_{x\to\infty}\left\langle\left|\frac{N_{I}(\pi,x)-\pi_{L}(x)\mu_{\infty}(I)-F^{\pm}(M,\pi)(x)}{\sqrt{\pi_{L}(x)(\mu_{\infty}(I)-\mu_{\infty}(I)^{2})}}\right|^{2}\right\rangle=0.$

This implies that under the conditions of the above proposition, the quantity

$\frac{F^{\pm}(M,\pi)(x)}{\sqrt{\pi_{L}(x)(\mu_{\infty}(I)-\mu_{\infty}(I)^{2})}}$

converges in mean square to

(4.2)
$\frac{N_{I}(\pi,x)-\pi_{L}(x)\mu_{\infty}(I)}{\sqrt{\pi_{L}(x)(\mu_{\infty}(I)-\mu_{\infty}(I)^{2})}}$

as $x\to\infty$. Since convergence in mean square implies convergence in
distribution (see [RES01, Chapter 6, Theorems 5 and 7]), and the normal
distribution is characterized by its moments, Theorem 4.1 immediately gives us
that the quantity in (4.2) follows the Gaussian distribution. This completes
the proof of Theorem 1.1.

##  5\. Higher Moments

In this section we give the details of the proof of Theorem 4.1. In order to
simplify calculations while computing higher moments, it is useful to express
the main term of the trace formula 2.3 using an integral. The following lemma
is Proposition 29.11 in [KL06], with the prime $p$ replaced with
$\mathrm{N}(\mathfrak{p})$. We prove it here for completeness.

######  Lemma 5.1.

For a non-archimedean valuation $\nu$ corresponding to a prime ideal
$\mathfrak{p}$, we define

(5.1)
$d\mu_{\nu}(\theta)=\frac{\mathrm{N}(\mathfrak{p})+1}{\left(\mathrm{N}(\mathfrak{p})^{\frac{1}{2}}+\mathrm{N}(\mathfrak{p})^{-\frac{1}{2}}\right)^{2}-4\cos^{2}\theta}d\mu_{\infty}(\theta),$

where $d\mu_{\infty}(\theta)=\frac{2}{\pi}\sin^{2}\theta~{}d\theta.$ Then,

(5.2)
$\int_{0}^{\pi}U_{m}(\cos\theta)d\mu_{\nu}(\theta)=\begin{cases}\mathrm{N}(\mathfrak{p})^{-\frac{m}{2}}&\text{
if $m$ is even}\\\ 0&\text{ if $m$ is odd}.\end{cases}$

###### Proof.

For $|t|<1$ and $x\in[-1,1]$, the generating function of the Chebyshev
polynomials of the second kind is given by

(5.3) $\sum_{n=0}^{\infty}U_{n}(x)t^{n}=\frac{1}{1-2xt+t^{2}}.$

Substituting $t=\pm\mathrm{N}(\mathfrak{p})^{-\frac{1}{2}}$ and adding, we
obtain

$\sum_{n=0}^{\infty}U_{2n}(x)\mathrm{N}(\mathfrak{p})^{-n}=\frac{\mathrm{N}(\mathfrak{p})+1}{\left(\mathrm{N}(\mathfrak{p})^{\frac{1}{2}}+\mathrm{N}(\mathfrak{p})^{-\frac{1}{2}}\right)^{2}-4x^{2}}.$

Denoting the above quantity by $u_{\mathfrak{p}}(x)$, and letting
$x=\cos\theta$, we see that

$\displaystyle\int_{0}^{\pi}U_{m}(\cos\theta)d\mu_{\nu}(\theta)$
$\displaystyle=\int_{0}^{\pi}U_{m}(\cos\theta)\left(\sum_{n=0}^{\infty}U_{2n}(x)\mathrm{N}(\mathfrak{p})^{-n}\right)d\mu_{\infty}(\theta)$
$\displaystyle=\begin{cases}\mathrm{N}(\mathfrak{p})^{-\frac{m}{2}}&\text{ if
$m$ is even}\\\ 0&\text{ if $m$ is odd},\end{cases}$

using (2.1), the orthogonality of the Chebyshev polynomials. ∎

As a consequence of Lemma 5.1, the following alternate expression for the
trace formula (2.3) holds.

######  Lemma 5.2.

Let $\underline{m}=(m_{1},\ldots,m_{h})$ be a tuple of non-negative integers.
Further, let $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{h}$ be distinct prime
ideals coprime to the squarefree ideal $\mathfrak{n}.$ Then,

(5.4)
$\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}\prod_{i=1}^{h}U_{m_{i}}(\cos\theta_{\pi}(\mathfrak{p}_{i}))=\prod_{i=1}^{h}\int_{0}^{\pi}U_{m_{i}}(\cos\theta)d\mu_{\nu_{i}}(\theta)+O_{L,\mathfrak{n}}\left(\frac{\prod_{i=1}^{h}\mathrm{N}(\mathfrak{p}_{i})^{\frac{3m_{i}}{2}}}{\prod_{i=1}^{d}k_{i}}\right).$

###### Proof.

This follows easily by using (5.2) in the main term of trace formula in (2.3)
and noting (2.4). ∎

We now proceed to calculate the higher moments
$\left\langle\left(\frac{F^{\pm}(M,\pi)(x))}{\sqrt{\pi_{L}(x)}}\right)^{n}\right\rangle$.
Observe that

(5.5)
$\displaystyle\left\langle\left(\frac{F^{\pm}(M,\pi)(x))}{\sqrt{\pi_{L}(x)}}\right)^{n}\right\rangle$
$\displaystyle=\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\frac{1}{\pi_{L}(x)^{\frac{n}{2}}}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}\left(\sum_{\mathrm{N}(\mathfrak{p})\leq
x}\sum_{m=1}^{M}\hat{F}^{\pm}_{M}(m)U_{m}(\cos\theta_{\pi}(\mathfrak{p}))\right)^{n}$

Let $Z^{\pm}_{M}(\theta)=\sum_{m=1}^{M}\hat{F}_{M}^{\pm}(m)U_{m}(\cos\theta)$.
Then

$\displaystyle\left(\sum_{\mathrm{N}(\mathfrak{p})\leq
x}\sum_{m=1}^{M}\hat{F}^{\pm}_{M}(m)U_{m}(\cos\theta_{\pi}(\mathfrak{p}))\right)^{n}$
$\displaystyle=\left(\sum_{\mathrm{N}(\mathfrak{p})\leq
x}Z_{M}^{\pm}(\theta_{\pi}(\mathfrak{p}))\right)^{n}$
$\displaystyle=\sum_{u=1}^{n}\sum_{(r_{1},\ldots,r_{u})}^{(1)}\frac{n!}{r_{1}!\cdots
r_{u}!}\frac{1}{u}\sum_{(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{u})}^{(2)}\prod_{i=1}^{u}Z_{M}^{\pm}(\theta_{\pi}(\mathfrak{p}_{i}))^{r_{i}},$

where

  * •

The sum $\sum\limits_{(r_{1},\ldots,r_{u})}^{(1)}$ is taken over tuples of
positive integers $(r_{1},\ldots,r_{u})$ such that $r_{1}+\cdots+r_{u}=n$.

  * •

The sum $\sum\limits_{(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{u})}^{(2)}$ is
taken over tuples of distinct prime ideals
$\mathfrak{p}_{1},\ldots,\mathfrak{p}_{u}$ coprime to $\mathfrak{n}$ with
$\mathrm{N}(\mathfrak{p}_{i})\leq x$ for each $i=1,\ldots,u$.

Let $[f(\theta)\bullet U_{n}(\cos\theta)]$ denote the $n$-th Fourier
coefficient when $f$ is expanded as a Fourier series using the orthogonal
basis of Chebyshev polynomials of the second kind. Explicitly,

$[f(\theta)\bullet
U_{n}(\cos\theta)]=\frac{2}{\pi}\int_{0}^{\pi}f(\theta)\,U_{n}(\cos\theta)\sin^{2}\theta~{}d\theta.$

Expanding each $Z_{M}^{\pm}(\theta_{\pi}(\mathfrak{p}_{i}))^{r_{i}}$ in this
way we have

$\displaystyle\prod_{i=1}^{u}Z_{M}^{\pm}(\theta_{\pi}(\mathfrak{p}_{i}))^{r_{i}}$
$\displaystyle=\prod_{i=1}^{u}\left(\sum_{m_{i}=1}^{Mr_{i}}\left[Z^{\pm}_{M}(\theta)^{r_{i}}\bullet
U_{m_{i}}(\cos\theta)\right]U_{m_{i}}(\cos\theta_{\pi}(\mathfrak{p}_{i}))\right)$
$\displaystyle=\sum_{(m_{1},\ldots,m_{u})}^{(3)}\prod_{i=1}^{u}\left[Z^{\pm}_{M}(\theta)^{r_{i}}\bullet
U_{m_{i}}(\cos\theta)\right]U_{m_{i}}(\cos\theta_{\pi}(\mathfrak{p}_{i})).$

Here, $\sum\limits_{(m_{1},\ldots,m_{u})}^{(3)}$ denote that the sum is taken
over tuples where each $m_{i}$ ranges from $1$ to $Mr_{i}$. Averaging over
$\pi\in\Pi_{\underline{k}}(\mathfrak{n})$, and using the trace formula given
in equation (5.4),

$\displaystyle\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}\prod_{i=1}^{u}Z_{M}^{\pm}(\theta_{\pi}(\mathfrak{p}_{i}))^{r_{i}}$
$\displaystyle=\sum_{(m_{1},\ldots,m_{u})}^{(3)}\prod_{i=1}^{u}\left[Z^{\pm}_{M}(\theta)^{r_{i}}\bullet
U_{m_{i}}(\cos\theta)\right]\frac{1}{\\#\Pi_{\underline{k}}(\mathfrak{n})}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}\prod_{i=1}^{u}U_{m_{i}}(\cos\theta_{\pi}(\mathfrak{p}_{i}))$
$\displaystyle=\sum_{(m_{1},\ldots,m_{u})}^{(3)}\prod_{i=1}^{u}\left[Z^{\pm}_{M}(\theta)^{r_{i}}\bullet
U_{m_{i}}(\cos\theta)\right]\left(\prod_{i=1}^{u}\int_{0}^{\pi}U_{m_{i}}(\cos\theta)d\mu_{\nu_{i}}(\theta)+O_{L,\mathfrak{n}}\left(\frac{\prod_{i=1}^{u}\mathrm{N}(\mathfrak{p}_{i})^{\frac{3}{2}m_{i}}}{\prod_{i=1}^{d}k_{i}}\right)\right)$
$\displaystyle=\prod_{i=1}^{u}\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{r_{i}}~{}d\mu_{\nu_{i}}(\theta)+O_{L,\mathfrak{n}}\left(\frac{1}{\prod_{i=1}^{d}k_{i}}\prod_{i=1}^{u}\sum_{m_{i}=1}^{Mr_{i}}\langle
Z_{M}^{\pm}(\cdot)^{r_{i}},U_{m_{i}}(\cos\cdot)\rangle\mathrm{N}(\mathfrak{p}_{i})^{\frac{3}{2}m_{i}}\right).$

Let us estimate the error term. Observe that
$Z_{M}^{\pm}(\theta)=F_{M}^{\pm}(\theta)-\hat{F}_{M}^{\pm}(0)$, so it is
absolutely bounded. Therefore, $[Z_{M}^{\pm}(\theta)^{r_{i}}\bullet
U_{m_{i}}(\cos\theta)]\ll m_{i}$ for each $i=1,\ldots,u$ using the trivial
bound $|U_{n}(\cos\theta)|\leq n+1.$ We also know that
$\mathrm{N}(\mathfrak{p}_{i})\leq x$, so

$\prod_{i=1}^{u}\sum_{m_{i}=1}^{Mr_{i}}\left[Z^{\pm}_{M}(\theta)^{r_{i}}\bullet
U_{m_{i}}(\cos\theta)\right]\mathrm{N}(\mathfrak{p}_{i})^{\frac{3}{2}m_{i}}\ll_{n}M^{2u}x^{\frac{3}{2}Mu}.$

Therefore, the higher moments

$\left\langle\left(\frac{F^{\pm}(M,\pi)(x))}{\sqrt{\pi_{L}(x)}}\right)^{n}\right\rangle$

are given by

(5.6)
$\displaystyle\frac{1}{\pi_{L}(x)^{\frac{n}{2}}}\sum_{u=1}^{n}\sum_{(r_{1},\ldots,r_{u})}^{(1)}\frac{n!}{r_{1}!\cdots
r_{u}!}\frac{1}{u}\sum_{(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{u})}^{(2)}\prod_{i=1}^{u}\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{r_{i}}~{}d\mu_{\nu}(\theta)+O_{L,n}\left(\frac{M^{2n}x^{\frac{3}{2}Mn}\pi_{L}(x)^{\frac{n}{2}}}{\prod_{i=1}^{d}k_{i}}\right).$

We now analyze the main term in the above equation. First, we prove:

######  Lemma 5.3.

Assume the notations introduced earlier. Then, the following hold.

(5.7)
$\displaystyle\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{r}~{}d\mu_{\nu}(\theta)$
$\displaystyle=\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{r}~{}d\mu_{\infty}(\theta)+O\left(\frac{1}{\mathrm{N}(\mathfrak{p})}\right)$
(5.8) $\displaystyle\int_{0}^{\pi}Z_{M}^{\pm}(\theta)~{}d\mu_{\nu}(\theta)$
$\displaystyle=O\left(\frac{1}{\mathrm{N}(\mathfrak{p})}\right)$ (5.9)
$\displaystyle\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{r}~{}d\mu_{\nu}(\theta)$
$\displaystyle=\begin{cases}\sum_{m=1}^{M}\hat{F}_{M}^{\pm}(m)^{2}+O\left(\frac{1}{\mathrm{N}(\mathfrak{p})}\right)&\text{
if }r=2\\\ O(1)&\text{ for }r>2.\end{cases}$

###### Proof.

Equation (5.7) follows from noting that $Z_{M}^{\pm}(\theta)$ is bounded and
that for any prime ideal $\mathfrak{p}$,

$\displaystyle\frac{2}{\pi}\frac{(\mathrm{N}(\mathfrak{p})+1)\sin^{2}\theta}{\left(\mathrm{N}(\mathfrak{p})^{\frac{1}{2}}+\mathrm{N}(\mathfrak{p})^{-\frac{1}{2}}\right)^{2}-4\cos^{2}\theta}$
$\displaystyle=\frac{2}{\pi}\sin^{2}\theta+O\left(\frac{1}{\mathrm{N}(\mathfrak{p})}\right).$

Equation (5.8) follows from letting $r=1$ in (5.7) and the orthogonality of
Chebyshev polynomials $U_{m}(\cos\theta)$. Finally,

$\displaystyle\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{2}~{}d\mu_{\nu}(\theta)$
$\displaystyle=\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{2}~{}d\mu_{\infty}(\theta)+O\left(\frac{1}{\mathrm{N}(\mathfrak{p})}\right)\quad\text{
using }\eqref{connecting mu_p to mu_infty}$
$\displaystyle=\int_{0}^{\pi}\left(\sum_{m=1}^{M}\hat{F}_{M}^{\pm}(m)U_{m}(\cos\theta)\right)^{2}~{}d\mu_{\infty}(\theta)+O\left(\frac{1}{\mathrm{N}(\mathfrak{p})}\right)$
$\displaystyle=\int_{0}^{\pi}\sum_{m_{1},m_{2}=1}^{M}\hat{F}_{M}^{\pm}(m_{1})\hat{F}_{M}^{\pm}(m_{2})\sum_{k=0}^{\min\\{m_{1},m_{2}\\}}U_{m_{1}+m_{2}-2k}(\cos\theta)~{}d\mu_{\infty}(\theta)+O\left(\frac{1}{\mathrm{N}(\mathfrak{p})}\right)$
$\displaystyle=\sum_{m=1}^{M}\hat{F}_{M}^{\pm}(m)^{2}+O\left(\frac{1}{\mathrm{N}(\mathfrak{p})}\right)$

using orthogonality. If $r>2$, the claimed estimate follows because
$Z^{\pm}_{M}$ is bounded. This proves (5.9). ∎

We now have the tools to work out the integrals in (5.6). We do so by dividing
the partitions into three types.
Case 1: If $(r_{1},\ldots,r_{u})=(2,\ldots,2)$, i.e., each part is equal to
$2$, then $n$ is even and $u=n/2$. In this case, the sum corresponding to this
partition is

$\displaystyle\frac{1}{\pi_{L}(x)^{\frac{n}{2}}}\frac{n!}{2^{\frac{n}{2}}\frac{n}{2}!}\sum_{(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{u})}^{(2)}\prod_{i=1}^{n/2}\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{2}~{}d\mu_{\nu}(\theta)$
$\displaystyle=\frac{1}{\pi_{L}(x)^{\frac{n}{2}}}\frac{n!}{2^{\frac{n}{2}}\frac{n}{2}!}\sum_{(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{u})}^{(2)}\prod_{i=1}^{n/2}\left(\sum_{m=1}^{M}\hat{F}_{M}^{\pm}(m)^{2}+O\left(\frac{1}{\mathrm{N}(\mathfrak{p}_{i})}\right)\right)$
$\displaystyle=\frac{n!}{2^{\frac{n}{2}}\frac{n}{2}!}\left(\sum_{m=1}^{M}\hat{F}_{M}^{\pm}(m)^{2}\right)^{\frac{n}{2}}+o(\pi_{L}(x)).$

Using equation (2.14) and noting that $M$ is an increasing function of $x$, we
conclude

(5.10)
$\lim\limits_{x\to\infty}\frac{1}{\pi_{L}(x)^{\frac{n}{2}}}\frac{n!}{2^{\frac{n}{2}}\frac{n}{2}!}\sum_{(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{u})}^{(2)}\prod_{i=1}^{n/2}\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{2}~{}d\mu_{\nu}(\theta)=\frac{n!}{2^{\frac{n}{2}}\frac{n}{2}!}(\mu_{\infty}(I)-\mu_{\infty}(I)^{2})^{\frac{n}{2}}.$

Case 2: If $(r_{1},\ldots,r_{u})$ has $\ell$ parts equal to $1$ with
$1\leq\ell\leq n$. Then

$n=r_{1}+\cdots+r_{u}\geq\ell+2(u-\ell).$

Therefore in this case,

$u-\ell\leq\frac{n-\ell}{2}\leq\frac{n-1}{2}.$

Using equations (5.8) and (5.9),

$\displaystyle\frac{1}{\pi_{L}(x)^{\frac{n}{2}}}\frac{n!}{2^{\frac{n}{2}}\frac{n}{2}!}\sum_{(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{u})}^{(2)}\prod_{i=1}^{u}\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{r_{i}}~{}d\mu_{\nu}(\theta)$
$\displaystyle\ll\frac{1}{\pi_{L}(x)^{\frac{n}{2}}}(\log\log
x)^{\ell}\pi_{L}(x)^{(u-\ell)}$
$\displaystyle\ll\pi_{L}(x)^{-\frac{1}{2}}(\log\log x)^{\ell}.$

Case 3: The remaining case is where $(r_{1},\ldots,r_{u})$ has all parts
$r_{i}\geq 2$ and at least one part greater than or equal to $3$. In this
case, it is easy to see that $u\leq\frac{n}{2}-1$. So we have

$\displaystyle\frac{1}{\pi_{L}(x)^{\frac{n}{2}}}\frac{n!}{2^{\frac{n}{2}}\frac{n}{2}!}\sum_{(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{u})}^{(2)}\prod_{i=1}^{u}\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{r_{i}}~{}d\mu_{\nu}(\theta)$
$\displaystyle\ll\pi_{L}(x)^{-1},$

using (5.9) for each part. Therefore, for partitions $(r_{1},\ldots,r_{u})$
described in Case 2 and Case 3,

(5.11)
$\lim\limits_{x\to\infty}\frac{1}{\pi_{L}(x)^{\frac{n}{2}}}\frac{n!}{2^{\frac{n}{2}}\frac{n}{2}!}\sum_{(\mathfrak{p}_{1},\ldots,\mathfrak{p}_{u})}^{(2)}\prod_{i=1}^{n/2}\int_{0}^{\pi}Z_{M}^{\pm}(\theta)^{r_{i}}~{}d\mu_{\nu}(\theta)=0.$

Gathering equations (5.10), (5.11) and choosing
$M=\lfloor\sqrt{\pi_{L}(x)}\log\log x\rfloor$, we have proved

$\lim_{x\to\infty}\left\langle\left(\frac{F^{\pm}(M,\pi)(x))}{\sqrt{\pi_{L}(x)}}\right)^{n}\right\rangle=\begin{cases}0&\text{
if }n\text{ is odd,}\\\
\frac{n!}{2^{\frac{n}{2}}\frac{n}{2}!}\left(\mu_{\infty}(I)-\mu_{\infty}(I)^{2}\right)^{\frac{n}{2}}&\text{
if }n\text{ is even.}\end{cases}$

This completes the proof.

##  6\. Improvements

We can obtain an improvement on the growth conditions on the weight vector
$\underline{k}=\underline{k}(x)$ in Theorem 4.1, if the indicator function
$\chi_{I}$ were to be replaced by a smooth test function, adapting the line of
proof in [BPS20, Theorem 1.6]. As in Theorem 1.1, $\underline{k}(x)$ is a
vector $(k_{1}(x),\ldots,k_{d}(x))$ where the components run over even
integers $\geq 4$. More precisely, the following holds.

######  Theorem 6.1.

Let $\Phi\in C^{\infty}(\mathbb{R})$ be a real-valued, even function in the
Schwartz class and $\widehat{\Phi}$ its Fourier transform. Fix a real number
$M\geq 1$ and define

$\phi_{M}(t)=\sum_{m\in\mathbb{Z}}\Phi(M(t+m))\text{ and
}V_{\Phi,M}=\int_{0}^{1}\phi_{M}(t)^{2}\mu_{\infty}(t)dt-\left(\int_{0}^{1}\phi_{M}(t)\mu_{\infty}(t)dt\right)^{2}.$

For $\pi\in\Pi_{\underline{k}}(\mathfrak{n}),$ define

$N_{\Phi,M,\pi}(x)=\sum_{\begin{subarray}{c}\mathrm{N}(\mathfrak{p})\leq x\\\
\mathfrak{p}\nmid\mathfrak{n}\end{subarray}}\phi_{M}(\theta_{\pi}(\mathfrak{p})).$

  1. (a)

Suppose $\widehat{\Phi}$ is compactly supported and
$\underline{k}=\underline{k}(x)$ satisfies $\frac{\sum_{i=1}^{d}\log
k_{i}}{\log x}\to\infty$ as $x\to\infty.$ Then, for any integer $r\geq 0,$

(6.1)
$\lim_{x\to\infty}\frac{1}{|\Pi_{\underline{k}}(\mathfrak{n})|}\sum_{\pi\in\Pi_{\underline{k}}(\mathfrak{n})}\left(\frac{N_{\Phi,M,\pi}(x)-{\pi}_{L}(x)\int_{0}^{1}\phi_{M}(t)\mu_{\infty}(t)dt}{\sqrt{{\pi}(x)V_{\Phi,M}}}\right)^{r}=\begin{cases}0&\text{
if }r\text{ is odd}\\\ \frac{r!}{\left(\frac{r}{2}\right)!2^{r/2}}&\text{ if
}r\text{ is even.}\end{cases}$

  2. (b)

For fixed $\lambda,\omega>0$, suppose the Fourier transform $\widehat{\Phi}$
satisfies $\widehat{\Phi}(t)\ll e^{-\lambda|t|^{\omega}}$, as $|t|\to\infty.$
Then, the asymptotic (6.1) holds if $\underline{k}=\underline{k}(x)$ satisfies
$\frac{\sum_{i=1}^{d}\log k_{i}}{(\log x)^{1+1/\omega}}\to\infty$ as
$x\to\infty.$

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# A tight lower bound on the minimal dispersion

M. Trödler Department of Mathematics, Faculty of Nuclear Sciences and Physical
Engineering, Czech Technical University, Trojanova 13, 12000 Praha, Czech
Republic.E-mail<EMAIL_ADDRESS>The work of this author has been
supported by the grant P202/23/04720S of the Grant Agency of the Czech
Republic. J. Volec11footnotemark: 1 E-mail<EMAIL_ADDRESS>The work
of this author has been supported by the grant 23-06815M of the Grant Agency
of the Czech Republic. and J. Vybíral11footnotemark: 1 Corresponding author.
E-mail<EMAIL_ADDRESS>The work of this author has been supported
by the grant P202/23/04720S of the Grant Agency of the Czech Republic.

###### Abstract

We give a new lower bound for the minimal dispersion of a point set in the
unit cube and its inverse function in the high dimension regime. This is done
by considering only a very small class of test boxes, which allows us to
reduce bounding the dispersion to a problem in extremal set theory.
Specifically, we translate a lower bound on the size of $r$-cover-free
families to a lower bound on the inverse of the minimal dispersion of a point
set. The lower bound we obtain matches the recently obtained upper bound on
the minimal dispersion up to logarithmic terms.

##  1 Introduction and the Main Result

For a given subset of the $d$-dimensional unit cube $X\subset[0,1]^{d}$, there
are various different ways how to measure whether $X$ is well spread over the
unit cube. One of the possibilities is a concept introduced in 1996 by Rote
and Tichy [13] called the dispersion of $X$. The _dispersion of $X$_ is the
volume of the largest axis-parallel box in $[0,1]^{d}$, which does not
intersect $X$, i.e.,

$\operatorname{disp}(X)=\sup_{B:B\cap X=\emptyset}|B|,$

where the supremum runs over all the boxes $B=\prod_{i=1}^{d}(a_{i},b_{i})$,
where $0\leq a_{i}<b_{i}\leq 1$ for all $i\in\\{1,\dots,d\\},$ and $|B|$
stands for the volume of $B$. Intuitively, the smaller the dispersion of a
point set is, the better spread over the unit cube the points have. We are
therefore interested in point sets that have simultaneously small cardinality
and small dispersion.

Given $n\in\mathbb{N}$, we denote the minimal dispersion of a point set that
has cardinality $n$ by

$\operatorname{disp}^{*}(n,d)=\inf_{\begin{subarray}{c}X\subset[0,1]^{d}\\\
\\#X=n\end{subarray}}\operatorname{disp}(X).$

For a fixed $d\in\mathbb{N}$, we define the inverse function of the minimal
dispersion in $[0,1]^{d}$ by

$\displaystyle N(\varepsilon,d)$
$\displaystyle=\min\\{n\in\mathbb{N}:\operatorname{disp}^{*}(n,d)\leq\varepsilon\\}$
$\displaystyle=\min\\{n\in\mathbb{N}:\exists X\subset[0,1]^{d}\ \text{with}\
\\#X=n\ \text{and}\ \operatorname{disp}(X)\leq\varepsilon\\}.$

The study of the minimal dispersion $\operatorname{disp}^{*}(n,d)$ as well as
its inverse function $N(\varepsilon,d)$ recently attracted a remarkable
attention in mathematical community. The pigeonhole principle implies the
trivial lower bound $\operatorname{disp}^{*}(n,d)\geq\frac{1}{n+1}$, which
translates into

$N(\varepsilon,d)\geq\frac{1}{\varepsilon}-1.$

This bound was complemented in [13] and then further improved by Bukh and Chao
[4] to

$N(\varepsilon,d)\leq C\cdot\frac{d^{2}\log d}{\varepsilon}.$ (1)

Let us note that further upper and lower bounds for different regimes of
$\varepsilon$ and $d$ were obtained also in [5, 10, 11, 12, 14], and the
dispersion of certain specific sets was studied in [3, 6, 7, 8, 9, 17, 18].

The first non-trivial lower bound on $N(\varepsilon,d)$ that exhibits a growth
if $d$ increases was achieved by Aistleitner, Hinrichs and Rudolf [1].
Specifically, they showed that

$N(\varepsilon,d)\geq\frac{\log_{2}d}{8\varepsilon}$ (2)

for every $d\geq 2$ and every $0<\varepsilon<1/4.$ The results mentioned so
far suggest that $N(\varepsilon,d)$ depends linearly on $1/\varepsilon$ and it
only remains to determine the $d$-dependence.

In another line of research, Sosnovec [16] showed that $N(\varepsilon,d)\leq
C_{\varepsilon}\log d.$ The dependence of $C_{\varepsilon}$ was not optimized
in [16], and a brief inspection of the proof reveals that it is actually
super-exponential in $1/\varepsilon$. Nevertheless, it matches (2) when it
comes to the dependence on $d$. Let us mention that the point set with small
dispersion was generated in [16] by sampling the coordinates of the points
independently at random from a uniform grid in $[0,1].$ An improved analysis
of the same method was given in [19], which together with a further
improvement due to Litvak [10] yields

$N(\varepsilon,d)\leq C\cdot\frac{\log
d\cdot\log(\frac{1}{\varepsilon})}{\varepsilon^{2}}.$ (3)

The comparison between (1) and (3) is not easy. Depending on the relation
between $d$ and $1/\varepsilon$, one or the other might perform better.
Although the analysis in [10, 19] regarding the dependence on $1/\varepsilon$
was much finer than the one in [16], it was quite natural to assume that the
second power of $1/\varepsilon$ in (3) is an artefact of the proof method and
that the bound could be potentially further improved even when $\varepsilon$
is moderately large.

The main result of this work is to show that, surprisingly, this is not the
case and the second power of $1/\varepsilon$ in (3) is optimal when
$\varepsilon$ is sufficiently large.

######  Theorem 1.

There is an absolute constant $c>0$, such that the following statement is
true. For any integer $d\geq{\color[rgb]{0,0,0}2}$ and any real $\varepsilon$
satisfying
$\frac{1}{4}\geq\varepsilon\geq\frac{1}{{\color[rgb]{0,0,0}4}\sqrt{d}}$, it
holds that

$N(\varepsilon,d)>\frac{c\,\log
d}{\varepsilon^{2}\cdot\log{\frac{1}{\varepsilon}}}\,.$ (4)

The proof of this result is motivated by the insight gained by a detailed
inspection of the proof given in [19]. For a fixed box $B$, the sampling from
a finite grid, which is bounded away of zero and one, naturally splits the
individual coordinates into two groups — those, where the corresponding
interval $(a_{i},b_{i})$ is large enough and covers the whole grid, and the
coordinates, where this is not the case. It turned out in [19] that the boxes
with a large number of the coordinates inside the second group are the most
difficult to hit by a randomly generated point. Therefore, we choose a very
small class of test boxes when compared to the number of all the parallel-axis
boxes, and consider any point set that hits all these test boxes. This
naturally reduces the question under study to a well-known combinatorial
problem called _$r$ -cover-free systems_, see Section 2 for the definition.
The key tool in our proof is a lower bound of Alon and Asodi [2] on the
minimum size of $r$-cover-free system.

Reformulated in the language of minimal dispersion, Theorem 1 states the
following.

######  Corollary 2.

There are two absolute constants $c_{1},c_{2}>0$ such that for every positive
integers $d$ and $n$ with $d\geq 2$ and $2\log d\leq n\leq c_{1}d$ it holds
that

$\operatorname{disp}^{*}(n,d)\geq c_{2}\biggl{(}\frac{\log
d}{n}\biggr{)}^{1/2}\cdot\biggl{(}\log\Bigl{(}\frac{n}{\log
d}\Bigr{)}\biggr{)}^{-1/2}.$

Before we present our proof of Theorem 1, let us add a few remarks.

######  Remark.

  1. 1.

It follows from the proof of Theorem 1 that one can actually choose
$c=1/{\color[rgb]{0,0,0}1920}$, but we did not attempt to optimize this
constant.

  2. 2.

The essentially tight factor $1/\varepsilon^{2}$ in the lower bound for
$N(\varepsilon,d)$ in (4) is rather surprising. It seems that a logarithmic
dependence on $d$ and a linear dependence on $1/\varepsilon$ exclude each
other. What might be even more surprising, this lower bound is obtained by
considering only a very small class of parallel-axis test boxes.

  3. 3.

We leave it as an open problem whether the proof method can be further
improved, possibly extending (4) to smaller values of $\varepsilon$.

##  2 Proof of Theorem 1

We start by fixing some notation that will be used in the proof. For a finite
set $X$, we denote its size by $\\#X$. For a positive integer $d$, we denote
by $[d]$ the set $\\{1,2,\ldots,d\\}$. Additionally, for a non-negative
integer $k$, we write $\binom{[d]}{k}$ to denote the collection of all the
$k$-element subsets of $[d]$. Given a point $x\in[0,1]^{d}$ and an integer
$i\in[d]$, we denote by $(x)_{i}$ the $i$-th coordinate of $x$.

By focusing on a very specific set of parallel-axis boxes of volume at least
$\varepsilon$, we are able to translate bounding $N(\varepsilon,d)$ from below
to a question in extremal set theory. We say that a family $\mathcal{F}$ of
subsets of a ground set $X$ is _$r$ -cover-free_ if no $F\in\mathcal{F}$ is
contained in the union of any $r$ sets from $\mathcal{F}\setminus\\{F\\}$.
Based on a previous work of Ruszinkó [15], Alon and Asodi [2] proved the
following lower-bound on the size of the ground set of an $r$-cover-free
family.

######  Theorem 3 ([2, Lemma 2.8]).

Let $\mathcal{F}$ be a family of $d$ subsets of a ground set $X$. If
$\mathcal{F}$ is $r$-cover-free, where $r\leq 2\sqrt{d}$, then

$\\#X>\frac{1}{10}\cdot\frac{r^{2}\log\left(d-\frac{r}{2}\right)}{\log r}\,.$

We now proceed with the proof of Theorem 1. Fix $\varepsilon$ and $d$, and let
$k$ be the unique integer satisfying $2^{-(k+1)}<\varepsilon\leq 2^{-k}$. Note
that $k\geq 2$. For a given a set $A\subseteq[d]$ and $j\in[d]\setminus A$,
let $B^{A,j}\subseteq[0,1]^{d}$ be the axes-parallel box defined as

$B^{A,j}:=I_{1}\times I_{2}\times\cdots\times I_{d},\quad\mbox{where
}\begin{cases}I_{j}=(0,2^{1-k}),\\\ I_{i}=(2^{1-k},1)\quad\mbox{for }i\in
A,\\\ I_{i}=(0,1)\quad\mbox{for
}i\in[d]\setminus\left(A\cup\\{j\\}\right).\end{cases}$

Let $\mathcal{B}:=\left\\{B^{A,j}:A\in\binom{[d]}{2^{k-2}}\mbox{ and
}j\in[d]\setminus A\right\\}.$ The following claim yields that any set of
points from $[0,1]^{d}$ with dispersion at most $\varepsilon$ must have at
least one point inside every $B\in\mathcal{B}$.

######  Claim 4.

$\left|B^{A,j}\right|\geq\varepsilon$ for any $A\subseteq[d]$ of size at most
$2^{k-2}$ and any $j\in[d]\setminus A$.

###### Proof.

Since $2^{-2x}$ is convex, it holds that $(1-x)\geq 2^{-2x}$ for every
$x\in[0,1/2]$. Therefore,

$\left|B^{A,j}\right|=\left(1-2^{1-k}\right)^{\\#A}\cdot
2^{1-k}\geq\left(2^{-2^{2-k}}\right)^{\\#A}\cdot 2^{1-k}\geq
2^{-k}\geq\varepsilon\,,$

which finishes the proof of the claim. ∎

Fix a finite set of points $X\subseteq[0,1]^{d}$ that intersects every
$B\in\mathcal{B}$. For every $j\in[d]$, we define an auxiliary set $F_{j}$ of
the points with a small $j$-th coordinate. Specifically, let $F_{j}:=\\{x\in
X:(x)_{j}<2^{1-k}\\}$. In order to lower bound $\\#X$ using Theorem 3, we
establish the following claim.

######  Claim 5.

The family $\left\\{F_{1},F_{2},\ldots,F_{d}\right\\}$ is $2^{k-2}$-cover-
free.

###### Proof.

Suppose there is a set $A\in\binom{[d]}{2^{k-2}}$ and an integer
$j\in[d]\setminus A$ such that

$F_{j}\subseteq\bigcup_{i\in A}F_{i}\,.$

However, there must exist a point $x\in X\cap B^{A,j}$. Since
$(x)_{j}<2^{1-k}$ and $(x)_{i}>2^{1-k}$ for all $i\in A$, we conclude that $x$
is an element of $F_{j}\setminus\bigcup_{i\in A}F_{i}$, which is a
contradiction. ∎

Since $2^{k-2}\leq\frac{1}{4\varepsilon}\leq{\color[rgb]{0,0,0}\sqrt{d}}$,
Theorem 3 applied to the family $\left\\{F_{1},\ldots,F_{d}\right\\}$ and
$r=2^{k-2}$ readily yields that

$\\#X>\frac{1}{10}\cdot\frac{2^{2k-4}\cdot\log\left(d-2^{k-3}\right)}{\log\left(2^{k-2}\right)}\geq\frac{1}{10}\cdot\frac{2^{2k-4}\cdot\log\left(d-\sqrt{d}{\color[rgb]{0,0,0}/2}\right)}{\log\left(2^{k-2}\right)}\,.$

Since $\frac{1}{8\varepsilon}<2^{k-2}<\frac{1}{\varepsilon}$ and
$\log(d-\sqrt{d}{\color[rgb]{0,0,0}/2})\geq\frac{\log{d}}{{\color[rgb]{0,0,0}3}}$
for all $d\geq{\color[rgb]{0,0,0}2}$, we conclude that

$\\#X>\frac{1}{1920}\cdot\frac{\log
d}{\varepsilon^{2}\cdot\log{\frac{1}{\varepsilon}}}\,.$

This finishes the proof of Theorem 1.

Acknowledgements. We are extremely grateful to Noga Alon for pointing out to
us the relevance of the notion of $r$-free-cover systems as well as the papers
[2] and [15]. We thank Noah Kravitz, Alexander Litvak, Vojtěch Rödl, Marcelo
Tadeu de Sá Oliveira Sales, and Stefan Steinerberger for fruitful discussions.
We also gratefully acknowledge the support of the Leibniz Center for
Informatics, where several discussions about this problem were held during the
Dagstuhl Seminar “Algorithms and Complexity for Continuous Problems” (Seminar
ID 23351).

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