# Hyperuniformity of random measures on Euclidean and hyperbolic spaces

Michael Björklund  and  Mattias Byléhn

###### Abstract.

We investigate lower asymptotic bounds of number variances for invariant
locally square-integrable random measures on Euclidean and real hyperbolic
spaces. In the Euclidean case we show that there are subsequences of radii for
which the number variance grows at least as fast as the volume of the boundary
of Euclidean balls, generalizing a classical result of Beck. With regards to
real hyperbolic spaces we prove that random measures are never geometrically
hyperuniform and if the random measure admits non-trivial complementary series
diffraction, then it is hyperfluctuating. Moreover, we define spectral
hyperuniformity and stealth of random measures on real hyperbolic spaces in
terms of vanishing of the complementary series diffraction and sub-Poissonian
decay of the principal series diffraction around the Harish-Chandra
$\Xi$-function.

###### Key words and phrases:

Hyperuniformity, spherical diffraction, asymptotic properties of spherical
functions

###### 2020 Mathematics Subject Classification:

Primary: 60G55. Secondary: 60G57, 43A90,

##  1\. Introduction

In the field of statistical mechanics for low-temperature states of matter,
the notion of _hyperuniformity_ , or _superhomogeneity_ , as introduced by
Stillinger and Torquato in the seminal paper [21] in 2003, quantifies long-
range order of an invariant point process in terms of the degree of
suppression of large-scale density fluctuations. Equivalently, hyperuniformity
is characterized by the suppression of small frequencies in the diffraction
picture. Hyperuniformity and the associated number variances of invariant
point processes have been studied extensively in Euclidean spaces, examples
including invariant random lattice shifts, i.i.d. perturbations of such,
certain determinantal point processes, most mathematical quasicrystals and
more. See [20] for an extensive survey and [8] for details on mathematical
quasicrystals.

In this article we investigate lower asymptotic bounds of number variances,
i.e. the variance of the measure of asymptotically large balls, for invariant
locally square-integrable random measures on Euclidean and real hyperbolic
spaces. We prove for general invariant locally square-integrable random
measures on Euclidean space that there are unbounded sequences of centered
Euclidean balls such that the number variance grows at least as fast as the
volume of the boundary of the balls, extending a Theorem of Beck in the case
of a locally finite point set [5, Theorem 2A]. Our methods are Fourier
theoretic and utilizes the _diffraction measure_ of a random measure,
generalizing the notion of the _structure factor_ of a point process.

As for random measures on real hyperbolic space, we investigate contending
notions of hyperuniformity. This is also done in terms of the diffraction
measure associated to such a random measure, which for hyperbolic spaces (and
more generally, commutative spaces) is defined using the available _spherical
transform_ , generalising the Hankel transform in Euclidean space. We show
that, contrary to the Euclidean setting, the large-scale number variance of a
random measure on real hyperbolic space grows at least as fast as the volume
of large metric balls along some unbounded sequence of metric balls.
Nevertheless, we are able to define a notion of spectral hyperuniformity on
the principal part of the relevant frequency domain that is distinctly
different from measuring large-scale density fluctuations. Instead, our
definition favours the property of having suppressed ”small scale” frequency
fluctuations to characterize hyperuniformity.

###  1.1. Random measures

Let $X$ denote the space $\mathbb{R}^{n}$ with the action of translations or
real hyperbolic space $\mathbb{H}^{n}$ with the action of orientation-
preserving isometries. By a _random measure_ on $X$ we mean a probability
measure $\mu$ on the cone $\mathscr{M}_{+}(X)$ of positive locally finite
measures on $X$, which can be thought of as the law of a
$\mathscr{M}_{+}(X)$-valued random variable. One usually studies such random
measures in terms of _linear statistics_
$Sf:\mathscr{M}_{+}(X)\rightarrow\mathbb{C}$ associated to bounded measurable
functions $f:X\rightarrow\mathbb{C}$ with bounded support, given by

$\displaystyle Sf(p)=\int_{X}f(x)dp(x)\,.$

Given an invariant locally square-integrable random measure $\mu$ on $X$, we
denote by $\mathrm{NV}_{\mu}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq
0}$ the _number variance_ with respect to $\mu$,

$\displaystyle\mathrm{NV}_{\mu}(r)=\mathrm{Var}_{\mu}(S\chi_{B_{r}})=\int_{\mathscr{M}_{+}(X)}|p(B_{r})-\iota_{\mu}\mathrm{Vol}_{X}(B_{r})|^{2}d\mu(p)$
(1.1)

where $B_{r}=B_{r}(o)$ is the metric ball of radius $r>0$ centered at a fixed
reference point $o\in X$ and $\iota_{\mu}>0$ is the intensity of $\mu$, i.e.
the mean measure of a unit volume Borel set. More generally, one can consider
variances of the form $\mathrm{Var}(S\chi_{rB})$ for $r>0$, where $B\subset X$
is any bounded Borel set with positive measure and $rB$ is the
$r$-neighbourhood of $B$ with respect to the metric on $X$. We will in this
paper only consider metric balls $B_{r}$.

###  1.2. Number variances and diffraction measures in $\mathbb{R}^{n}$

A uniform upper bound on the number variance of a random measure $\mu$ on
$\mathbb{R}^{n}$ is obtained from an application of the mean ergodic theorem,
yielding

$\displaystyle\limsup_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{\mathrm{Vol}_{\mathbb{R}^{n}}(B_{R})^{2}}=0\,.$

We provide a general lower bound of the number variance for maximizing
sequences of radii, extending a celebrated result of Beck [5, Theorem 2A].

######  Theorem 1.1.

Let $\mu$ be an invariant locally square-integrable random measure on
$\mathbb{R}^{n}$. Then

$\displaystyle\limsup_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{\mathrm{Vol}_{n-1}(\partial
B_{R})}>0\,.$ (1.2)

######  Remark 1.2.

The question of whether there is a uniform lower bound on this quotient for a
given random measure $\mu$ is in general unclear. In Subsections 5.2 and 5.3
however, we show that the randomly shifted standard lattice in
$\mathbb{R}^{5}$ satisfies

$\displaystyle\liminf_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu_{\mathbb{Z}^{5}}}(R)}{\mathrm{Vol}_{4}(\partial
B_{R})}=0\,.$ (1.3)

The proof of Theorem 1.1 relies on a spectral formulation of the number
variances. More precisely, the _diffraction measure_ $\widehat{\eta}_{\mu}$ of
an invariant locally square-integrable random measure $\mu$ is the unique
positive Radon measure on $\mathbb{R}^{n}$ satisfying

$\displaystyle\mathrm{Cov}_{\mu}(Sf_{1},Sf_{2})=\int_{\mathbb{R}^{n}}\widehat{f}_{1}(\xi)\overline{\widehat{f}_{2}(\xi)}d\widehat{\eta}_{\mu}(\xi)$

for all bounded measurable functions $f_{1},f_{2}$ on $\mathbb{R}^{n}$ with
bounded support, where $\widehat{f}$ denotes the Fourier transform of $f$. In
particular, one computes the number variance to be

$\displaystyle\mathrm{NV}_{\mu}(r)=(2\pi)^{n}r^{n}\int_{\mathbb{R}^{n}}J_{\frac{n}{2}}(r\lVert\xi\rVert)^{2}\frac{d\widehat{\eta}_{\mu}(\xi)}{\lVert\xi\rVert^{n}}\,,$

where $J_{n/2}$ denotes a Bessel function of the first kind. Classical
asymptotic expansions of such Bessel functions allow for a proof of Theorem
1.1.

###  1.3. Hyperuniformity and stealth in $\mathbb{R}^{n}$

An invariant locally square-integrable random measure $\mu$ on
$\mathbb{R}^{n}$ is _geometrically hyperuniform_ if

$\displaystyle\limsup_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{\mathrm{Vol}_{\mathbb{R}^{n}}(B_{R})}=0\,.$

The volume term in the denominator coincides with the variance of the
invariant Poisson point process on $\mathbb{R}^{n}$ with unit intensity, and
one can interpret a hyperuniform random measure $\mu$ as having ”sub-
Poissonian mass fluctuations”. If the above upper limit is infinite, the
random measure is _hyperfluctuating_. In [8], the first author and Hartnick
prove that a random measure $\mu$ is geometrically hyperuniform if and only if
it is _spectrally hyperuniform_ , meaning that

$\displaystyle\limsup_{\varepsilon\rightarrow
0^{+}}\frac{\widehat{\eta}_{\mu}(B_{\varepsilon}(0))}{\mathrm{Vol}_{\mathbb{R}^{n}}(B_{\varepsilon}(0))}=0\,.$

The denominator here can be interpreted as the diffraction measure of the
invariant Poisson point process in $\mathbb{R}^{n}$ with unit intensity.

A particularly rigid class of hyperuniform invariant random measures $\mu$ are
_stealthy_ random measures, meaning random measures for which the diffraction
measure $\widehat{\eta}_{\mu}$ vanishes in a neighbourhood of the origin in
$\mathbb{R}^{n}$. Examples include invariant random shifts of lattices, and in
upcoming work with A. Fish we will provide new families of stealthy random
measures and point processes on $\mathbb{R}^{n}$.

###  1.4. Number variances and diffraction measures for real hyperbolic spaces

For $n$-dimensional real hyperbolic space
$\mathbb{H}^{n}=\mathrm{SO}^{\circ}(1,n)/\mathrm{SO}(n)$, we consider locally
square-integrable random measures $\mu$ that are invariant under the action of
the orientation-preserving isometry group $\mathrm{SO}^{\circ}(1,n)$ in
analogy with isotropic (i.e. translation and rotation invariant) random
measures on $\mathbb{R}^{n}$.

On $\mathbb{H}^{n}$, there is a well-behaved generalization of the Fourier
transform, the _spherical transform_ , available for
bi-$\mathrm{SO}(n)$-invariant functions
$\varphi:\mathrm{SO}^{\circ}(1,n)\rightarrow\mathbb{C}$. It is given by

$\displaystyle\widehat{\varphi}(\lambda)=\int_{\mathrm{SO}^{\circ}(1,n)}\varphi(g)\omega_{\lambda}^{(n)}(g)dg\,,$

where $dg$ denotes a fixed Haar measure and $\omega^{(n)}_{\lambda}$ are
$\mathrm{SO}(n)$-_spherical functions_ for $\mathrm{SO}^{\circ}(1,n)$, that
is, the bi-$\mathrm{SO}(n)$-invariant functions on $\mathrm{SO}^{\circ}(1,n)$
given by

$\displaystyle\omega_{\lambda}^{(n)}(g)=\frac{2^{\frac{n-1}{2}}\Gamma(\frac{n}{2})}{\sqrt{\pi}\,\Gamma(\frac{n-1}{2})}\sinh(d(g.o,o))^{2-n}\int_{0}^{d(g.o,o)}(\cosh(d(g.o,o))-\cosh(s))^{\frac{n-3}{2}}\cos(\lambda
s)ds\,,$

where $d$ denotes the hyperbolic metric. With respect to the spherical
transform, the diffraction measure of an invariant locally square-integrable
random measure $\mu$ is a pair of measures defined by the equation

$\displaystyle\mathrm{Cov}_{\mu}(S\varphi_{1},S\varphi_{2})=\int_{0}^{\infty}\widehat{\varphi}_{1}(\lambda)\overline{\widehat{\varphi}_{2}(\lambda)}d\widehat{\eta}_{\mu}^{(p)}(\lambda)+\int_{0}^{\frac{n-1}{2}}\widehat{\varphi}_{1}(i\lambda)\overline{\widehat{\varphi}_{2}(i\lambda)}d\widehat{\eta}_{\mu}^{(c)}(\lambda)\,.$

We refer to the measures $\widehat{\eta}_{\mu}^{(p)}$ on $(0,+\infty)$ and
$\widehat{\eta}_{\mu}^{(c)}$ on $i[0,\frac{n-1}{2})$ as the _principal-_ and
_complementary diffraction measure_ respectively. We leave out $\lambda=0$
corresponding to the _Harish-Chandra $\Xi$-function_ $\omega_{0}$ from the
support of the principal diffraction measure for reasons that will become
clear from our results. The existence and uniqueness of such diffraction
measures can be seen as part of important work by Krein, Gelfand-Vilenkin and
later by Bopp in [9, Part II, Theorem 3].

######  Remark 1.3.

Diffraction measures can more generally be defined for random measures on a
large class of _commutative spaces_ $X=G/K$ with $K$ compact, which includes
higher rank symmetric spaces, regular trees, Bruhat-Tits buildings and
products of such. These are positive measures on the $K$-spherical unitary
dual $\widehat{G}^{K}$ of the pair $(G,K)$. We investigate the general
framework and prove the existence and uniqueness of diffraction measures in an
upcoming article.

In terms of the diffraction measure, the number variance can be written as

$\displaystyle\mathrm{NV}_{\mu}(r)=\frac{\pi^{n}}{\Gamma(\frac{n}{2}+1)^{2}}\,\sinh(r)^{2n}\Big{(}\int_{0}^{\infty}|\omega^{(n+2)}_{\lambda}(a_{r})|^{2}d\widehat{\eta}_{\mu}^{(p)}(\lambda)+\int_{0}^{\frac{n-1}{2}}|\omega^{(n+2)}_{i\lambda}(a_{r})|^{2}d\widehat{\eta}_{\mu}^{(c)}(\lambda)\Big{)}\,.$

In this picture, it is crucial to understand the asymptotic behaviour of the
spherical functions in order to understand the asymptotics of the number
variance.

Also, the asymptotics of the number variance of a random measure $\mu$ depends
significantly on whether the complementary diffraction measure is trivial or
not.

A first naive definition of geometric hyperuniformity of a random measure
$\mu$ on $\mathbb{H}^{n}$ is to require that

$\displaystyle\limsup_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{\mathrm{Vol}_{\mathbb{H}^{n}}(B_{R})}=0\,.$

However, since $\mathrm{SO}^{\circ}(1,n)$ is non-amenable then
$\mathrm{Vol}(\partial B_{R})\asymp\mathrm{Vol}_{\mathbb{H}^{n}}(B_{R})$ as
$R\rightarrow+\infty$, so it is a priori unclear whether this definition
admits examples or not. We prove that no random measures on $\mathbb{H}^{n}$
are geometrically hyperuniform in the following hyperbolic analogue of Theorem
1.1.

######  Theorem 1.4.

Let $\mu$ be a locally square-integrable invariant random measure on
$\mathbb{H}^{n}$. Then

$\displaystyle\limsup_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{\mathrm{Vol}_{\mathbb{H}^{n}}(B_{R})}>0\,.$
(1.4)

Moreover,

  1. (1)

if $\widehat{\eta}_{\mu}^{(c)}(\\{0\\})>0$, then

$\displaystyle\liminf_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{R^{2}\mathrm{Vol}_{\mathbb{H}^{n}}(B_{R})}>0\,.$
(1.5)

  2. (2)

if $\widehat{\eta}_{\mu}^{(c)}([\delta\frac{n-1}{2},\frac{n-1}{2}))>0$ for
some $\delta\in(0,1)$, then

$\displaystyle\liminf_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{\mathrm{Vol}_{\mathbb{H}^{n}}(B_{R})^{1+\delta}}>0\,.$
(1.6)

For the special case of a randomly shifted lattice orbit in $\mathbb{H}^{n}$,
Theorem 1.4 implies that there is a sequence of radii such that the number
variance grows at least as fast as the hyperbolic volume of large metric
balls. Moreover, when the complementary diffraction measure is non-trivial,
the random measure is hyperfluctuating in analogy with the Euclidean
terminology.

There are related works to the above result: In [12, Theorem 2], Hill and
Parnovski derive explicit asymptotics of the number variance for randomly
shifted lattice orbits in $\mathbb{H}^{n}$. Theorem 1.4 implies that the error
term in their result is sharp. In [16, Theorem 2], Magyar provides a positive
lower bound on the $L^{\infty}$-norm of the discrepancy of infinite point sets
over balls in $\mathbb{H}^{n}$ that is proportional to the volume of the ball.
For number variances (the $L^{2}$-norm of the discrepancy), we expect that
there are examples of translation bounded point processes $\mu$ in
$\mathbb{H}^{n}$ for which

$\displaystyle\liminf_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{\mathrm{Vol}_{\mathbb{H}^{n}}(B_{R})}=0\,.$

###  1.5. Hyperuniformity in real hyperbolic space

In light of Theorem 1.4, there is no canonical ”geometric” definition of
hyperuniformity of a random measure $\mu$ on $\mathbb{H}^{n}$. We instead
formulate a notion of _spectral hyperuniformity_ in terms of the diffraction
measure of $\mu$ that is distinctly different from the geometric one.

######  Definition 1.5.

An invariant locally square-integrable random measure on $\mathbb{H}^{n}$ is
_spectrally hyperuniform_ if $\widehat{\eta}_{\mu}^{(c)}=0$ and

$\displaystyle\limsup_{\varepsilon\rightarrow
0^{+}}\frac{\widehat{\eta}_{\mu}^{(p)}((0,\varepsilon])}{\varepsilon^{3}}=0\,.$

The denominator $\varepsilon^{3}$ is asymptotic to the diffraction measure of
the interval $(0,\varepsilon]$ for the unit intensity invariant Poisson point
process as $\varepsilon\rightarrow 0^{+}$, so this definition of
hyperuniformity is in that way consistent with spectral hyperuniformity on
$\mathbb{R}^{n}$. It is worth noting that, with respect to this definition, it
is no longer the case that all invariant randomly shifted lattice orbits are
spectrally hyperuniform as there are lattices that admit complementary
spectrum, even co-compact ones, see for example [13, Section 1.2, Eq.B]. With
this notion of hyperuniformity in mind, we define what it means for a random
measure to be stealthy.

######  Definition 1.6.

An invariant locally square-integrable random measure on $\mathbb{H}^{n}$ is
_stealthy_ if $\widehat{\eta}_{\mu}^{(c)}=0$ and there is a $\lambda_{o}>0$
such that $\widehat{\eta}_{\mu}^{(p)}((0,\lambda_{o}))=0$.

Stealthy random measures are spectrally hyperuniform. In [13, Section 1.2,
Eq.B], Jenni constructs a lattice
$\Gamma<\mathrm{SL}_{2}(\mathbb{R})\cong\mathrm{SO}^{\circ}(1,2)$ that does
not admit complementary spectrum nor principal spectrum around $\lambda=0$ in
the sense that the smallest non-zero eigenvalue of the Laplace operator on
$\Gamma\backslash\mathbb{H}^{2}$ is strictly larger than $1/4$. The eigenvalue
$1/4$ corresponds to $\lambda=0$, so the resulting invariant random lattice
orbit of $\Gamma$ in $\mathbb{H}^{2}$ is stealthy, and in particular
spectrally hyperuniform.

The lack of other examples of spectrally hyperuniform point processes and
random measures in $\mathbb{H}^{n}$ is however quite severe. In an upcoming
article we will prove that determinantal point processes in $\mathbb{H}^{n}$
(and more generally in any commutative space associated with a non-amenable
Gelfand pair) are not spectrally hyperuniform, in contrast to the Euclidean
case where determinantal point processes associated to kernels which define
orthogonal projections in $L^{2}(\mathbb{R}^{n})$ are always hyperuniform.
Here we show that locally square-integrable i.i.d. perturbations of an
invariant random lattice orbit in $\mathbb{H}^{n}$ are not hyperuniform. We
note the apparent contrast to the setting in $\mathbb{R}^{n}$, where i.i.d.
perturbations of an invariant random lattice are always locally square-
integrable and always preserves hyperuniformity, see [1, Appendix B] and
Proposition 4.5. The following Proposition is proved in Section 10.

######  Proposition 1.7.

Let $\Gamma<\mathrm{SO}^{\circ}(1,n)$ be a lattice and $d\nu(g)=\beta(g)dg$
for some positive measurable bi-$\mathrm{SO}(n)$-invariant function $\beta$
with $\int\beta(g)dg=1$ such that

$\displaystyle\beta(g)\leq C\mathrm{e}^{-2(n-1+\varepsilon)d(g.o,o)}$

for all $g\in G$ and some $C>0,\varepsilon>0$. Then the invariant random
lattice orbit of $\Gamma$ in $\mathbb{H}^{n}$ with independent
$\nu$-distributed perturbations is locally square-integrable and not
spectrally hyperuniform.

######  Remark 1.8 (Heat kernel hyperuniformity in Euclidean and hyperbolic
spaces).

In future work, we will prove that spectral hyperuniformity is equivalent to
what we refer to as _heat kernel hyperuniformity_. Denote the heat kernel by
$h_{\tau}(x)$, $\tau>0$, the fundamental solution of the heat operator
$\partial_{\tau}-\Delta$ in the respective geometries. In $\mathbb{R}^{n}$, we
define $\mu$ to be heat kernel hyperuniform if

$\displaystyle\limsup_{\tau\rightarrow+\infty}\,\tau^{\frac{n}{2}}\,\mathrm{Var}_{\mu}(Sh_{\tau})=0\,,$

and similarly in $\mathbb{H}^{n}$, if

$\displaystyle\limsup_{\tau\rightarrow+\infty}\,\tau^{\frac{3}{2}}\mathrm{e}^{\frac{(n-1)^{2}}{2}\tau}\,\mathrm{Var}_{\mu}(Sh_{\tau})=0\,.$

One perk of this formulation is that the smoothness of $h_{\tau}$ allows us to
extend the notion of hyperuniformity to invariant random _distributions_ on
these spaces as well, for which we expect that one can construct hyperuniform
and stealthy examples. Moreover, by uniform estimates of the heat kernel due
to Anker and Ostellari in [3], one should be able to formulate heat kernel
hyperunifomity for a large class of non-compact symmetric spaces.

Acknowledgement: This paper is part of the second author’s doctoral thesis at
the University of Gothenburg under the supervision of the first author. The
first author was supported by the grants 11253320 and 2023-03803 from the
Swedish Research Council. We thank Tobias Hartnick and Günter Last for
providing valuable insights into the making of the paper. The second author is
grateful to Tobias Hartnick for many useful discussions and hospitality during
several visits to the Karlsruhe Institute of Technology, and also to Morten
Risager for providing valuable information regarding number variances of
lattices during a stay at the Mittag-Leffler Analytic Number Theory Program,
spring 2024.

###### Contents

  1. 1 Introduction
  2. 2 Random measures on homogeneous spaces
  3. 3 Euclidean harmonic analysis
  4. 4 Euclidean diffraction measures
  5. 5 Beck’s Theorem
  6. 6 Spherical harmonic analysis on real hyperbolic spaces
  7. 7 Hyperbolic diffraction measures
  8. 8 A hyperbolic analouge of Beck’s theorem
  9. 9 Spectral hyperuniformity and stealth
  10. 10 Random perturbed lattice orbits in hyperbolic spaces

##  2\. Random measures on homogeneous spaces

We introduce invariant locally square-integrable random measures and their
autocorrelation measures. The constructions work for a large family of proper
homogeneous metric spaces, but we will only apply them to Euclidean and
hyperbolic spaces in the subsequent sections.

In Subsection 2.1 we set up the metric spaces over which we will consider
random measures. We proceed by defining invariant locally square-integrable
random measures, related objects and provide examples in Subsection 2.2. In
Subsection 2.3, we define the autocorrelation measure of an invariant locally
square-integrable random measure and compute it for the examples given in the
previous subsection.

###  2.1. Homogeneous metric spaces

Let $(X,d)$ be a proper non-compact metric space and fix a point $o\in X$.
Suppose that there is a closed and unimodular subgroup $G$ of the isometry
group $\mathrm{Isom}(X,d)$ whose action on $X$ is transitive. The action of
$g\in G$ on $x\in X$ will be denoted by $g.x$. If we denote the $G$-stabilizer
of the point $o$ by $K$ then the orbit map $\pi:G\rightarrow X$, $g\mapsto
g.o$, descends to a bijection $G/K\rightarrow X$. We will assume that $K$ is
_compact_.

We will frequently make use of the fact that there is a Borel measurable
_section_ $\varsigma:X\rightarrow G$ of the orbit map $\pi$, that is, a Borel
measurable map such that $\varsigma(x).o=x$ for all $x\in X$ and such that
$\varsigma(B)\subset G$ is pre-compact for every bounded Borel set $B\subset
X$, see [15, Lemma 1.1]. One shows that such a section satisfies the
$\varsigma(g.o)\in gK$ for all $g\in G$. For Euclidean and real hyperbolic
spaces, we will construct _continuous_ sections.

Fix a Haar measure $m_{G}$ on $G$ and define a positive $G$-invariant measure
$m_{X}$ on $X$ as the push-forward of $m_{G}$ along $\pi$. Explicitly,

$\displaystyle m_{X}(B)=m_{G}\big{(}\big{\\{}g\in G:g.o\in B\big{\\}}\big{)}$
(2.1)

for all bounded Borel sets $B\subset X$. Since the orbit map $\pi:G\rightarrow
X$ is proper, $m_{X}$ defines a positive Radon measure on $X$. Also, since the
Haar measure $m_{G}$ is unique up to scaling, then the measure $m_{X}$ is
unique up to scaling. We will moreover denote by $m_{K}$ the unique Haar
probability measure on $K$.

On the level of functions, if $M$ is a measurable space we denote by
$\mathscr{L}^{\infty}(M)$ the vector space of bounded measurable complex
valued functions on $M$. With regards to the proper metric space $X$, we
denote by
$\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)\subset\mathscr{L}^{\infty}(X)$ the
subspace of functions which vanish outside of a bounded set, or equivalently,
vanish outside a compact subset. The Borel section $\varsigma$ then
establishes a bijection between $f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$
and right-$K$-invariant functions
$\varphi_{f}\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G)$ by
$f(x)=\varphi_{f}(\varsigma(x))$ for all $x\in X$. The vector space
$\mathscr{L}^{\infty}_{\mathrm{bnd}}(G)$ defines an involutive algebra over
the complex numbers under the convolution operation

$\displaystyle(\varphi*\psi)(g)=\int_{G}\varphi(gh^{-1})\psi(h)dm_{G}(h)\,,\quad\varphi,\psi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G)\,,$

and the involution $\varphi^{*}(g)=\overline{\varphi(g^{-1})}$. If
$\varphi,\psi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G)$ are
right-$K$-invariant then $\varphi^{*}$ is left-$K$-invariant, and one checks
that $\varphi^{*}*\psi$ is bi-$K$-invariant, so that the subspace
$\mathscr{L}^{\infty}_{\mathrm{bnd}}(G,K)\subset\mathscr{L}^{\infty}_{\mathrm{bnd}}(G)$
of bi-$K$-invariant functions defines an involutive subalgebra. We can
moreover identify $\mathscr{L}^{\infty}_{\mathrm{bnd}}(G,K)$ with the subspace
$\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)^{K}\subset\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$
of left-$K$-invariant functions on $X$ using the section $\varsigma$ as
mentioned. We say that a function on $X$ is _radial_ if it is
left-$K$-invariant. More generally, if $\nu,\eta$ are Radon measures on $G$
with $\nu$ finite, their convolution is given by

$(\nu*\eta)(Q)=\int_{G}\int_{G}\chi_{Q}(gh)d\nu(g)d\eta(h)$

for Borel sets $Q\subset G$.

###  2.2. Random measures

Let $\mathscr{M}_{+}(X)$ denote the set of positive locally finite Borel
measures on $X$, endowed with the smallest $\sigma$-algebra such that for
every $f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$, the _linear statistic_
$Sf:\mathscr{M}_{+}(X)\rightarrow\mathbb{C}$ given by

$\displaystyle Sf(p)=p(f)=\int_{X}f(x)dp(x)\,,\quad p\in\mathscr{M}_{+}(X)$

is measurable. Then $\mathscr{M}_{+}(X)$ is a Polish space when endowed with
the vague topology with respect to the subspace
$C_{c}(X)\subset\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$ of compactly supported
continuous complex valued functions on $X$. By a _random measure on $X$_ we
will mean a probability measure $\mu$ on $\mathscr{M}_{+}(X)$. We say that
$\mu$ is a _point process_ if it is supported on the subspace

$\mathscr{M}_{+}^{\mathrm{LF}}(X)=\Big{\\{}p\in\mathscr{M}_{+}(X):\mathrm{supp}(p)\mbox{
is locally finite}\Big{\\}}$

of measures with locally finite support. A point process is _simple_ if it is
supported on

$\mathscr{M}_{+}^{\mathrm{LFS}}(X)=\Big{\\{}\delta_{\Xi}=\sum_{x\in\Xi}\delta_{x}\in\mathscr{M}_{+}(X):\Xi\subset
X\mbox{ is locally finite}\Big{\\}}$

of locally finite measures with unit masses. The action of $G$ on $X$ lifts to
an action of $G$ on $\mathscr{M}_{+}(X)$ by push-forward and leaves the
mentioned subspaces invariant. Moreover, an additional lifting by push-forward
yields an action of $G$ on the space of random measures on $X$. A random
measure $\mu$ on $X$ is

  * •

_invariant_ if $g_{*}\mu=\mu$ for all $g\in G$.

  * •

_locally $k$-integrable_, $k\in\mathbb{N}$, if for every bounded Borel set
$B\subset X$,

$\int_{\mathscr{M}_{+}(X)}p(B)^{k}d\mu(p)<+\infty\,.$

In terms of linear statistics, $S\chi_{B}\in L^{k}(\mathscr{M}_{+}(X),\mu)$
for all bounded Borel $B\subset X$. Note that a locally $k$-integrable random
measure $\mu$ is locally $k^{\prime}$-integrable for every $1\leq
k^{\prime}\leq k$.

We will from now on assume that all random measures are invariant and locally
square-integrable, and we will simply refer to them as random measures, if not
stated otherwise.

By local square-integrability, the linear statistics $Sf$,
$f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$, have a well-defined _$\mu$
-expectation_

$\mathbb{E}_{\mu}(Sf)=\int_{\mathscr{M}_{+}(X)}p(f)d\mu(p),$

_$\mu$ -covariance_,

$\displaystyle\mathrm{Cov}_{\mu}(Sf_{1},Sf_{2})$
$\displaystyle=\mathbb{E}_{\mu}((Sf_{1}-\mathbb{E}_{\mu}(Sf_{2}))\overline{(Sf_{2}-\mathbb{E}_{\mu}(Sf_{2}))})$
$\displaystyle=\mathbb{E}_{\mu}(Sf_{1}\overline{Sf_{2}})-\mathbb{E}_{\mu}(Sf_{1})\overline{\mathbb{E}_{\mu}(Sf_{2})}\,,$

and _$\mu$ -variance_

$\mathrm{Var}_{\mu}(Sf)=\mathrm{Cov}_{\mu}(Sf,Sf)=\mathbb{E}_{\mu}(|Sf-\mathbb{E}_{\mu}(Sf)|^{2})=\mathbb{E}_{\mu}(|Sf|^{2})-|\mathbb{E}_{\mu}(Sf)|^{2}\,.$

By invariance of $\mu$, the linear functional $f\mapsto\mathbb{E}_{\mu}(Sf)$
on $\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$ is $G$-invariant, and since
$m_{X}$ is the only $G$-invariant positive Radon measure on $X$ up to scaling
there is a unique scalar $\iota_{\mu}>0$, the _intensity_ of $\mu$, such that

$\mathbb{E}_{\mu}(Sf)=\iota_{\mu}m_{X}(f)\,.$

The $\mu$-variance of $Sf$ can thus be written as

$\mathrm{Var}_{\mu}(Sf)=\mathbb{E}_{\mu}(|Sf|^{2})-\iota_{\mu}^{2}|m_{X}(f)|^{2}\,.$

In the case that $f=\chi_{B_{r}}$ is the indicator function of the metric ball
$B_{r}$ centered at $o\in X$, we arrive at the number variance.

######  Definition 2.1.

Let $\mu$ be an invariant locally square-integrable random measure on $X$. The
_number variance_ of $\mu$ is the function $\mathrm{NV}_{\mu}:\mathbb{R}_{\geq
0}\rightarrow\mathbb{R}_{\geq 0}$ given by

$\displaystyle\mathrm{NV}_{\mu}(r)=\mathrm{Var}_{\mu}(S\chi_{B_{r}})\,.$

######  Example 2.1 (Invariant Poisson point processes).

The _$m_{X}$ -Poisson point process_ is the random measure $\mu$ on $X$
satisfying

  1. (1)

for every bounded Borel set $B\subset X$, the linear statistic $S\chi_{B}$ is
Poisson distributed with intensity $m_{X}(B)$. In other words, for every non-
negative integer $j$,

$\displaystyle\mu\Big{(}\Big{\\{}p\in\mathscr{M}_{+}(X):p(B)=j\Big{\\}}\Big{)}=\mathrm{e}^{-m_{X}(B)}\frac{m_{X}(B)^{j}}{j!}\,.$

  2. (2)

for every finite collection $B_{1},...,B_{N}$ of disjoint bounded Borel sets
in $X$, the linear statistics $S\chi_{B_{1}},...,S\chi_{B_{N}}$ are
$\mu$-independent.

One shows that the two conditions above ensure the existence and uniqueness of
$\mu$ up to equivalence and that $\mu$ defines a locally square-integrable
simple point process [14, Theorem 3.6 + Prop. 3.2]. Moreover, since $m_{X}$ is
$G$-invariant, $\mu$ is invariant. We will write $\mu_{\mathrm{Poi}}$ for the
$m_{X}$-Poisson point process and $\mathrm{Var}_{\mathrm{Poi}}$ for the
associated variance.

######  Example 2.2 (Random lattice orbits).

Let $\Gamma<G$ be a lattice, i.e. a discrete subgroup such that there is a
(necessarily unique) $G$-invariant probability measure $m_{G/\Gamma}$ on
$G/\Gamma$. Since $\Gamma$ is a subgroup it is also uniformly discrete in $G$
in the sense that there is a (symmetric) open neighbourhood $U\subset G$ of
the identity such that $|\Gamma\cap gU|\leq 1$ for all $g\in G$. In
particular, if $Q\subset G$ is compact then a standard covering argument for
$Q$ using left translates of $U$ yields

$\sup_{g\in G}|g\Gamma\cap Q|=\sup_{g\in G}|\Gamma\cap gQ|<+\infty\,.$

We let $\Gamma_{o}=\mathrm{Stab}_{\Gamma}(o)=\Gamma\cap K$ be the stabilizer
of $o\in X$ under the action of $\Gamma$, which is finite since $\Gamma$ is
discrete and $K$ compact. The map
$G/\Gamma\rightarrow\mathscr{M}_{+}^{\mathrm{LFS}}(X)$ sending a coset
$g\Gamma$ to the measure

$\displaystyle\delta_{g\Gamma.o}=\sum_{\gamma\Gamma_{o}\in\Gamma/\Gamma_{o}}\delta_{g\gamma.o}=\frac{1}{|\Gamma_{o}|}\sum_{\gamma\in\Gamma}\delta_{g\gamma.o}$

induces a simple point process $\mu_{\Gamma}$ on $X$ as the pushforward of
$m_{G/\Gamma}$ via said map, in other words

$\displaystyle\int_{\mathscr{M}_{+}(X)}\Psi(p)d\mu_{\Gamma}(p)=\int_{G/\Gamma}\Psi(\delta_{g\Gamma.o})dm_{G/\Gamma}(g\Gamma)$

for all $\Psi\in\mathscr{L}^{\infty}(\mathscr{M}_{+}(X))$. This point process
will be invariant as $m_{G/\Gamma}$ is a $G$-invariant measure and it is
locally $k$-integrable for every $k\in\mathbb{N}$ as

$\displaystyle\int_{\mathscr{M}_{+}(X)}p(B)^{k}d\mu_{\Gamma}(p)=\int_{G/\Gamma}|g\Gamma.o\cap
B|^{k}dm_{G/\Gamma}(g\Gamma)\leq\sup_{g\in G}|g\Gamma.o\cap B|^{k}<+\infty$

by uniform discreteness of $\Gamma$. We say that $\mu_{\Gamma}$ is a _random
lattice orbit_ in $X$. In the case that $X=G$ is a proper metric group then we
say that $\mu_{\Gamma}$ is a _random lattice_.

The description of the probability measure $m_{G/\Gamma}$ can be made
explicit. A subset $F_{\Gamma}\subset G$ is a (left) _fundamental domain_ for
$\Gamma$ if it is measurable and

$\displaystyle G=\bigsqcup_{\gamma\in\Gamma}F_{\Gamma}\gamma\,.$

If $F_{\Gamma}$ is pre-compact then we say that $\Gamma$ is _cocompact_. It
follows from uniform discreteness of $\Gamma$ that $Q\cap
F_{\Gamma}\gamma=\varnothing$ for all but finitely many $\gamma\in\Gamma$
whenever $Q\subset G$ is compact. Given a fundamental domain $F_{\Gamma}$ of
$\Gamma$ in $G$ we define the _covolume_ of $\Gamma$ to be

$\mathrm{covol}(\Gamma)=m_{G}(F_{\Gamma})\,.$

The $G$-invariant probability measure on $G/\Gamma$ is then given by

$m_{G/\Gamma}=\mathrm{covol}(\Gamma)^{-1}(\pi_{\Gamma})_{*}m_{G}|_{F_{\Gamma}}\,,$

where $\pi_{\Gamma}:G\rightarrow G/\Gamma$ is the canonical quotient map. The
above definitions can be shown to be independent of the choice of fundamental
domain.

Next we construct new invariant point processes by applying random i.i.d.
perturbations to each point in the lattice orbit.

######  Example 2.3 (Random perturbed lattice orbits).

Let $\Gamma<G$ be a lattice and let $\nu\in\mathrm{Prob}(G)$ a probability
measure.

Consider the space $Z=G^{\Gamma}$ of $\Gamma$-indexed sequences
$z=(z_{\gamma})_{\gamma\in\Gamma}$ in $G$, endowed with the product topology.
The group $\Gamma$ acts on $Z$ by right translation,
$\gamma.(z_{\gamma^{\prime}})_{\gamma^{\prime}\in\Gamma}=(z_{\gamma^{\prime}\gamma})_{\gamma\in\Gamma}$
and it also acts on $G$ canonically from the right by $g\mapsto g\gamma^{-1}$
for each $\gamma\in\Gamma$. The action of $G\times\Gamma$ on $G\times Z$ given
by $(g,\gamma).(z,h)=(gh\gamma^{-1},\gamma.z)$ for $g,h\in G$,
$\gamma\in\Gamma$ and $z\in Z$ is well-defined and continuous, and we let
$Y=(G\times Z)/\Gamma$ be the space of $\Gamma$-orbits. Explicitly, each $y\in
Y$ is of the form

$\displaystyle y=(g,z)\Gamma=\big{\\{}(g\gamma,\gamma.z)\in G\times
G^{\Gamma}:\gamma\in\Gamma\big{\\}}$

for some $g\in G$, $z\in G^{\Gamma}$. On $Y$ we define a $G$-invariant
positive probability measure $m_{Y}$ by

$\displaystyle
m_{Y}(F)=\int_{G/\Gamma}\Big{(}\int_{Z}F((g,z)\Gamma)d\nu^{\otimes\Gamma}(z)\Big{)}dm_{G/\Gamma}(g\Gamma)$

for all $F\in\mathscr{L}^{\infty}(Y)$, where $\nu^{\otimes\Gamma}$ is the
product probability measure on $Z$.

To construct a $\nu$-perturbed random $\Gamma$-orbit in $X$ we first define a
$\nu$-perturbed random $\Gamma$-orbit in $G$ by considering the measures

$\displaystyle\delta_{(g,z)\Gamma}=\sum_{\gamma\in\Gamma}\delta_{g\gamma
z_{\gamma}}\,.$

######  Lemma 2.2.

Let $\Gamma<G$ be a lattice and $\nu\in\mathrm{Prob}(G)$. Then

  1. (1)

$\delta_{(g,z)\Gamma}$ is locally finite
$m_{G/\Gamma}\otimes\nu^{\otimes\Gamma}$-almost everywhere.

  2. (2)

If $\nu$ is absolutely continuous with respect to the Haar measure $m_{G}$,
then $\delta_{(g,z)\Gamma}$ is simple
$m_{G/\Gamma}\otimes\nu^{\otimes\Gamma}$-almost everywhere.

###### Proof.

For (1), let $Q\subset G$ be compact. We want to show that

$\displaystyle\nu^{\otimes\Gamma}\Big{(}\Big{\\{}z\in Z:\gamma z_{\gamma}\in
Q\mbox{ for infinitely many }\gamma\in\Gamma\Big{\\}}\Big{)}=0\,.$

By the Borel-Cantelli Lemma, it suffices to show that

$\displaystyle\sum_{\gamma\in\Gamma}\nu^{\otimes\Gamma}\Big{(}\Big{\\{}z\in
Z:\gamma z_{\gamma}\in
Q\Big{\\}}\Big{)}=\sum_{\gamma\in\Gamma}\nu(\gamma^{-1}Q)<+\infty\,,$

which follows from a standard covering argument by picking $\Gamma$-injective
open sets $U_{1},...,U_{N}\subset G$ that cover $Q$ and using that $\nu$ is a
probability measure.

For (2) it is enough to show that the event
$\gamma_{1}z_{\gamma_{1}}=\gamma_{2}z_{\gamma_{2}}$ is a null set in $Z$ with
respect to $\nu^{\otimes\Gamma}$ for all $\gamma_{1}\neq\gamma_{2}$.
Explicitly, if we set $B_{\gamma_{1},\gamma_{2}}=\\{z\in
Z:\gamma_{1}z_{\gamma_{1}}=\gamma_{2}z_{\gamma_{2}}\\}$ then we aim to show
that $\nu^{\otimes\Gamma}(B_{\gamma_{1},\gamma_{2}})=0$ for
$\gamma_{1}\neq\gamma_{2}$ in $\Gamma$. Since $\Gamma$ is countable it will
follow by $\sigma$-subadditivity that

$\displaystyle\nu^{\otimes\Gamma}\Big{(}\Big{\\{}z\in
Z:\gamma_{1}z_{\gamma_{1}}=\gamma_{2}z_{\gamma_{2}}\,\,\mbox{for
some}\,\gamma_{1}\neq\gamma_{2}\Big{\\}}\Big{)}\leq\sum_{\gamma_{1}\neq\gamma_{2}}\nu^{\otimes\Gamma}(B_{\gamma_{1},\gamma_{2}})=0\,.$

Using that $\nu^{\otimes\Gamma}$ is a product measure, we get

$\displaystyle\nu^{\otimes\Gamma}(B_{\gamma_{1},\gamma_{2}})$
$\displaystyle=(\nu\otimes\nu)\Big{(}\Big{\\{}(z_{1},z_{2})\in G\times
G:\gamma_{1}z_{1}=\gamma_{2}z_{2}\Big{\\}}\Big{)}$
$\displaystyle=(\nu\otimes\nu)\Big{(}\Big{\\{}(z_{1},\gamma_{2}^{-1}\gamma_{1}z_{1})\in
G\times G:z_{1}\in G\Big{\\}}\Big{)}\,.$

By Fubini’s Theorem,

$\displaystyle\nu^{\otimes\Gamma}(B_{\gamma_{1},\gamma_{2}})$
$\displaystyle=\int_{G}\nu(\\{\gamma_{2}^{-1}\gamma_{1}z_{1}\\})d\nu(z_{1})\,,$

so if $\nu$ is absolutely continuous with respect to the Haar measure $m_{G}$
then $\nu(\\{\gamma_{2}^{-1}\gamma_{1}z_{1}\\})=0$ for all $z_{1}\in G$ and
$\nu^{\otimes\Gamma}(B_{\gamma_{1},\gamma_{2}})=0$. ∎

We now assume that $\nu\in\mathrm{Prob}(G)$ is absolutely continuous with
respect to the Haar measure $m_{G}$. By Lemma 2.2 the invariant random measure
$\widetilde{\mu}_{\Gamma,\nu}$ obtained as the push-forward of $m_{Y}$ via the
map $(g,z)\Gamma\mapsto\delta_{(g,z)\Gamma}$ is an invariant simple point
process on $G$, explicitly given by

$\displaystyle\int_{\mathscr{M}_{+}(G)}\Psi(q)d\widetilde{\mu}_{\Gamma,\nu}(q)=\int_{G/\Gamma}\Big{(}\int_{Z}\Psi(\delta_{(g,z)\Gamma})d\nu^{\otimes\Gamma}(z)\Big{)}dm_{G/\Gamma}(g\Gamma)$

for all $\Psi\in\mathscr{L}^{\infty}(\mathscr{M}_{+}(G))$. In order to
construct a locally finite simple $\nu$-perturbed random $\Gamma$-orbit in $X$
we consider the measures

$p_{(g,z)\Gamma}=\pi_{*}\delta_{(g,z)\Gamma}=\sum_{\gamma\in\Gamma}\delta_{g\gamma
z_{\gamma}.o}\,,$

where $\pi:G\rightarrow X$ is the orbit map $g\mapsto g.o$. The following
Proposition provides conditions for when these measures are almost everywhere
locally finite and simple with respect to
$m_{G/\Gamma}\otimes\nu^{\otimes\Gamma}$.

######  Proposition 2.3.

Suppose that $\nu\in\mathrm{Prob}(G)$ is right-$K$-invariant and absolutely
continuous with respect to the Haar measure $m_{G}$. Moreover, assume that
$m_{G}(K)=0$. Then the measures $p_{(g,z)\Gamma}$ are locally finite and
simple $m_{G/\Gamma}\otimes\nu^{\otimes\Gamma}$-almost everywhere.

Under these assumptions we get an invariant simple point process in $X$ as the
push-forward of $\widetilde{\mu}_{\Gamma,\nu}$ along the map
$\mathscr{M}_{+}(G)\ni q\mapsto\pi_{*}q\in\mathscr{M}_{+}(X)$.

######  Definition 2.4 (Random perturbed lattice orbit).

Let $\Gamma<G$ be a lattice and $\nu\in\mathrm{Prob}(G)$ a right-$K$-invariant
probability measure absolutely continuous with respect to the Haar measure
$m_{G}$. The _$\nu$ -perturbed $\Gamma$-orbit_ in $X=G/K$ is the simple point
process $\mu_{\Gamma,\nu}$ given by

$\displaystyle\int_{\mathscr{M}_{+}(X)}\Psi(p)d\mu_{\Gamma,\nu}(p)=\int_{G/\Gamma}\Big{(}\int_{Z}\Psi(p_{(g,z)\Gamma})d\nu^{\otimes\Gamma}(z)\Big{)}dm_{G/\Gamma}(g\Gamma)$

for all $\Psi\in\mathscr{L}^{\infty}(\mathscr{M}_{+}(X))$.

######  Remark 2.5.

If $K$ is compact and open in $G$, for example when $X$ is the homogeneous
$p$-regular tree and $G=\mathrm{PGL}_{2}(\mathbb{Q}_{p})$,
$K=\mathrm{PGL}_{2}(\mathbb{Z}_{p})$, then $m_{G}(K)>0$. In this case there
are lattices $\Gamma$ and probability measures $\nu$ absolutely continuous
with respect to the Haar measure on $G$ for which the associated random
perturbed lattice orbit is not a simple point process.

###### Proof of Proposition 2.3.

Similarly to the proof of item (2) in Lemma 2.2, it is enough to show that
$\nu^{\otimes\Gamma}(B_{\gamma_{1},\gamma_{2}})=0$, where

$\displaystyle B_{\gamma_{1},\gamma_{2}}$ $\displaystyle=\Big{\\{}z\in
Z:\gamma_{1}z_{\gamma_{1}}.o=\gamma_{2}z_{\gamma_{2}}.o\Big{\\}}=\Big{\\{}z\in
Z:\gamma_{2}z_{\gamma_{2}}\in\gamma_{1}z_{\gamma_{1}}K\Big{\\}}\,.$

Using that $\nu^{\otimes\Gamma}$ is a product measure, we get

$\displaystyle\nu^{\otimes\Gamma}(B_{\gamma_{1},\gamma_{2}})$
$\displaystyle=(\nu\otimes\nu)\Big{(}\Big{\\{}(z_{1},z_{2})\in G\times
G:\gamma_{2}z_{2}\in\gamma_{1}z_{1}K\Big{\\}}\Big{)}$
$\displaystyle=(\nu\otimes\nu)\Big{(}\Big{\\{}(z_{1},\gamma_{2}^{-1}\gamma_{1}z_{1}k)\in
G\times G:z_{1}\in G\,,\,k\in K\Big{\\}}\Big{)}\,.$

By Fubini’s Theorem,

$\displaystyle\nu^{\otimes\Gamma}(B_{\gamma_{1},\gamma_{2}})$
$\displaystyle=\int_{G}\nu(\gamma_{2}^{-1}\gamma_{1}z_{1}K)d\nu(z_{1})\,.$

Now, if $\nu$ is absolutely continuous with respect to the Haar measure
$m_{G}$ then $\nu(\gamma_{2}^{-1}\gamma_{1}z_{1}K)$ vanishes for all $z_{1}\in
G$ if and only if $m_{G}(K)=0$, so
$\nu^{\otimes\Gamma}(B_{\gamma_{1},\gamma_{2}})=0$. ∎

Next we address local square-integrability of $\mu_{\Gamma,\nu}$, which in the
general setting requires the following conditions on either $\Gamma$ or $\nu$.

######  Lemma 2.6.

Let $\Gamma<G$ be a lattice with fundamental domain $F_{\Gamma}$ and $\nu$ be
a right-$K$-invariant probability measure on $G$. If $\Gamma$ is cocompact or
the support $\mathrm{supp}(\nu)$ of $\nu$ is compact then the random perturbed
lattice orbit $\mu_{\Gamma,\nu}$ on $X$ is locally square-integrable.

###### Proof.

Let $B\subset\mathbb{R}^{n}$ be a bounded Borel set. Then by definition of
$\mu_{\Gamma,\nu}$,

$\displaystyle\mathbb{E}_{\mu_{\Gamma,\nu}}(|S\chi_{B}|^{2})=\int_{G/\Gamma}\Big{(}\sum_{\gamma_{1},\gamma_{2}\in\Gamma}\int_{Z}$
$\displaystyle\chi_{B}(g\gamma_{1}z_{\gamma_{1}}.o)\chi_{B}(g\gamma_{2}z_{\gamma_{2}}.o)d\nu^{\otimes\Gamma}(z)\Big{)}dm_{G/\Gamma}(g\Gamma)\,.$

Note how

$\displaystyle\int_{Z}\chi_{B}(g\gamma_{1}z_{\gamma_{1}}.o)$
$\displaystyle\chi_{B}(g\gamma_{2}z_{\gamma_{2}}.o)d\nu^{\otimes\Gamma}(z)=\begin{dcases}\nu(\gamma^{-1}g^{-1}Q)&\mbox{
if }\gamma_{1}=\gamma_{2}=\gamma\\\
\nu(\gamma_{1}^{-1}g^{-1}Q)\nu(\gamma_{2}^{-1}g^{-1}Q)&\mbox{ if
}\gamma_{1}\neq\gamma_{2}\end{dcases}$

where $Q=\pi^{-1}(B)=\varsigma(B)K$ is measurable and pre-compact. Splitting
into these two cases we have that

$\displaystyle\mathbb{E}_{\mu_{\Gamma,\nu}}(|S\chi_{B}|^{2})$
$\displaystyle=\int_{G/\Gamma}\Big{(}\sum_{\gamma\in\Gamma}\nu(\gamma^{-1}g^{-1}Q)\Big{)}dm_{G/\Gamma}(g\Gamma)$
$\displaystyle+\int_{G/\Gamma}\Big{(}\sum_{\gamma_{1}\neq\gamma_{2}}\nu(\gamma_{1}^{-1}g^{-1}Q)\nu(\gamma_{2}^{-1}g^{-1}Q)\Big{)}dm_{G/\Gamma}(g\Gamma)\,.$

The first term can be computed using Fubini and the $G$-invariance of
$m_{G/\Gamma}$ as

$\displaystyle\int_{G/\Gamma}\Big{(}\sum_{\gamma\in\Gamma}\nu(\gamma^{-1}g^{-1}Q)\Big{)}dm_{G/\Gamma}(g\Gamma)=\frac{m_{G}(Q)}{\mathrm{covol}(\Gamma)}<+\infty\,.$

We postpone the details of this computation to Example 2.6.

For the second term, it suffices to show that

$\displaystyle
I_{\Gamma,\nu}:=\int_{G/\Gamma}\Big{(}\sum_{\gamma\in\Gamma}\nu(\gamma^{-1}g^{-1}Q)\Big{)}^{2}dm_{G/\Gamma}(g\Gamma)<+\infty\,.$
(2.2)

First, if $\Gamma$ is cocompact with fundamental domain $F_{\Gamma}$ then we
use Fubini to write

$\displaystyle
I_{\Gamma,\nu}=\frac{1}{\mathrm{covol}(\Gamma)}\sum_{\gamma_{1},\gamma_{2}}\int_{G}\int_{G}\Big{(}\int_{F_{\Gamma}}\chi_{Q}(g\gamma_{1}h_{1})\chi_{Q}(g\gamma_{2}h_{2})dm_{G}(g)\Big{)}d\nu(h_{1})d\nu(h_{2})\,.$

If we let $Q^{\prime}=F_{\Gamma}^{-1}Q$, then $Q^{\prime}$ is measurable and
pre-compact in $G$ and $\chi_{Q}(g\gamma h)\leq\chi_{Q^{\prime}}(\gamma h)$
for all $g\in F_{\Gamma},h\in G,\gamma\in\Gamma$, so

$\displaystyle\frac{1}{\mathrm{covol}(\Gamma)}\int_{F_{\Gamma}}\chi_{Q}(g\gamma_{1}h_{1})\chi_{Q}(g\gamma_{2}h_{2})dm_{G}(g)\leq\chi_{Q^{\prime}}(\gamma_{1}h_{1})\chi_{Q^{\prime}}(\gamma_{2}h_{2})\,.$
(2.3)

Thus the expression in Equation 2.2 is bounded by

$\displaystyle
I_{\Gamma,\nu}\leq\sum_{\gamma_{1},\gamma_{2}}\int_{G}\int_{G}\chi_{Q^{\prime}}(\gamma_{1}h_{1})\chi_{Q^{\prime}}(\gamma_{2}h_{2})d\nu(h_{1})d\nu(h_{2})=\Big{(}\sum_{\gamma\in\Gamma}\nu(\gamma^{-1}Q^{\prime})\Big{)}^{2}\,,$

which is finite by the same argument as in the proof of item (1) in Lemma 2.2.

Secondly, if $\mathrm{supp}(\nu)$ is compact and we let
$Q^{\prime\prime}=Q\,\mathrm{supp}(\nu)^{-1}$, then $Q^{\prime\prime}$ is
measurable and pre-compact in $G$ and by the same argument as for Equation 2.3
we have

$\displaystyle\int_{G}\int_{G}\chi_{Q}(g\gamma_{1}h_{1})\chi_{Q}(g\gamma_{2}h_{2})d\nu(h_{1})d\nu(h_{2})\leq\chi_{Q^{\prime\prime}}(g\gamma_{1})\chi_{Q^{\prime\prime}}(g\gamma_{2})\,.$

This bound along with a standard computation yields

$\displaystyle I_{\Gamma,\nu}$
$\displaystyle\leq\frac{1}{\mathrm{covol}(\Gamma)}\int_{F_{\Gamma}}\sum_{\gamma_{1},\gamma_{2}}\chi_{Q^{\prime\prime}}(g\gamma_{1})\chi_{Q^{\prime\prime}}(g\gamma_{2})dm_{G}(g)$
$\displaystyle=\frac{1}{\mathrm{covol}(\Gamma)}\sum_{\gamma_{1}\in\Gamma}\int_{G}\chi_{Q^{\prime\prime}}(g\gamma_{1})\chi_{Q^{\prime\prime}}(g)dm_{G}(g)=\frac{1}{\mathrm{covol}(\Gamma)}\sum_{\gamma_{1}\in\Gamma}m_{G}(Q^{\prime\prime}\gamma_{1}^{-1}\cap
Q^{\prime\prime})\,.$

Since $Q^{\prime\prime}$ is pre-compact, the latter series is a sum over
finitely many $\gamma_{1}\in\Gamma$ and hence finite. ∎

We will give more examples of perturbations $\nu$ for which random perturbed
lattice orbits on real hyperbolic spaces are locally square-integrable in
section 10.

###  2.3. Autocorrelation measures

The autocorrelation measure of a random measure is a signed Radon measure on
$G$ that uniquely determines the $\mu$-covariance of linear statistics,

$\displaystyle\mathrm{Cov}_{\mu}(Sf_{1},Sf_{2})$
$\displaystyle=\int_{\mathscr{M}_{+}(X)}p(f_{1})\overline{p(f_{2})}d\mu(p)-\iota_{\mu}^{2}m_{X}(f_{1})\overline{m_{X}(f_{2})}\,,\quad
f_{1},f_{2}\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)\,.$

The definition is the following.

######  Definition 2.7 (Autocorrelation measures).

Let $b\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$ be a positive function with
$m_{X}(b)=1$. The _autocorrelation measure_ of an invariant locally square-
integrable random measure $\mu$ on $X$ is the bi-$K$-invariant signed Radon
measure $\eta_{\mu}$ on $G$ defined by

$\displaystyle\eta_{\mu}(\varphi)=\int_{\mathscr{M}_{+}(X)}\Big{(}\int_{X}\int_{X}\varphi(\varsigma(x)^{-1}\varsigma(y))b(x)dp(x)dp(y)\Big{)}d\mu(p)-\iota_{\mu}^{2}\int_{G}\varphi(g)dm_{G}(g)\,.$

for all $\varphi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G)$.

The truncation function $b\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$ is
necessary for the definition as the bivariante functions
$(x,y)\mapsto\varphi(\varsigma(x)^{-1}\varsigma(y))$ do not have bounded
support. However, the definition does not depend on the choice of $b$ by the
following Lemma.

######  Lemma 2.8.

The autocorrelation measure $\eta_{\mu}$ of an invariant locally square-
integrable random measure $\mu$ satisfies

$\displaystyle\eta_{\mu}(\psi^{*}*\varphi)=\mathrm{Cov}_{\mu}(S(\varphi\circ\varsigma),S(\psi\circ\varsigma))$

for all right-$K$-invariant
$\varphi,\psi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G)$. In particular, the
definition of $\eta_{\mu}$ is independent of the choice of
$b\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$.

###### Proof.

Assume without loss of generality that $m_{G}(\varphi)=m_{G}(\psi)=0$, so that
$m_{G}(\psi^{*}*\varphi)=\overline{m_{G}(\psi)}m_{G}(\varphi)=0$. Then it
suffices to show that

$\displaystyle\eta_{\mu}(\psi^{*}*\varphi)=\int_{\mathscr{M}_{+}(X)}\Big{(}\int_{X}\int_{X}\overline{\psi(\varsigma(x))}\varphi(\varsigma(y))dp(x)dp(y)\Big{)}d\mu(p)\,.$

Given this identity, the subspace of functions of the form $\psi^{*}*\varphi$
are enough to uniquely determine $\eta_{\mu}$, so $\eta_{\mu}$ will be
independent of the choice of $b$.

First, we write

$\displaystyle(\psi^{*}*\varphi)(\varsigma(x)^{-1}\varsigma(y))=\int_{G}\overline{\psi(h\varsigma(x))}\varphi(h\varsigma(y))dm_{G}(h)$

and using Fubini, we get that

$\displaystyle\eta_{\mu}(\psi^{*}*\varphi)=\int_{G}\int_{\mathscr{M}_{+}(X)}\Big{(}\int_{X}\int_{X}\overline{\psi(h\varsigma(x))}\varphi(h\varsigma(y))b(x)dp(x)dp(y)\Big{)}d\mu(p)dm_{G}(h)\,.$

Since $h\varsigma(x).o=h.x=\varsigma(h.x).o$ for all $x\in X$ then
$\psi(h\varsigma(x))=\psi(\varsigma(h.x))$ and
$\varphi(h\varsigma(y))=\varphi(\varsigma(h.y))$ by right-$K$-invariance. Thus

$\displaystyle\int_{X}\int_{X}\overline{\psi(h\varsigma(x))}\varphi(h\varsigma(y))b(x)dp(x)dp(y)=\int_{X}\int_{X}\overline{\psi(\varsigma(x))}\varphi(\varsigma(y))b(h^{-1}.x)dh_{*}p(x)dh_{*}p(y)$

and by the $G$-invariance of $\mu$, we see that

$\displaystyle\eta_{\mu}(\psi^{*}*\varphi)$
$\displaystyle=\int_{G}\int_{\mathscr{M}_{+}(X)}\Big{(}\int_{X}\int_{X}\overline{\psi(\varsigma(x))}\varphi(\varsigma(y))b(h^{-1}.x)dp(x)dp(y)\Big{)}d\mu(p)dm_{G}(h)$
$\displaystyle=\int_{\mathscr{M}_{+}(X)}\Big{(}\int_{X}\int_{X}\Big{(}\int_{G}b(h^{-1}.x)dm_{G}(h)\Big{)}\overline{\psi(\varsigma(x))}\varphi(\varsigma(y))dp(x)dp(y)\Big{)}d\mu(p)$
$\displaystyle=m_{X}(b)\int_{\mathscr{M}_{+}(X)}\Big{(}\int_{X}\int_{X}\overline{\psi(\varsigma(x))}\varphi(\varsigma(y))dp(x)dp(y)\Big{)}d\mu(p)\,.$

Since $b\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$ was assumed to satisfy
$m_{X}(b)=1$ then we are done. ∎

######  Example 2.4 (Autocorrelation of the invariant Poisson point process).

Let $f=\sum_{j=1}^{N}a_{j}\chi_{B_{j}}$ be a simple function for disjoint and
bounded Borel sets $B_{1},...,B_{N}$ in $X$. Since the linear statistics
$S\chi_{B_{1}},...,S\chi_{B_{N}}$ are Poisson distributed and independent, the
variance of the linear statistic $Sf$ with respect to the invariant Poisson
point process $\mu_{\mathrm{Poi}}$ is

$\displaystyle\mathrm{Var}_{\mathrm{Poi}}(Sf)$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{N}a_{i}\overline{a_{j}}\mathrm{Cov}_{\mathrm{Poi}}(S\chi_{B_{i}},S\chi_{B_{j}})=\sum_{j=1}^{N}|a_{j}|^{2}\mathrm{Var}_{\mathrm{Poi}}(S\chi_{B_{j}})$
$\displaystyle=\sum_{j=1}^{N}|a_{j}|^{2}m_{X}(B_{j})=\int_{X}|f(x)|^{2}dm_{X}(x)\,.$

The latter can be rewritten as
$\int_{X}|f(x)|^{2}dm_{X}(x)=\int_{G}|\varphi_{f}(g)|^{2}dm_{G}(g)=(\varphi_{f}^{*}*\varphi_{f})(e)$,
so that

$\displaystyle\mathrm{Var}_{\mathrm{Poi}}(Sf)=\delta_{e}(\varphi_{f}^{*}*\varphi_{f})\,.$

This identity extends to every $f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$
and by polarization we find that the autocorrelation measure of the
$m_{X}$-Poisson point process is $\eta_{\mathrm{Poi}}=\delta_{e}$ as a linear
functional on $\mathscr{L}^{\infty}_{\mathrm{bnd}}(G,K)$. In particular, the
number variance of the process is

$\displaystyle\mathrm{NV}_{\mathrm{Poi}}(R)=\int_{X}|\chi_{B_{R}}(x)|^{2}dm_{X}(x)=\mathrm{Vol}_{X}(B_{R})\,.$

######  Example 2.5 (Intensity and autocorrelation of a random lattice orbit).

Using the explicit description for the probability measure $m_{G/\Gamma}$ from
Example 2.2, we now compute the autocorrelation
$\eta_{\Gamma}:=\eta_{\mu_{\Gamma}}$. The expectation of linear statistics can
be computed using a fundamental domain $F_{\Gamma}\subset G$ of $\Gamma$ to be

$\displaystyle\mathbb{E}_{\mu_{\Gamma}}(Sf)$
$\displaystyle=|\Gamma_{o}|^{-1}\mathrm{covol}(\Gamma)^{-1}m_{G}(\varphi_{f})$

for every $f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$, where
$\varphi_{f}(g)=f(g.o)$. In particular, the intensity is
$\iota_{\Gamma}:=\iota_{\mu_{\Gamma}}=|\Gamma_{o}|^{-1}\mathrm{covol}(\Gamma)^{-1}$.
The second correlation is computed in a similar way to be

$\displaystyle\mathbb{E}_{\mu_{\Gamma}}(Sf_{1}\overline{Sf}_{2})$
$\displaystyle=|\Gamma_{o}|^{-2}\mathrm{covol}(\Gamma)^{-1}\sum_{\gamma\in\Gamma}\int_{G}f_{1}(g\gamma.o)\overline{f_{2}(g.o)}dm_{G}(g)$
$\displaystyle=|\Gamma_{o}|^{-2}\mathrm{covol}(\Gamma)^{-1}\sum_{\gamma\in\Gamma}(\varphi_{f_{2}}^{*}*\varphi_{f_{1}})(\gamma)$

for all $f_{1},f_{2}\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$. With this the
autocorrelation measure can be identified as

$\displaystyle\eta_{\Gamma}(\varphi^{*}*\varphi)$
$\displaystyle=\frac{1}{|\Gamma_{o}|\mathrm{covol}(\Gamma)}\Big{(}\frac{1}{|\Gamma_{o}|}\sum_{\gamma\in\Gamma}(\varphi^{*}*\varphi)(\gamma)-\frac{1}{|\Gamma_{o}|\mathrm{covol}(\Gamma)}\Big{|}\int_{G}\varphi(g)dm_{G}(g)\Big{|}^{2}\Big{)}$

for all right-$K$-invariant
$\varphi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G)$. In shorter notation,

$\displaystyle\eta_{\Gamma}(\varphi)=\iota_{\Gamma}\Big{(}\frac{1}{|\Gamma_{o}|}\delta_{\Gamma}(\varphi)-\iota_{\Gamma}m_{G}(\varphi)\Big{)}$
(2.4)

for all bi-$K$-invariant functions
$\varphi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G,K)$.

######  Example 2.6 (Intensity and autocorrelation of a random perturbed
lattice orbit).

Let $G$ be such that $m_{G}(K)=0$, $\Gamma<G$ a lattice and
$\nu\in\mathrm{Prob}(G)$ a right-$K$-invariant measure on $G$ absolutely
continuous with respect to the Haar measure $m_{G}$. The expected value of a
linear statistic $Sf$, $f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(X)$, with
respect to the random perturbed lattice orbit $\mu_{\Gamma,\nu}$ can be
computed using the $\Gamma$-invariance of $\nu^{\otimes\Gamma}$ and the same
computation as for the random lattice orbit. Fubini yields

$\displaystyle\mathbb{E}_{\mu_{\Gamma,\nu}}(Sf)$
$\displaystyle=\int_{G/\Gamma}\Big{(}\int_{Z}\sum_{\gamma\in\Gamma}f(g\gamma
z_{\gamma}.o)d\nu^{\otimes\Gamma}(z)\Big{)}dm_{G/\Gamma}(g\Gamma)$
$\displaystyle=\int_{G/\Gamma}\sum_{\gamma\in\Gamma}\Big{(}\int_{G}f(g\gamma
h.o)d\nu(h)\Big{)}dm_{G/\Gamma}(g\Gamma)$
$\displaystyle=\int_{G}\Big{(}\int_{G/\Gamma}\sum_{\gamma\in\Gamma}f(g\gamma
h.o)dm_{G/\Gamma}(g\Gamma)\Big{)}d\nu(h)\,.$

By the computation for the intensity of random lattice orbits and the
unimodularity of $G$ we get

$\displaystyle\mathbb{E}_{\mu_{\Gamma,\nu}}(Sf)=\frac{1}{\mathrm{covol}(\Gamma)}\int_{G}\Big{(}\int_{G}f(gh.o)dm_{G}(g)\Big{)}d\nu(h)=\frac{m_{G}(\varphi_{f})}{\mathrm{covol}(\Gamma)}\,.$

Thus the intensity of $\mu_{\Gamma,\nu}$ is
$\iota_{\Gamma,\nu}=\mathrm{covol}(\Gamma)^{-1}$, in particular independent of
$\nu$.

For the second moment of the process $\mu_{\Gamma,\nu}$, we assume that
$\Gamma$ is cocompact or $\mathrm{supp}(\nu)$ is compact, so that
$\mu_{\Gamma,\nu}$ is locally square-integrable by Lemma 2.6. Using Fubini,

$\displaystyle\mathbb{E}_{\mu_{\Gamma,\nu}}(Sf_{1}\overline{Sf_{2}})=\int_{G/\Gamma}\Big{(}\sum_{\gamma_{1},\gamma_{2}\in\Gamma}\int_{Z}f_{1}(g\gamma_{1}z_{\gamma_{1}}.o)\overline{f_{2}(g\gamma_{2}z_{\gamma_{2}}.o)}d\nu^{\otimes\Gamma}(z)\Big{)}dm_{G/\Gamma}(g\Gamma)\,.$

Note how

$\displaystyle\int_{Z}f_{1}(g\gamma_{1}z_{\gamma_{1}}.o)\overline{f_{2}(g\gamma_{2}z_{\gamma_{2}}.o)}d\nu^{\otimes\Gamma}(z)=\begin{dcases}(\varphi_{f_{1}}\overline{\varphi}_{f_{2}}*\check{\nu})(g\gamma)&\mbox{
if }\gamma_{1}=\gamma_{2}=\gamma\\\
(\varphi_{f_{1}}*\check{\nu})(g\gamma_{1})\overline{(\varphi_{f_{2}}*\check{\nu})(g\gamma_{2})}&\mbox{
if }\gamma_{1}\neq\gamma_{2}\,,\end{dcases}$

where $\varphi_{f}(g)=f(g.o)$ as before and $\check{\nu}$ is the
left-$K$-invariant probability measure on $G$ defined by
$\check{\nu}(Q)=\nu(Q^{-1})$ for Borel sets $Q\subset G$. From this we get
that

$\displaystyle\mathbb{E}_{\mu_{\Gamma,\nu}}(Sf_{1}\overline{Sf_{2}})$
$\displaystyle=\int_{G/\Gamma}\Big{(}\sum_{\gamma_{1}\neq\gamma_{2}}(\varphi_{f_{1}}*\check{\nu})(g\gamma_{1})\overline{(\varphi_{f_{2}}*\check{\nu})(g\gamma_{2})}\Big{)}dm_{G/\Gamma}(g\Gamma)$
$\displaystyle\quad\quad\quad+\int_{G/\Gamma}\Big{(}\sum_{\gamma\in\Gamma}(\varphi_{f_{1}}\overline{\varphi}_{f_{2}}*\check{\nu})(g\gamma)\Big{)}dm_{G/\Gamma}(g\Gamma)$
$\displaystyle=\int_{G/\Gamma}\Big{(}\sum_{\gamma_{1},\gamma_{2}\in\Gamma}(\varphi_{f_{1}}*\check{\nu})(g\gamma_{1})\overline{(\varphi_{f_{2}}*\check{\nu})(g\gamma_{2})}\Big{)}dm_{G/\Gamma}(g\Gamma)$
$\displaystyle\quad\quad\quad-\int_{G/\Gamma}\Big{(}\sum_{\gamma\in\Gamma}(\varphi_{f_{1}}*\check{\nu})(g\gamma)\overline{(\varphi_{f_{2}}*\check{\nu})(g\gamma)}\Big{)}dm_{G/\Gamma}(g\Gamma)$
$\displaystyle\quad\quad\quad+\int_{G/\Gamma}\Big{(}\sum_{\gamma\in\Gamma}(\varphi_{f_{1}}\overline{\varphi}_{f_{2}}*\check{\nu})(g\gamma)\Big{)}dm_{G/\Gamma}(g\Gamma)\,.$

From the computations made for the random lattice orbits,

$\displaystyle\int_{G/\Gamma}\Big{(}\sum_{\gamma_{1},\gamma_{2}\in\Gamma}(\varphi_{f_{1}}*\check{\nu})(g\gamma_{1})$
$\displaystyle\overline{(\varphi_{f_{2}}*\check{\nu})(g\gamma_{2})}\Big{)}dm_{G/\Gamma}(g\Gamma)=$
$\displaystyle=\frac{1}{\mathrm{covol}(\Gamma)}\sum_{\gamma\in\Gamma}(\nu*\varphi_{f_{2}}^{*}*\varphi_{f_{1}}*\check{\nu})(\gamma)\,,$
$\displaystyle\int_{G/\Gamma}\Big{(}\sum_{\gamma\in\Gamma}(\varphi_{f_{1}}*\check{\nu})(g\gamma)$
$\displaystyle\overline{(\varphi_{f_{2}}*\check{\nu})(g\gamma)}\Big{)}dm_{G/\Gamma}(g\Gamma)=$
$\displaystyle=\frac{1}{\mathrm{covol}(\Gamma)}\int_{G}(\varphi_{f_{1}}*\check{\nu})(g)\overline{(\varphi_{f_{2}}*\check{\nu})(g)}dm_{G}(g)\,,$

and by unimodularity of $G$,

$\displaystyle\int_{G/\Gamma}\Big{(}\sum_{\gamma\in\Gamma}(\varphi_{f_{1}}\overline{\varphi}_{f_{2}}*\check{\nu})(g\gamma)\Big{)}dm_{G/\Gamma}(g\Gamma)$
$\displaystyle=\frac{1}{\mathrm{covol}(\Gamma)}\int_{G}(\varphi_{f_{1}}\overline{\varphi}_{f_{2}}*\check{\nu})(g)dm_{G}(g)$
$\displaystyle=\frac{1}{\mathrm{covol}(\Gamma)}\int_{G}\Big{(}\int_{G}\varphi_{f_{1}}(gh)\overline{\varphi_{f_{2}}(gh)}dm_{G}(g)\Big{)}d\nu(h)$
$\displaystyle=\frac{1}{\mathrm{covol}(\Gamma)}\int_{G}\varphi_{f_{1}}(g)\overline{\varphi_{f_{2}}(g)}dm_{G}(g)\,.$

Thus

$\displaystyle\mathbb{E}_{\mu_{\Gamma,\nu}}(Sf_{1}\overline{Sf_{2}})=\frac{1}{\mathrm{covol}(\Gamma)}\Big{(}$
$\displaystyle\sum_{\gamma\in\Gamma}(\nu*\varphi_{f_{2}}^{*}*\varphi_{f_{1}}*\check{\nu})(\gamma)$
$\displaystyle-\int_{G}(\varphi_{f_{1}}*\check{\nu})(g)\overline{(\varphi_{f_{2}}*\check{\nu})(g)}dm_{G}(g)$
$\displaystyle+\int_{G}\varphi_{f_{1}}(g)\overline{\varphi_{f_{2}}(g)}dm_{G}(g)\Big{)}\,,$

so the autocorrelation measure $\eta_{\Gamma,\nu}$ of $\mu_{\Gamma,\nu}$ is

$\displaystyle\eta_{\Gamma,\nu}(\varphi^{*}*\varphi)$
$\displaystyle=\frac{1}{\mathrm{covol}(\Gamma)}\Big{(}\sum_{\gamma\in\Gamma}(\nu*\varphi^{*}*\varphi*\check{\nu})(\gamma)-\int_{G}|(\varphi*\check{\nu})(g)|^{2}dm_{G}(g)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\int_{G}|\varphi(g)|^{2}dm_{G}(g)-\frac{1}{\mathrm{covol}(\Gamma)}\Big{|}\int_{G}\varphi(g)dm_{G}(g)\Big{|}^{2}\Big{)}$
$\displaystyle=|\Gamma_{o}|^{2}\mathrm{Var}_{\mu_{\Gamma}}(S((\varphi*\check{\nu})\circ\varsigma)+\mathrm{covol}(\Gamma)^{-1}\big{(}m_{G}(|\varphi|^{2})-m_{G}(|\varphi*\check{\nu}|^{2})\big{)}\,.$

In short hand notation, considering the autocorrelation measure as a
functional on the sub-algebra $\mathscr{L}^{\infty}_{\mathrm{bnd}}(G,K)$ of
bi-$K$-invariant functions on $G$, we have

$\displaystyle\eta_{\Gamma,\nu}=|\Gamma_{o}|^{2}\,\check{\nu}*\eta_{\Gamma}*\nu+\frac{1}{\mathrm{covol}(\Gamma)}\big{(}\delta_{e}-\check{\nu}*\nu\big{)}\,,$

where $\eta_{\Gamma}$ is the random lattice orbit autocorrelation of $\Gamma$
from equation 2.4. In particular, if $\nu=m_{K}$ then
$\eta_{\Gamma,m_{K}}=\eta_{\Gamma}$.

##  3\. Euclidean harmonic analysis

In Subsection 3.2 we introduce the Fourier transform and compute the Fourier
transform of the indicator function on a centered Euclidean ball in terms of
Bessel functions of the first kind. An asymptotic expansion and an integral
formula for the relevant Bessel functions are surveyed in Subsection 3.3.

###  3.1. Euclidean space

Let $n\in\mathbb{N}$ and consider $\mathbb{R}^{n}$ with the metric induced by
the Euclidean norm $\lVert\cdot\rVert$ and the origin $o\in\mathbb{R}^{n}$ as
a reference point. For $r>0$ we let $B_{r}$ denote the Euclidean ball centered
at $o$. We will write $dx=dm_{\mathbb{R}^{n}}(x)$ for the Lebesgue measure on
$\mathbb{R}^{n}$. If we denote by $\sigma_{n-1}$ the canonical surface measure
on the $(n-1)$-dimensional unit sphere $\mathbb{S}^{n-1}$ with mass
$\sigma_{n-1}(\mathbb{S}^{n-1})=2\pi^{n/2}/\Gamma(n/2)$, then integration
against the Lebesgue measure can be written as

$\displaystyle
m_{\mathbb{R}^{n}}(f)=\int_{0}^{\infty}\int_{\mathbb{S}^{n-1}}f(tu)\,d\sigma_{n-1}(u)\,t^{n-1}dt\,,\quad
f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{R}^{n})\,.$

With this measure we have that
$\mathrm{Vol}_{\mathbb{R}^{n}}(B_{r})=\pi^{n/2}\Gamma(\frac{n}{2}+1)^{-1}r^{n}$.

We consider the transitive action of $\mathbb{R}^{n}$ on itself by translation
and emphasize that invariance of a random measure $\mu$ on $\mathbb{R}^{n}$
refers to invariance under this action, which corresponds to
$G=\mathbb{R}^{n}$ and $K=\\{0\\}$ in the setting of the previous section.

###  3.2. The Fourier transform on $\mathbb{R}^{n}$

We take the Fourier transform of a function
$f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{R}^{n})$ to mean the function
$\widehat{f}\in C_{0}(\mathbb{R}^{n})$ given by

$\displaystyle\widehat{f}(\xi)=\int_{\mathbb{R}^{n}}f(x)\mathrm{e}^{-i\langle
x,\xi\rangle}dx\,.$

In particular, $\widehat{f}_{1}=\widehat{f}_{2}$ if and only if $f_{1}=f_{2}$
Lebesgue almost everywhere. The classical Plancherel formula reads as

$\displaystyle\int_{\mathbb{R}^{n}}|f(x)|^{2}dx=(2\pi)^{-n}\int_{\mathbb{R}^{n}}|\widehat{f}(\xi)|^{2}d\xi\,.$

In particular, the Fourier transform extends to a unitary map

$L^{2}(\mathbb{R}^{n})\longrightarrow
L^{2}(\mathbb{R}^{n},(2\pi)^{-n}d\xi)\,.$

######  Example 3.1 (Fourier transform of the indicator function on centered
Euclidean balls).

Let us compute the spherical transform of the indicator function
$\chi_{B_{r}}$. Since $\chi_{B_{r}}$ is radial, averaging over the unit sphere
$\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$ yields

$\displaystyle\widehat{\chi}_{B_{r}}(\xi)=\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}\int_{B_{r}}\omega^{(n)}_{\lVert\xi\rVert}(x)\,dx$

where

$\displaystyle\omega_{\lVert\xi\rVert}^{(n)}(x)$
$\displaystyle=\frac{\Gamma(\frac{n}{2})}{2\pi^{\frac{n}{2}}}\int_{\mathbb{S}^{n-1}}\mathrm{e}^{-i\lVert\xi\rVert\langle
v,x\rangle}d\sigma_{n-1}(v)\,.$

These are the $\mathrm{SO}(n)$-_spherical functions_ of the Euclidean group
$\mathrm{SO}(n)\ltimes\mathbb{R}^{n}$ of rotations and translations, and they
can be expressed in terms of _Bessel functions of the first kind_ ,

$\displaystyle
J_{\alpha}(z)=\frac{2^{1-\alpha}z^{\alpha}}{\pi^{\frac{1}{2}}\Gamma(\alpha+\frac{1}{2})}\int_{0}^{1}(1-s^{2})^{\alpha-\frac{1}{2}}\cos(zs)ds\,,\quad\alpha,z\in\mathbb{C}\,,\,\,\mathrm{Re}(\alpha)>-\frac{1}{2}\,.$

The following formulas can be derived from [18, Lemma 4.13 + Theorem 4.15, p.
170-171].

######  Lemma 3.1.

Let $\xi\in\mathbb{R}^{n}$ and $x\in\mathbb{R}^{n}$. Then

$\displaystyle\omega^{(n)}_{\lVert\xi\rVert}(x)$
$\displaystyle=\frac{2^{\frac{n-2}{2}}\Gamma(\frac{n}{2})}{(\lVert\xi\rVert\lVert
x\rVert)^{\frac{n-2}{2}}}J_{\frac{n-2}{2}}(\lVert\xi\rVert\lVert
x\rVert)\quad\mbox{and}\quad\widehat{\chi}_{B_{r}}(\xi)=(2\pi)^{\frac{n}{2}}\Big{(}\frac{r}{\lVert\xi\rVert}\Big{)}^{\frac{n}{2}}J_{\frac{n}{2}}(r\lVert\xi\rVert)\,.$

In particular, the Fourier transform satisfies the homogeneity
$\widehat{\chi}_{B_{r}}(\xi)=r^{n}\widehat{\chi}_{B_{1}}(r\xi)$ as expected,
and $\widehat{\chi}_{B_{r}}(0)=\mathrm{Vol}_{\mathbb{R}^{n}}(B_{r})$.

###  3.3. Some properties of Bessel functions

A crucial tool for the following sections will be a first order asymptotic
approximation of the (half-)integer Bessel functions $J_{\frac{n}{2}}$, and
hence for the spherical functions $\omega_{\lambda}^{(n)}$.

######  Lemma 3.2 ([22], p.199).

Let $\alpha>-1/2$ and let $\phi_{\alpha}=(2\alpha+1)\pi/4$. As
$t\rightarrow+\infty$, the Bessel function $J_{\alpha}(t)$ admits the
asymptotic expansion

$\displaystyle J_{\alpha}(t)\sim\Big{(}\frac{2}{\pi
t}\Big{)}^{1/2}\Big{(}\cos(t-\phi_{\alpha})\sum_{j=0}^{\infty}\frac{(-1)^{j}A_{2j}}{t^{2j}}-\sin(t-\phi_{\alpha})\sum_{j=0}^{\infty}\frac{(-1)^{j}A_{2j+1}}{t^{2j+1}}\Big{)}\,,$

where

$\displaystyle
A_{j}=\frac{\Gamma(\alpha+j+\frac{1}{2})}{2^{j}j!\Gamma(\alpha-j+\frac{1}{2})}\,,\quad
j\in\mathbb{Z}_{\geq 0}\,.$

In particular, there is a constant $\kappa_{\alpha}>0$ such that

$\displaystyle\Big{|}J_{\alpha}(t)-\Big{(}\frac{2}{\pi
t}\Big{)}^{1/2}\cos(t-\phi_{\alpha})\Big{|}\leq\frac{\kappa_{\alpha}}{t^{3/2}}\,,\quad\forall\,t\geq
1\,.$

######  Corollary 3.3.

Let $\alpha\geq 1/2$. Then there is a constant $C_{\alpha}>0$ such that

$\displaystyle|J_{\alpha}(t)|\leq C_{\alpha}(1+t)^{-1/2}\,,\quad\forall\,t\geq
0\,.$

###### Proof.

If $0\leq t\leq 1$, then $t^{\alpha}\leq 2(1+t)^{-1/2}$, and we bound
$J_{\alpha}$ from above and below by

$\displaystyle|J_{\alpha}(t)|=\frac{2^{1-\alpha}t^{\alpha}}{\pi^{\frac{1}{2}}\Gamma(\alpha+\frac{1}{2})}\Big{|}\int_{0}^{1}(1-s^{2})^{\alpha-\frac{1}{2}}\cos(ts)ds\Big{|}\leq\frac{2^{1-\alpha}t^{\alpha}}{\pi^{\frac{1}{2}}\Gamma(\alpha+\frac{1}{2})}\leq\frac{2^{2-\alpha}}{\pi^{\frac{1}{2}}\Gamma(\alpha+\frac{1}{2})(1+t)^{1/2}}\,.$

If $t\geq 1$ then we let $M_{\alpha}(t)=(\frac{2}{\pi
t})^{1/2}\cos(t-\phi_{\alpha})$. We use that $t^{-1/2}\leq 2(1+t)^{-1/2}$ for
all $t\geq 1$ and Lemma 3.2 to get the bound

$\displaystyle|J_{\alpha}(t)|\leq|M_{\alpha}(t)|+|J_{\alpha}(t)-M_{\alpha}(t)|\leq\Big{(}\frac{2}{\pi
t}\Big{)}^{1/2}+\frac{\kappa_{\alpha}}{t^{3/2}}\leq\frac{\big{(}\frac{2}{\pi}\big{)}^{1/2}+\kappa_{\alpha}}{t^{1/2}}\leq
2\,\frac{\big{(}\frac{2}{\pi}\big{)}^{1/2}+\kappa_{\alpha}}{(1+t)^{1/2}}\,.$

To finish the proof, we take

$\displaystyle
C_{\alpha}=\max\Big{(}\frac{2^{2-\alpha}}{\pi^{\frac{1}{2}}\Gamma(\alpha+\frac{1}{2})},\,2\Big{(}\Big{(}\frac{2}{\pi}\Big{)}^{1/2}+\kappa_{\alpha}\Big{)}\Big{)}\,.$

∎

In proving Theorem 1.1, we will encounter an integral of the following type.

######  Lemma 3.4.

Let $\alpha\geq 1/2$ and $\beta\geq 1$. Then for every $T\geq 1$ there is a
$\delta_{T}=\delta_{T}(\alpha,\beta)$ with $\delta_{T}\rightarrow 0$ as
$T\rightarrow+\infty$ such that

$\displaystyle\int_{0}^{T}t^{\beta}J_{\alpha}(t)^{2}dt=\Big{(}\frac{1}{\pi\beta}+\delta_{T}\Big{)}T^{\beta}\,.$

###### Proof of Lemma 3.4.

Let $M_{\alpha}(t)=(\frac{2}{\pi t})^{1/2}\cos(t-\phi_{\alpha})$, so that
$|J_{\alpha}(t)-M_{\alpha}(t)|\leq\kappa_{\alpha}t^{-3/2}$ for all $t\geq 1$
as in Lemma 3.2. We bound the integral in question from above and below by

$\displaystyle\int_{0}^{T}t^{\beta}M_{\alpha}(t)^{2}dt-\int_{0}^{T}t^{\beta}|J_{\alpha}(t)^{2}$
$\displaystyle-M_{\alpha}(t)^{2}|dt$
$\displaystyle\leq\int_{0}^{T}t^{\beta}J_{\alpha}(t)^{2}dt$
$\displaystyle\leq\int_{0}^{T}t^{\beta}M_{\alpha}(t)^{2}dt+\int_{0}^{T}t^{\beta}|J_{\alpha}(t)^{2}-M_{\alpha}(t)^{2}|dt\,.$

First,

$\displaystyle\int_{0}^{T}t^{\beta}M_{\alpha}(t)^{2}dt$
$\displaystyle=\frac{2}{\pi}\int_{0}^{T}t^{\beta-1}\cos^{2}(t-\phi_{\alpha})dt$
$\displaystyle=\frac{1}{\pi}\Big{(}\int_{0}^{T}t^{\beta-1}dt+\int_{0}^{T}t^{\beta-1}\cos(2(t-\phi_{\alpha}))dt\Big{)}$
$\displaystyle=\frac{1}{\pi}\Big{(}\frac{T^{\beta}}{\beta}+T^{\beta}\int_{0}^{1}s^{\beta-1}\cos(2(Ts-\phi_{\alpha}))ds\Big{)}\,.$

Note that $\int_{0}^{1}s^{\beta-1}\cos(2(Ts-\phi_{\alpha}))ds\rightarrow 0$ as
$T\rightarrow+\infty$ by the Riemann-Lebesgue Lemma.

Secondly, by Lemma 3.2 and Corollary 3.3 we have

$\displaystyle|J_{\alpha}(t)^{2}-M_{\alpha}(t)^{2}|$
$\displaystyle\leq|J_{\alpha}(t)|^{2}+|M_{\alpha}(t)|^{2}\leq\frac{C_{\alpha}^{2}}{1+t}+\frac{2}{\pi
t}\leq\frac{C_{\alpha}^{2}+1}{t}\quad\mbox{ if }\,0\leq t\leq 1\,,$ (3.1)
$\displaystyle|J_{\alpha}(t)^{2}-M_{\alpha}(t)^{2}|$
$\displaystyle=|J_{\alpha}(t)+M_{\alpha}(t)||J_{\alpha}(t)-M_{\alpha}(t)|$
(3.2)
$\displaystyle\leq\Big{(}\frac{C_{\alpha}}{(1+t)^{1/2}}+\Big{(}\frac{2}{\pi
t}\Big{)}^{1/2}\Big{)}\frac{\kappa_{\alpha}}{t^{3/2}}\leq\frac{(C_{\alpha}+1)\kappa_{\alpha}}{t^{2}}\quad\mbox{
if }\,t\geq 1\,,$ (3.3)

so that

$\displaystyle\int_{0}^{T}t^{\beta}|J_{\alpha}(t)^{2}-M_{\alpha}(t)^{2}|dt$
$\displaystyle=\int_{0}^{1}t^{\beta}|J_{\alpha}(t)^{2}-M_{\alpha}(t)^{2}|dt+\int_{1}^{T}t^{\beta}|J_{\alpha}(t)^{2}-M_{\alpha}(t)^{2}|dt$
$\displaystyle\leq(C^{2}_{\alpha}+1)\int_{0}^{1}t^{\beta-1}dt+(C_{\alpha}+1)\kappa_{\alpha}\int_{1}^{T}t^{\beta-2}dt$
$\displaystyle=\frac{C_{\alpha}^{2}+1}{\beta}+(C_{\alpha}+1)\kappa_{\alpha}\frac{T^{\beta-1}-1}{\beta-1}\,,\quad
T\geq 1\,.$

Note that the right hand side remains finite in the limiting case of
$\beta=1^{+}$. Putting everything together, we define

$\displaystyle\delta_{T}=T^{-\beta}\int_{0}^{T}t^{\beta}J_{\alpha}(t)^{2}dt-\frac{1}{\pi\beta}\,,$

so that

$\displaystyle\int_{0}^{T}t^{\beta}J_{\alpha}(t)^{2}dt=\Big{(}\frac{1}{\pi\beta}+\delta_{T}\Big{)}T^{\beta}\,,\quad
T\geq 1$

and

$\displaystyle|\delta_{T}|$ $\displaystyle\leq
T^{-\beta}\Big{(}\int_{0}^{T}t^{\beta}M_{\alpha}(t)^{2}dt-\frac{T^{\beta}}{\pi\beta}+\int_{0}^{T}t^{\beta}|J_{\alpha}(t)^{2}-M_{\alpha}(t)^{2}|dt\Big{)}$
$\displaystyle\leq\frac{1}{\pi}\Big{|}\int_{0}^{1}s^{\beta-1}\cos(2(Ts-\phi_{\alpha}))ds\Big{|}+\frac{C_{\alpha}^{2}+1}{\beta
T^{\beta}}+(C_{\alpha}+1)\kappa_{\alpha}\frac{T^{\beta-1}-1}{(\beta-1)T^{\beta}}\longrightarrow
0$

as $T\rightarrow+\infty$. We also note that this limit is preserved in the
limiting case $\beta=1^{+}$. ∎

##  4\. Euclidean diffraction measures

We define the diffraction measure of an invariant locally square-integrable
random measure on $\mathbb{R}^{n}$ and recall a known upper bound for its
asymptotic volume growth.

The autocorrelation measure $\eta_{\mu}$ of an invariant locally square-
integrable random measure $\mu$ on $\mathbb{R}^{n}$ is _positive-definite_ in
the sense of measures, that is

$\displaystyle\eta_{\mu}(f^{*}*f)\geq
0\,,\quad\forall\,f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{R}^{n})\,.$

The positive-definiteness of $\eta_{\mu}$ is the crucial property that will
allow us to define the diffraction measure of $\mu$. Recall that a Radon
measure $\eta$ is _tempered_ if it extends to a continuous linear functional
on the Schwartz space $\mathscr{S}(\mathbb{R}^{n})$ of smooth functions
$f:\mathbb{R}^{n}\rightarrow\mathbb{C}$ with sub-polynomial decay, that is,

$\displaystyle\sup_{x\in\mathbb{R}^{n}}(1+\lVert
x\rVert)^{\alpha}|\partial^{\beta}f(x)|<+\infty$

for all $\alpha\in\mathbb{Z}_{\geq 0}$ and all multi-indices
$\beta\in\mathbb{Z}_{\geq 0}^{n}$. The following classical result establishes
the existence and uniqueness of a measure-theoretic Fourier transform of
positive-definite measures.

######  Theorem 4.1 (Bochner-Schwartz).

Let $\eta$ be a positive-definite signed Radon measure on $\mathbb{R}^{n}$.
Then there is a unique positive tempered Radon measure $\widehat{\eta}$ on
$\mathbb{R}^{n}$ such that

$\eta(f)=\int_{\mathbb{R}^{n}}\widehat{f}(\xi)d\widehat{\eta}(\xi)$

for all functions $f\in\mathscr{S}(\mathbb{R}^{n})$. In particular, $\eta$ is
tempered.

We refer to $\widehat{\eta}$ as the _Fourier transform_ of $\eta$.

######  Definition 4.2 (Euclidean diffraction measures).

The _diffraction measure_ of an invariant locally square-integrable random
measure $\mu$ on $\mathbb{R}^{n}$ is the unique Fourier transform
$\widehat{\eta}_{\mu}\in\mathscr{M}_{+}(\mathbb{R}^{n})$ of the
autocorrelation measure $\eta_{\mu}$ on $\mathbb{R}^{n}$.

In particular, the $\mu$-variance of linear statistics $Sf$ with
$f\in\mathscr{S}(\mathbb{R}^{n})$ can written in terms of the diffraction
measure as

$\displaystyle\mathrm{Var}_{\mu}(Sf)=\eta_{\mu}(f^{*}*f)=\int_{\mathbb{R}^{n}}|\widehat{f}(\xi)|^{2}d\widehat{\eta}_{\mu}(\xi)\,.$

The linear span of the functions $f^{*}*f\in\mathscr{S}(\mathbb{R}^{n})$ are
enough to uniquely determine the autocorrelation measure $\eta_{\mu}$, so the
above formula for the $\mu$-variance uniquely determines the diffraction
measure $\widehat{\eta}_{\mu}$. Moreover, one can make use of the mean ergodic
theorem to see that the diffraction measure satisfies
$\widehat{\eta}_{\mu}(\\{0\\})=0$.

We now justify the extension of this formula to all functions
$f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{R}^{n})$, in particular for
indicator functions $\chi_{B_{r}}$ with $r>0$.

######  Lemma 4.3.

Let $\mu$ be an invariant locally square-integrable random measure on
$\mathbb{R}^{n}$ and
$f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{R}^{n})$. Then

$\displaystyle\eta_{\mu}(f^{*}*f)=\int_{\mathbb{R}^{n}}|\widehat{f}(\xi)|^{2}d\widehat{\eta}_{\mu}(\xi)\,.$

In particular, $\widehat{f}\in L^{2}(\mathbb{R}^{n},\widehat{\eta}_{\mu})$.

###### Proof.

For every $\varepsilon>0$, let
$\beta_{\varepsilon}\in\mathscr{S}(\mathbb{R}^{n})$ be the function satisfying
$\widehat{\beta}_{\varepsilon}(\xi)=\mathrm{e}^{-\varepsilon^{2}\lVert\xi\rVert^{2}}$
for all $\xi\in\mathbb{R}^{n}$. Setting
$f_{\varepsilon}=\beta_{\varepsilon}*f\in\mathscr{S}(\mathbb{R}^{n})$ then a
standard argument shows that $f_{\varepsilon}$ converges to $f$ Lebesgue-
almost everywhere. Since $f$ is measurable and bounded with bounded support,
the convolution $f^{*}*f$ defines a continuous compactly supported function on
$\mathbb{R}^{n}$. It follows that $f_{\varepsilon}^{*}*f_{\varepsilon}$
converges _uniformly_ to $f^{*}*f$. In particular, since $\eta_{\mu}$ is a
Radon measure then

$\displaystyle\lim_{\varepsilon\rightarrow
0^{+}}\eta_{\mu}(f_{\varepsilon}^{*}*f_{\varepsilon})=\eta_{\mu}(f^{*}*f)\,.$

On the other hand, by definition of $\beta_{\varepsilon}$ and the diffraction
measure we have that

$\displaystyle\eta_{\mu}(f_{\varepsilon}^{*}*f_{\varepsilon})=\int_{\mathbb{R}^{n}}|\widehat{f}(\xi)|^{2}\mathrm{e}^{-2\varepsilon^{2}\lVert\xi\rVert^{2}}d\widehat{\eta}_{\mu}(\xi)\leq\int_{\mathbb{R}^{n}}|\widehat{f}(\xi)|^{2}d\widehat{\eta}_{\mu}(\xi)\,.$

Since $\mathrm{e}^{-2\varepsilon^{2}\lVert\xi\rVert^{2}}$ is monotonely
increasing as $\varepsilon\rightarrow 0^{+}$ with limit $1$ then the monotone
convergence Theorem tells us that

$\displaystyle\eta_{\mu}(f^{*}*f)=\lim_{\varepsilon\rightarrow
0^{+}}\int_{\mathbb{R}^{n}}|\widehat{f}(\xi)|^{2}\mathrm{e}^{-2\varepsilon^{2}\lVert\xi\rVert^{2}}d\widehat{\eta}_{\mu}(\xi)=\int_{\mathbb{R}^{n}}|\widehat{f}(\xi)|^{2}d\widehat{\eta}_{\mu}(\xi)\,.$

∎

The above Lemma imposes a growth condition on the diffraction measure
$\widehat{\eta}_{\mu}$, which one can give a more specific description of.

######  Lemma 4.4 ([6], Proposition 4.9, p.21).

Let $\mu$ be a random measure on $\mathbb{R}^{n}$. Then the diffraction
measure of $\mu$ satisfies

$\displaystyle\widehat{\eta}_{\mu}(B_{L})\ll_{n,\mu}\mathrm{Vol}_{\mathbb{R}^{n}}(B_{L})$

for all sufficiently large $L>0$.

######  Example 4.1 (Poisson diffraction).

In Example 2.4 we computed the variance of linear statistics for the unit
intensity invariant Poisson point process, which in the Euclidean case reads
as

$\mathrm{Var}_{\mathrm{Poi}}(Sf)=(f^{*}*f)(0)=\int_{\mathbb{R}^{n}}|f(x)|^{2}dx\,.$

By the Plancherel formula,

$\displaystyle\mathrm{Var}_{\mathrm{Poi}}(Sf)=\int_{\mathbb{R}^{n}}|f(x)|^{2}dx=(2\pi)^{-n}\int_{\mathbb{R}^{n}}|\widehat{f}(\xi)|^{2}d\xi\,,$

so the diffraction measure of the Poisson process is
$d\widehat{\eta}_{\mathrm{Poi}}(\xi)=(2\pi)^{-n}d\xi$.

######  Example 4.2 (Diffraction of a random lattice).

Let $\Gamma<\mathbb{R}^{n}$ be a lattice. Then there is a matrix
$g_{\Gamma}\in\mathrm{GL}_{n}(\mathbb{R})$ such that
$\Gamma=g_{\Gamma}\mathbb{Z}^{n}$. The covolume of $\Gamma$ can then be
written as $\mathrm{covol}(\Gamma)=\det(g_{\Gamma})$ and the _dual lattice of
$\Gamma$_ is the lattice

$\displaystyle\Gamma^{\perp}=\Big{\\{}\xi\in\mathbb{R}^{n}:\langle\gamma,\xi\rangle\in
2\pi\mathbb{Z}\quad\forall\,\gamma\in\Gamma\Big{\\}}\,.$

The _Poisson summation formula_ states that

$\displaystyle\sum_{\gamma\in\Gamma}f(\gamma)=\frac{1}{\mathrm{covol}(\Gamma)}\sum_{\xi\in\Gamma^{\perp}}\widehat{f}(\xi)$

for $f\in\mathscr{S}(\mathbb{R}^{n})$. By the formula for the autocorrelation
measure in Example 2.5 and the Poisson summation formula,

$\displaystyle\eta_{\Gamma}(f)=\frac{1}{\mathrm{covol}(\Gamma)}\sum_{\gamma\in\Gamma}f(\gamma)-\frac{1}{\mathrm{covol}(\Gamma)^{2}}\int_{\mathbb{R}^{n}}f(x)dx=\frac{1}{\mathrm{covol}(\Gamma)^{2}}\sum_{\xi\in\Gamma^{\perp}\backslash\\{0\\}}\widehat{f}(\xi)$

for all $f\in\mathscr{S}(\mathbb{R}^{n})$. Thus the diffraction measure of the
random lattice $\mu_{\Gamma}$ is

$\widehat{\eta}_{\Gamma}=\frac{1}{\mathrm{covol}(\Gamma)^{2}}\sum_{\xi\in\Gamma^{\perp}\backslash\\{0\\}}\delta_{\xi}\,.$

######  Example 4.3 (Diffraction of a random perturbed lattice).

Since all lattices in Euclidean spaces are cocompact, random perturbed
lattices are locally square-integrable by Lemma 2.6 with a well-defined
diffraction measure.

If $\nu\in\mathrm{Prob}(\mathbb{R}^{n})$ we define its Fourier transform
$\widehat{\nu}:\mathbb{R}^{n}\rightarrow\mathbb{C}$ to be

$\displaystyle\widehat{\nu}(\xi)=\int_{\mathbb{R}^{n}}\mathrm{e}^{-i\langle
x,\xi\rangle}d\nu(x)\,.$

Then $\widehat{\nu}$ is a continuous function, but does not necessarily vanish
at infinity as in the absolutely continuous case. Moreover, one computes
$\widehat{\check{\nu}*\nu}=|\widehat{\nu}|^{2}$. With this Fourier transform
in mind and the formula for the autocorrelation of a random perturbed lattice
$(\Gamma,\nu)$ from Example 2.6,

$\displaystyle\eta_{\Gamma,\nu}=\check{\nu}*\eta_{\Gamma}*\nu+\iota_{\Gamma}(\delta_{0}-\check{\nu}*\nu)=\check{\nu}*\nu*\eta_{\Gamma}+\iota_{\Gamma}(\delta_{0}-\check{\nu}*\nu)$

where $\iota_{\Gamma}=\mathrm{covol}(\Gamma)^{-1}$, the corresponding
diffraction measure is

$\displaystyle
d\widehat{\eta}_{\Gamma,\nu}(\xi)=|\widehat{\nu}(\xi)|^{2}d\widehat{\eta}_{\Gamma}(\xi)+\iota_{\Gamma}(1-|\widehat{\nu}(\xi)|^{2})\frac{d\xi}{(2\pi)^{n}}\,.$

######  Proposition 4.5.

Let $\Gamma<\mathbb{R}^{n}$ be a lattice and
$\nu\in\mathrm{Prob}(\mathbb{R}^{n})$ a probability measure. Then the
perturbed lattice point process $\mu_{\Gamma,\nu}$ is hyperuniform.

###### Proof.

By the formula for the diffraction measure we have

$\displaystyle\widehat{\eta}_{\Gamma,\nu}(B_{\varepsilon})=\int_{B_{\varepsilon}}|\widehat{\nu}(\xi)|^{2}d\widehat{\eta}_{\Gamma}(\xi)+\frac{\iota_{\Gamma}}{(2\pi)^{n}}\int_{B_{\varepsilon}}(1-|\widehat{\nu}(\xi)|^{2})d\xi\,.$

Note that $\widehat{\nu}(0)=\nu(\mathbb{R}^{n})=1$, so
$1-|\widehat{\nu}(\xi)|^{2}=o(1)$ as $\xi\rightarrow 0$. Moreover, since
$\mu_{\Gamma}$ is stealthy, then
$\widehat{\eta}_{\Gamma}|_{B_{\varepsilon}}=0$ for sufficiently small
$\varepsilon>0$. Finally, as $\varepsilon\rightarrow 0$ we then have that

$\displaystyle\widehat{\eta}_{\Gamma,\nu}(B_{\varepsilon})=\frac{\iota_{\Gamma}}{(2\pi)^{n}}\int_{B_{\varepsilon}}(1-|\widehat{\nu}(\xi)|^{2})d\xi=\frac{\iota_{\Gamma}}{(2\pi)^{n}}o(\varepsilon^{n})\,,$

so $\mu_{\Gamma,\nu}$ is spectrally hyperuniform and hence hyperuniform. ∎

##  5\. Beck’s Theorem

We will in this section investigate lower bounds for number variance
$\mathrm{NV}_{\mu}(r)$ as $R\rightarrow+\infty$ for invariant locally square-
integrable random measures $\mu$ on $\mathbb{R}^{n}$. In Subsection 5.1 we
refine and prove Theorem 1.1, and in Subsections 5.2 and 5.3 we arrive at a
proof of Equation 1.3. Finally, we formulate the contents of Subsection 5.1
for isotropic random measures in terms of the associated powder diffraction
measure in Subsection 5.4.

###  5.1. Proof of Theorem 1.1

To prove Theorem 1.1, we will prove a lower bound on the average of the number
variance that is similar to the inequality that Beck and Chen prove in [5,
p.4].

######  Theorem 5.1.

Let $\mu$ be an invariant locally square-integrable random measure on
Euclidean space $\mathbb{R}^{n}$. Then there is a constant $C=C(n)>0$ such
that for every $R_{o}>0$,

$\displaystyle\frac{1}{R}\int_{0}^{R}\mathrm{NV}_{\mu}(r)dr\geq
C\Big{(}\int_{\lVert\xi\rVert\geq
R_{o}^{-1}}\frac{d\widehat{\eta}_{\mu}(\xi)}{\lVert\xi\rVert^{n+1}}\Big{)}R^{n-1}\,,\quad\forall\,R\geq
R_{o}\,.$

###### Proof.

From Lemma 3.1 we have that

$\displaystyle\widehat{\chi}_{B_{r}}(\xi)=(2\pi)^{\frac{n}{2}}\Big{(}\frac{r}{\lVert\xi\rVert}\Big{)}^{\frac{n}{2}}J_{\frac{n}{2}}(r\lVert\xi\rVert)\,.$

By Lemma 3.4 with $\alpha=n/2$ and $\beta=n$, there is for every $T\geq 1$ a
$\delta_{T}\in\mathbb{R}$ with $\delta_{T}\rightarrow 0$ as
$T\rightarrow+\infty$ such that

$\displaystyle\int_{0}^{R}|\widehat{\chi}_{B_{r}}(\xi)|^{2}dr$
$\displaystyle=\frac{(2\pi)^{n}}{\lVert\xi\rVert^{n}}\int_{0}^{R}r^{n}J_{\frac{n}{2}}(r\lVert\xi\rVert)^{2}dr=\frac{b_{n}^{2}}{\lVert\xi\rVert^{2n+1}}\int_{0}^{R\lVert\xi\rVert}t^{n}J_{\frac{n}{2}}(t)^{2}dt$
(5.1)
$\displaystyle=\frac{(2\pi)^{n}}{\lVert\xi\rVert^{n+1}}\Big{(}\frac{1}{\pi
n}+\delta_{R\lVert\xi\rVert}\Big{)}R^{n}>0$ (5.2)

for every $R\geq R_{o}$ and every $\lVert\xi\rVert\geq R_{o}^{-1}$. Define

$\displaystyle C=(2\pi)^{n}\inf_{T\geq 1}\Big{(}\frac{1}{\pi
n}+\delta_{T}\Big{)}=(2\pi)^{n}\inf_{T\geq
1}T^{-n}\int_{0}^{T}t^{n}J_{\frac{n}{2}}(t)^{2}dt\,,$

which is positive since

$\displaystyle\int_{0}^{T}t^{n}J_{\frac{n}{2}}(t)^{2}dt>0\quad\forall\,T\geq
1\quad\mbox{ and
}\quad\lim_{T\rightarrow+\infty}T^{-n}\int_{0}^{T}t^{n}J_{\frac{n}{2}}(t)^{2}dt=\frac{1}{\pi
n}>0$

by Lemma 3.4. An application of Fubini provides the lower bound

$\displaystyle\frac{1}{R}\int_{0}^{R}\mathrm{NV}_{\mu}(r)dr$
$\displaystyle=\int_{\mathbb{R}^{n}}\Big{(}\frac{1}{R}\int_{0}^{R}|\widehat{\chi}_{B_{r}}(\xi)|^{2}dr\Big{)}d\widehat{\eta}_{\mu}(\xi)$
$\displaystyle\geq\int_{\lVert\xi\rVert\geq
R_{o}^{-1}}\Big{(}\frac{1}{R}\int_{0}^{R}|\widehat{\chi}_{B_{r}}(\xi)|^{2}dr\Big{)}d\widehat{\eta}_{\mu}(\lambda)\geq
R^{n-1}C\int_{\lVert\xi\rVert\geq
R_{o}^{-1}}\frac{d\widehat{\eta}_{\mu}(\xi)}{\lVert\xi\rVert^{n+1}}$

as desired. ∎

######  Remark 5.2.

Since $\widehat{\eta}_{\mu}$ satisfies
$\widehat{\eta}_{\mu}(B_{L})\ll_{n,\mu}L^{n}$ for sufficiently large $L>0$ by
Lemma 4.4, then

$\displaystyle\int_{\lVert\xi\rVert\geq
R_{o}^{-1}}\frac{d\widehat{\eta}_{\mu}(\xi)}{\lVert\xi\rVert^{n+1}}\ll_{n}\int_{R_{o}^{-1}}^{\infty}\frac{\widehat{\eta}_{\mu}(B_{u})}{u^{n+2}}du\ll_{n,\mu,R_{o}}\int_{R^{-1}_{o}}^{\infty}\frac{du}{u^{2}}=R_{o}<+\infty\,.$

Thus the right hand side in Theorem 5.1 is indeed finite.

Theorem 1.1 now follows form the fact that

$\displaystyle\limsup_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{R^{n-1}}\geq\limsup_{R\rightarrow+\infty}\frac{1}{R^{n}}\int_{0}^{R}\mathrm{NV}_{\mu}(r)dr\geq
C\int_{\mathbb{R}^{n}}\frac{d\widehat{\eta}_{\mu}(\xi)}{\lVert\xi\rVert^{n+1}}>0\,,$

which may or may not be infinite.

Before moving on we emphasize that if the diffraction measure vanishes in a
neighbourhood around $\xi=0$, then the limit as $R\rightarrow+\infty$ in
Theorem 5.1 can be normalized and computed.

######  Definition 5.3 (Euclidean stealthy random measures).

An invariant locally square-integrable random measure $\mu$ on
$\mathbb{R}^{n}$ is _stealthy_ if there is a $\lambda_{o}>0$ such that
$\widehat{\eta}_{\mu}(B_{\lambda_{o}})=0$.

######  Corollary 5.4.

Let $\mu$ be a stealthy random measure on $\mathbb{R}^{n}$. Then

$\displaystyle\lim_{R\rightarrow+\infty}\frac{1}{R}\int_{0}^{R}\frac{\mathrm{NV}_{\mu}(r)}{r^{n-1}}dr=2^{n}\pi^{n-1}\int_{\mathbb{R}^{n}}\frac{d\widehat{\eta}_{\mu}(\xi)}{\lVert\xi\rVert^{n+1}}\,.$

###### Proof.

Let $\lambda_{o}>0$ such that $\widehat{\eta}_{\mu}(B_{\lambda_{o}})=0$.
Similarly to the proof of Theorem 5.1 we have

$\displaystyle\frac{1}{R}\int_{0}^{R}\frac{\mathrm{NV}_{\mu}(r)}{r^{n-1}}dr$
$\displaystyle=\int_{\lVert\xi\rVert\geq\lambda_{o}}\Big{(}\frac{1}{R}\int_{0}^{R}\frac{|\widehat{\chi}_{B_{r}}(\xi)|^{2}}{r^{n-1}}dr\Big{)}d\widehat{\eta}_{\mu}(\xi)$
$\displaystyle=(2\pi)^{n}\int_{\lVert\xi\rVert\geq\lambda_{o}}\Big{(}\frac{1}{R}\int_{0}^{R}rJ_{\frac{n}{2}}(r\lVert\xi\rVert)^{2}dr\Big{)}\frac{d\widehat{\eta}_{\mu}(\xi)}{\lVert\xi\rVert^{n}}\,.$

If $R\geq\lambda_{o}^{-1}$, then Lemma 3.4 with $\alpha=n/2$ and $\beta=1$
gives us

$\displaystyle\frac{1}{R}\int_{0}^{R}rJ_{\frac{n}{2}}(r\lVert\xi\rVert)^{2}dr=\frac{1}{R\lVert\xi\rVert^{2}}\int_{0}^{R\lVert\xi\rVert}tJ_{\frac{n}{2}}(t)^{2}dt\longrightarrow\frac{1}{\lVert\xi\rVert\pi}$

as $R\rightarrow+\infty$. By Remark 5.2 we know that
$\int_{\lVert\xi\rVert\geq\lambda_{o}}\lVert\xi\rVert^{-(n+1)}d\widehat{\eta}_{\mu}(\xi)$
is finite, so by dominated convergence we see that

$\displaystyle\lim_{R\rightarrow+\infty}(2\pi)^{n}\int_{\lVert\xi\rVert\geq\lambda_{o}}\Big{(}\frac{1}{R}\int_{0}^{R}rJ_{\frac{n}{2}}(r\lVert\xi\rVert)^{2}dr\Big{)}\frac{d\widehat{\eta}_{\mu}(\xi)}{\lVert\xi\rVert^{n}}=2^{n}\pi^{n-1}\int_{\lVert\xi\rVert\geq\lambda_{o}}\frac{d\widehat{\eta}_{\mu}(\xi)}{\lVert\xi\rVert^{n+1}}\,.$

∎

###  5.2. Dynamics for the invariant random $\mathbb{Z}^{n}$-lattice

We consider the invariant random $\mathbb{Z}^{n}$-lattice and compute its
diffraction measure. We also provide dynamical results on square roots of
natural numbers that will be a key ingredient in proving the lower limit

$\displaystyle\liminf_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu_{\mathbb{Z}^{5}}}(R)}{R^{4}}=0\,.$

####  5.2.1. Number variance of the invariant random $\mathbb{Z}^{n}$-lattice

Consider the standard lattice $\mathbb{Z}^{n}<\mathbb{R}^{n}$ and the random
lattice $\mu_{\mathbb{Z}^{n}}$. Its dual lattice is
$(\mathbb{Z}^{n})^{\perp}=2\pi\mathbb{Z}^{n}$, so by Example 4.2, the
diffraction measure $\widehat{\eta}_{\mathbb{Z}^{n}}$ of the random lattice
$\mu_{\mathbb{Z}^{n}}$ is

$\displaystyle\widehat{\eta}_{\mathbb{Z}^{n}}(\widehat{f})=\sum_{\gamma\in\mathbb{Z}^{n}\backslash\\{0\\}}\widehat{f}(2\pi\gamma)$

for any $f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{R}^{n})$. In
particular, $\mu_{\mathbb{Z}^{n}}$ is stealthy. The number variance of the
linear statistic $S\chi_{B_{R}}$ can then be written using Example 3.1 as

$\displaystyle\frac{\mathrm{NV}_{\mu_{\mathbb{Z}^{n}}}(R)}{R^{n-1}}=\frac{1}{R^{n-1}}\sum_{\gamma\in\mathbb{Z}^{n}\backslash\\{0\\}}|\widehat{\chi}_{B_{R}}(2\pi\gamma)|^{2}=(2\pi)^{n}R\sum_{\ell=1}^{\infty}\frac{r_{n}(\ell)}{\ell^{n/2}}J_{\frac{n}{2}}(2\pi
R\sqrt{\ell})^{2}$ (5.3)

where
$r_{n}(\ell)=\\#\\{\gamma\in\mathbb{Z}^{n}:\lVert\gamma\rVert^{2}=\ell\\}$ is
the sum-of-squares function. By Corollary 5.4,

$\displaystyle\lim_{R\rightarrow+\infty}\frac{1}{R}\int_{0}^{R}\frac{\mathrm{NV}_{\mu_{\mathbb{Z}^{n}}}(r)}{r^{n-1}}dr=2^{n}\pi^{n-1}\sum_{\ell=1}^{\infty}\frac{r_{n}(\ell)}{\ell^{\frac{n+1}{2}}}\,.$

Moreover, Remark 5.2 implies that the right hand side is finite. We now show
the slightly stronger statement that the map $R\mapsto
R^{-(n-1)}\mathrm{NV}_{\mu_{\mathbb{Z}^{n}}}(R)$ is uniformly bounded in
$R\geq 0$.

######  Lemma 5.5.

Let $n\geq 1$. Then the Dirichlet series

$\displaystyle\mathfrak{D}(s;r_{n})=\sum_{\ell=1}^{\infty}\frac{r_{n}(\ell)}{\ell^{s}}$

is absolutely convergent whenever $\mathrm{Re}(s)>\frac{n}{2}$.

###### Proof.

Without loss of generality, we assume that $s$ is real. Rewriting
$\mathfrak{D}(s,r_{n})$ as a series over the integer lattice and partitioning
into unit annuli, we get

$\displaystyle\mathfrak{D}(s;r_{n})=\sum_{\gamma\in\mathbb{Z}^{n}\backslash\\{0\\}}\frac{1}{\lVert\gamma\rVert^{2s}}=\sum_{k=1}^{\infty}\Big{(}\sum_{k\leq\lVert\gamma\rVert<k+1}\frac{1}{\lVert\gamma\rVert^{2s}}\Big{)}\,.$

An upper bound is then

$\displaystyle\mathfrak{D}(s;r_{n})\leq\sum_{k=1}^{\infty}\frac{\\#(\mathbb{Z}^{n}\cap
B_{k+1}\backslash B_{k})}{k^{2s}}\,.$

The number of lattice points in a ball can be bounded from above and below
using a standard box covering argument,

$\displaystyle\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}(k-\sqrt{n})^{n}\leq\\#(\mathbb{Z}^{n}\cap
B_{k})\leq\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}(k+\sqrt{n})^{n}$

for all sufficiently large $k\geq 1$. Thus the number of lattice points in the
annulus $B_{k+1}\backslash B_{k}$ is bounded as

$\displaystyle\\#(\mathbb{Z}^{n}\cap B_{k+1}\backslash B_{k})$
$\displaystyle=\\#(\mathbb{Z}^{n}\cap B_{k+1})-\\#(\mathbb{Z}^{n}\cap B_{k})$
$\displaystyle\leq\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}\Big{(}(k+1+\sqrt{n})^{n}-(k-\sqrt{n})^{n}\Big{)}\ll_{n}k^{n-1}$

for all sufficiently large $k\geq 1$, and we extend this to a uniform bound
$\\#(\mathbb{Z}^{n}\cap B_{k+1}\backslash B_{k})\ll_{n}k^{n-1}$ for all $k\geq
1$. This means that

$\displaystyle\mathfrak{D}(s;r_{n})\ll_{n}\sum_{k=1}^{\infty}\frac{1}{k^{2s-(n-1)}}\,.$

The latter series is convergent if and only if $s>\frac{n}{2}$. ∎

######  Corollary 5.6.

The number variance of the invariant random $\mathbb{Z}^{n}$-lattice satisfies

$\displaystyle\mathrm{NV}_{\mu_{\mathbb{Z}^{n}}}(R)\ll_{n}R^{n-1}$

for all $R\geq 0$.

###### Proof.

By Corollary 3.3, there is a constant $C_{n/2}>0$ such that
$J_{\frac{n}{2}}(t)\leq C_{n/2}(1+t)^{-1/2}$ for all $t\geq 0$. Thus

$\displaystyle\mathrm{NV}_{\mu_{\mathbb{Z}^{n}}}(R)$
$\displaystyle=(2\pi)^{n}R^{n}\sum_{\ell=1}^{\infty}\frac{r_{n}(\ell)}{\ell^{n/2}}J_{\frac{n}{2}}(2\pi
R\sqrt{\ell})^{2}$ $\displaystyle\leq
C_{n/2}^{2}(2\pi)^{n}R^{n}\sum_{\ell=1}^{\infty}\frac{r_{n}(\ell)}{\ell^{n/2}(1+2\pi
R\sqrt{\ell})}\leq
C_{n/2}^{2}(2\pi)^{n-1}R^{n-1}\sum_{\ell=1}^{\infty}\frac{r_{n}(\ell)}{\ell^{\frac{n+1}{2}}}\,.$

By Lemma 5.5, the right hand side is convergent and so we’re done. ∎

######  Example 5.1 (Vanishing of the number variance for random shifts of
$\mathbb{Z}$).

When $n=1$ we have $r_{1}(\ell)=2$ if $\ell=m^{2}$ for some integer $m\geq 1$
and $0$ else. Moreover, $\widehat{\chi}_{[-R,R]}(2\pi m)=\frac{\sin(2\pi
Rm)}{\pi m}$, so

$\displaystyle\mathrm{NV}_{\mu_{\mathbb{Z}}}(R)=\frac{2}{\pi^{2}}\sum_{m=1}^{\infty}\frac{\sin^{2}(2\pi
Rm)}{m^{2}}\,.$

Thus the number variance is identically zero along the sequence
$R=0,1/2,1,3/2,2,...$ of half-integers. In dimension $n=5$ we can not find
simultaneous zeroes of the Bessel function $J_{5/2}(2\pi R\sqrt{\ell})$ for
all relevant positive integers $\ell$, but we will provide a sequence of radii
for which the number variance is asymptotically dominated by $R^{4}$.

####  5.2.2. Rational independence of square roots and ergodicity

The support of the diffraction measure $\widehat{\eta}_{\mathbb{Z}^{n}}$ is
$\\{2\pi\xi:\xi\in\mathbb{Z}^{n}\\}\subset\mathbb{R}^{n}$, and in light of
Example 5.1 we will show that one can scale norms
$\lVert\xi_{1}\rVert,...,\lVert\xi_{N}\rVert$ for any finite collection
$\xi_{1},...,\xi_{N}\in\mathbb{Z}^{n}$ such that they are simultaneously close
to the integers. We will make crucial use of the following is a classical
result due to Besicovitch.

######  Lemma 5.7 ([7], Theorem 2, p.4).

Let $\ell_{1},...,\ell_{J}$ be distinct positive square-free integers. Then
the roots $\sqrt{\ell_{1}},...,\sqrt{\ell_{J}}$ are linearly independent over
$\mathbb{Q}$.

A simultaneous consideration of the fractional parts of such square roots give
rise to the following uniquely ergodic dynamical system.

######  Lemma 5.8.

Let $\alpha_{1},...,\alpha_{J}\in\mathbb{R}$ be linearly independent over
$\mathbb{Q}$. Then the translation action

$\displaystyle
v+\mathbb{Z}^{J}\longmapsto(v+\alpha)+\mathbb{Z}^{J}\,,\quad\alpha=(\alpha_{1},...,\alpha_{J})\in\mathbb{R}^{J}$

on the torus $\mathbb{R}^{J}/\mathbb{Z}^{J}$ is uniquely ergodic with the
Lebesgue measure as the invariant measure. In particular, every orbit of the
action is dense.

###### Proof.

Suppose that $\theta$ is an invariant probability measure on
$\mathbb{R}^{J}/\mathbb{Z}^{J}$ with respect to the prescribed action. It
suffices to show that $\theta$ is the Lebesgue measure. By invariance, the
Fourier transform $\widehat{\theta}:\mathbb{Z}^{J}\rightarrow\mathbb{C}$ of
$\theta$ satisfies

$\displaystyle\widehat{\theta}(\gamma)=\int_{\mathbb{R}^{J}/\mathbb{Z}^{J}}\mathrm{e}^{-2\pi
i\langle\gamma,x\rangle}d\theta(x)=\int_{\mathbb{R}^{J}/\mathbb{Z}^{J}}\mathrm{e}^{-2\pi
i\langle\gamma,x+\alpha\rangle}d\theta(x)=\mathrm{e}^{-2\pi
i\langle\gamma,\alpha\rangle}\widehat{\theta}(\gamma)$

for all $\gamma\in\mathbb{Z}^{J}$. This forces $\widehat{\theta}(\gamma)=0$
for all $\gamma\neq 0$, since otherwise there would be a non-zero $\gamma$ for
which $\langle\gamma,\alpha\rangle\in\mathbb{Z}$, contradicting the assumed
$\mathbb{Q}$-linear independence of $\alpha_{1},...,\alpha_{J}$. Thus
$\widehat{\theta}$ is the unit mass on $0\in\mathbb{Z}^{J}$ and hence $\theta$
is the Lebesgue measure on $\mathbb{R}^{J}/\mathbb{Z}^{J}$ by injectivity of
the Fourier transform. ∎

######  Corollary 5.9.

Let $N\in\mathbb{N}$. Then there is a positive sequence
$R_{k}\rightarrow+\infty$ such that

$\displaystyle R_{k}.(1,\sqrt{2},...,\sqrt{N})+\mathbb{Z}^{N}\longrightarrow
0+\mathbb{Z}^{N}$

as $k\rightarrow+\infty$ in $\mathbb{R}^{N}/\mathbb{Z}^{N}$.

###### Proof.

Take distinct positive square-free integers $\ell_{1},...,\ell_{J}$ and finite
subsets $A_{1},...,A_{J}\subset\mathbb{N}$ such that

$\displaystyle\big{\\{}1,\sqrt{2},...,\sqrt{N}\big{\\}}=\bigsqcup_{j=1}^{J}\big{\\{}k\sqrt{\ell_{j}}:k\in
A_{j}\big{\\}}\,.$ (5.4)

By Lemma 5.7, $\sqrt{\ell_{1}},...,\sqrt{\ell_{J}}$ are linearly independent
over $\mathbb{Q}$, and by Lemma 5.8 the action of translation by
$\alpha=(\sqrt{\ell_{1}},...,\sqrt{\ell_{J}})$ on the torus
$\mathbb{R}^{J}/\mathbb{Z}^{J}$ has dense orbits. Thus we may take a sequence
$R_{k}\rightarrow+\infty$ such that $R_{k}.\alpha+\mathbb{Z}^{J}\rightarrow
0+\mathbb{Z}^{J}$ in $\mathbb{R}^{J}/\mathbb{Z}^{J}$. It follows by the
decomposition in Equation (5.4) that
$R_{k}\sqrt{\ell}+\mathbb{Z}^{N}\rightarrow 0+\mathbb{Z}^{N}$ for all
$\ell=1,...,N$. ∎

###  5.3. Proof of Equation 1.3

From Equation (5.3) for the number variance of $\mu_{\mathbb{Z}^{n}}$, we have

$\displaystyle\frac{\mathrm{NV}_{\mu_{\mathbb{Z}^{n}}}(R)}{R^{n-1}}=(2\pi)^{n}R\sum_{\ell=1}^{\infty}\frac{r_{n}(\ell)}{\ell^{n/2}}J_{\frac{n}{2}}(2\pi
R\sqrt{\ell})^{2}\,.$

This is uniformly bounded in $R\geq 0$ by Corollary 5.6, so we fix
$\varepsilon>0$ and take $N\geq 1$ to be an integer such that

$\displaystyle
R\sum_{\ell=N+1}^{\infty}\frac{r_{n}(\ell)}{\ell^{n/2}}J_{\frac{n}{2}}(2\pi
R\sqrt{\ell})^{2}<\varepsilon\,,\quad\forall\,R\geq 0\,.$

It remains to find a suitable sequence of radii such that the first $N$ terms
of the series tend to $0$. To do this we make use of the approximation
$M_{\frac{n}{2}}(t)=(\frac{2}{\pi t})^{1/2}\cos(t-\phi_{n/2})$ of $J_{n/2}(t)$
in Lemma 3.2 along with the bound in Equation (3.1) for $R\geq 1$ to find that

$\displaystyle
R\sum_{\ell=1}^{N}\frac{r_{n}(\ell)}{\ell^{\frac{n}{2}}}J_{\frac{n}{2}}(2\pi
R\sqrt{\ell})^{2}\leq R$
$\displaystyle\sum_{\ell=1}^{N}\frac{r_{n}(\ell)}{\ell^{\frac{n}{2}}}M_{\frac{n}{2}}(2\pi
R\sqrt{\ell})^{2}$
$\displaystyle+R\sum_{\ell=1}^{N}\frac{r_{n}(\ell)}{\ell^{\frac{n}{2}}}|J_{\frac{n}{2}}(2\pi
R\sqrt{\ell})^{2}-M_{\frac{n}{2}}(2\pi R\sqrt{\ell})^{2}|$
$\displaystyle\leq\frac{1}{\pi^{2}}$
$\displaystyle\sum_{\ell=1}^{N}\frac{r_{n}(\ell)}{\ell^{\frac{n+1}{2}}}\cos^{2}(2\pi
R\sqrt{\ell}-\phi_{\frac{n}{2}})+\frac{2\kappa_{\frac{n}{2}}}{R^{\frac{1}{2}}}\sum_{\ell=1}^{N}\frac{r_{n}(\ell)}{\ell^{\frac{n}{2}}}\,,$

where again $\phi_{n/2}=\frac{n+1}{4}\pi$. Fix $\varepsilon>0$ and take
$R_{o}\geq 1$ such that

$\displaystyle\frac{2\kappa_{\frac{n}{2}}}{R^{\frac{1}{2}}}\sum_{\ell=1}^{N}\frac{r_{n}(\ell)}{\ell^{\frac{n}{2}}}<\varepsilon\quad\quad\forall\,R\geq
R_{o}\,.$

Then

$\displaystyle\frac{\mathrm{NV}_{\mu_{\mathbb{Z}^{n}}}(R)}{R^{n-1}}$
$\displaystyle=(2\pi)^{n}\Big{(}R\sum_{\ell=1}^{N}\frac{r_{n}(\ell)}{\ell^{\frac{n}{2}}}J_{\frac{n}{2}}(2\pi
R\sqrt{\ell})^{2}+R\sum_{\ell=N+1}^{\infty}\frac{r_{n}(\ell)}{\ell^{\frac{n}{2}}}J_{\frac{n}{2}}(2\pi
R\sqrt{\ell})^{2}\Big{)}$
$\displaystyle<\frac{(2\pi)^{n}}{\pi^{2}}\sum_{\ell=1}^{N}\frac{r_{n}(\ell)}{\ell^{\frac{n+1}{2}}}\cos^{2}(2\pi
R\sqrt{\ell}-\phi_{\frac{n}{2}})+2(2\pi)^{n}\varepsilon\quad\quad\forall\,R\geq
R_{o}\,.$

For the case $n=5$ we get

$\displaystyle\frac{\mathrm{Var}_{\mu_{\mathbb{Z}^{5}}}(S\chi_{B_{R}})}{R^{4}}<32\pi^{3}\sum_{\ell=1}^{N}\frac{r_{5}(\ell)}{\ell^{3}}\sin^{2}(2\pi
R\sqrt{\ell})+64\pi^{5}\varepsilon\quad\quad\forall\,R\geq R_{o}\,.$

By Corollary 5.9 there is an sequence $(R_{k})_{k\geq 1}$ with
$R_{k}\rightarrow+\infty$ as $k\rightarrow+\infty$ and such that
$R_{k}\sqrt{\ell}\rightarrow 0$ in $\mathbb{R}^{N}/\mathbb{Z}^{N}$ for every
$1\leq\ell\leq N$, hence $\sin^{2}(R_{k}\sqrt{\ell})\rightarrow 0$ for every
$1\leq\ell\leq N$. This means that

$\displaystyle\limsup_{k\rightarrow+\infty}\frac{\mathrm{NV}_{\mu_{\mathbb{Z}^{5}}}(R_{k})}{R_{k}^{4}}\leq
64\pi^{5}\varepsilon\,,$

and since $\varepsilon>0$ was chosen arbitrarily we are done.

######  Remark 5.10.

The key feature of dimension $n=5$ is that $\phi_{5/2}=\frac{3\pi}{2}$ is an
odd multiple of $\frac{\pi}{2}$, so that $\cos^{2}(t-\phi_{5/2})=\sin^{2}(t)$
for all $t\geq 0$. Thus the same proof works for dimensions $n=4m+1$, $m\geq
0$.

###  5.4. Isotropic random measures on $\mathbb{R}^{n}$

Given a random measure $\mu$ on $\mathbb{R}^{n}$, one can require invariance
under other subgroups of the Euclidean isometries rather than just
translations. The special Euclidean group $G_{n}=K_{n}\ltimes\mathbb{R}^{n}$
with $K_{n}=\mathrm{SO}(n)$ is the group of rotations and translations, and
contains the subgroup of translations considered in the previous subsections.
Random measures that are invariant under $G_{n}$ are commonly referred to as
_isotropic_.

Given a translation invariant random measure $\mu$ on $\mathbb{R}^{n}$, one
can construct an isotropic random measure $\mu^{\mathrm{iso}}$ as the push-
forward of $\mu$ along the $K_{n}$-averaging map
$\mathrm{Av}_{K_{n}}:\mathscr{M}_{+}(\mathbb{R}^{n})\rightarrow\mathscr{M}_{+}(\mathbb{R}^{n})^{K_{n}}$
defined by

$\mathrm{Av}_{K_{n}}(p)(B)=\int_{K_{n}}p(k.B)dm_{K_{n}}(k)$

for every bounded Borel set $B\subset\mathbb{R}^{n}$, where $m_{K_{n}}$ is the
Haar probability measure on $K_{n}$. Note however that this specific procedure
does not produce isotropic point processes from translation invariant point
processes.

If $\mu$ is an isotropic and locally square-integrable random measure on
$\mathbb{R}^{n}$, then the autocorrelation measure $\eta_{\mu}$ is a
bi-$K_{n}$-invariant measure on $G_{n}$ which canonically descends to a radial
measure on $\mathbb{R}^{n}$. Thus it is uniquely determined by its restriction
to the subalgebra
$\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{R}^{n})^{\mathrm{rad}}\subset\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{R}^{n})$
of radial functions. On this subalgebra the Fourier transform restricts to
what we call the _Hankel transform_. Given
$f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{R}^{n})^{\mathrm{rad}}$, its
Hankel transform is the function $\widetilde{f}:\mathbb{R}_{\geq
0}\rightarrow\mathbb{C}$ given by

$\displaystyle\widetilde{f}(\lambda)=\int_{\mathbb{R}^{n}}f(x)\omega_{\lambda}^{(n)}(x)dx\,,$

where $\omega_{\lambda}^{(n)}$ are the $\mathrm{SO}(n)$-spherical functions
from Lemma 3.1,

$\displaystyle\omega^{(n)}_{\lambda}(x)$
$\displaystyle=\frac{2^{\frac{n-2}{2}}\Gamma(\frac{n}{2})}{(\lambda\lVert
x\rVert)^{\frac{n-2}{2}}}J_{\frac{n-2}{2}}(\lambda\lVert x\rVert)\,.$

By construction, $\widehat{f}(\xi)=\widetilde{f}(\lVert\xi\rVert)$ for all
$\xi\in\mathbb{R}^{n}$ and dualizing this to measures, we are able to define
the Hankel transform of a positive-definite Radon measure on $\mathbb{R}^{n}$.

######  Definition 5.11.

Let $\mu$ be an isotropic locally square-integrable random measure on
$\mathbb{R}^{n}$. The _powder diffraction measure_ of $\mu$ is the Hankel
transform $\widetilde{\eta}_{\mu}\in\mathscr{M}_{+}(\mathbb{R}_{\geq 0})$ of
the autocorrelation measure $\eta_{\mu}$.

By the existence and uniqueness of the diffraction measure in Theorem 4.1,
$\widetilde{\eta}_{\mu}$ exists and is unique. In particular,

$\displaystyle\mathrm{Var}_{\mu}(Sf)=\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}\int_{0}^{\infty}|\widetilde{f}(\lambda)|^{2}d\widetilde{\eta}_{\mu}(\lambda)$

for all
$f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{R}^{n})^{\mathrm{rad}}$. With
this notation, the lower bound in Theorem 5.1 is

$\displaystyle\frac{1}{R}\int_{0}^{R}\mathrm{NV}_{\mu}(r)dr\geq
C\Big{(}\int_{R_{o}^{-1}}^{\infty}\frac{d\widetilde{\eta}_{\mu}(\lambda)}{\lambda^{n+1}}\Big{)}R^{n-1}\,,\quad\forall\,R\geq
R_{o}$

for $R_{o}>0$ and some $C=C(n,R_{o})>0$. If $\mu$ is in addition stealthy,
then the limit formula in Corollary 5.4 becomes

$\displaystyle\lim_{R\rightarrow+\infty}\frac{1}{R}\int_{0}^{R}\frac{\mathrm{NV}_{\mu}(r)}{r^{n-1}}dr=\frac{2^{n+1}\pi^{\frac{3n}{2}-1}}{\Gamma(\frac{n}{2})}\int_{0}^{\infty}\frac{d\widetilde{\eta}_{\mu}(\lambda)}{\lambda^{n+1}}\,.$

##  6\. Spherical harmonic analysis on real hyperbolic spaces

We introduce real hyperbolic spaces, Cartan coordinates and invariant
reference measures in Subsection 6.1 and the spherical functions in Subsection
6.2. We give a careful treatment of the spherical functions, providing a
trigonometric expansion in Subsection 6.3 and as a consequence some bounds and
an asymptotic mean in Subsection 6.4, which are crucial ingredients in the
proof of Theorem 1.4. Lastly, we survey some properties of the spherical
transform in Subsection 6.5.

###  6.1. Cartan decomposition and invariant reference measures

Consider the bilinear form $[x,y]=x_{0}y_{0}-x_{1}y_{1}-...-x_{n}y_{n}$ for
$x,y\in\mathbb{R}^{1+n}$ and the one-sheeted paraboloid
$\mathbb{H}^{n}=\\{x\in\mathbb{R}^{1+n}:[x,x]=1,\,x_{0}>0\\}$ with the
hyperbolic metric $d(x,y)=\mathrm{arccosh}([x,y])$. On $\mathbb{R}^{1+n}$, the
group $G_{n}=\mathrm{SO}^{\circ}(1,n)$ of orientation preserving linear maps
preserving $[\cdot,\cdot]$ restricts to a transitive isometric action on
$(\mathbb{H}^{n},d)$ and fixing the reference point
$o=(1,0,...,0)\in\mathbb{H}^{n}$, its stabilizer is

$K_{n}=\Big{\\{}\left(\begin{array}[]{@{}c|c@{}}1&0\\\ \hline\cr
0&k\end{array}\right):k\in\mathrm{SO}(n)\Big{\\}}<G_{n}\,.$

With this we obtain a proper pointed metric space $(\mathbb{H}^{n},d,o)$ as in
Subsection 2.1.

####  6.1.1. Cartan decomposition

A polar decomposition for $\mathbb{H}^{n}$ is available via the length map
$\ell:G_{n}\rightarrow\mathbb{R}_{\geq 0}$, $\ell(g)=d(g.o,o)$, which is a
bi-$K_{n}$-invariant map and descends to a bijection $K_{n}\backslash
G_{n}/K_{n}\rightarrow\mathbb{R}_{\geq 0}$. If we consider the one parameter
subgroup

$\displaystyle
A_{n}=\Big{\\{}a_{t}=\left(\begin{array}[]{@{}c|c|c@{}}\cosh(t)&0&\sinh(t)\\\
\hline\cr 0&\mathrm{Id}_{\mathbb{R}^{n-1}}&0\\\
\hline\cr\sinh(t)&0&\cosh(t)\end{array}\right):t\in\mathbb{R}\Big{\\}}<G_{n}$

then $\ell(a_{t})=|t|$, and we obtain the _Cartan decomposition_
$G_{n}=K_{n}A_{n}^{+}K_{n}$, where $A_{n}^{+}=\\{a_{t}:t\geq 0\\}$. From this
decomposition we get for every $R>0$ an open subset

$B_{R}=\ell^{-1}([0,R))=\\{k_{1}a_{t}k_{2}\in G_{n}:t\in[0,R),k_{1},k_{2}\in
K_{n}\\}\,,$

which we refer to as the centered open Cartan ball of radius $R>0$. Moreover,
we note that the orbit of the Cartan ball $B_{R}\subset G_{n}$ in
$\mathbb{H}^{n}$ coincides with the open metric ball, $B_{R}.o=B_{R}(o)$. We
will interchangeably use the notation $B_{R}$ for the centered Cartan ball in
$G_{n}$ and the metric ball $B_{R}(o)$ in $\mathbb{H}^{n}$.

####  6.1.2. Continuous sections

Note that if $g\in G_{n}$ with Cartan decomposition $g=k_{1}a_{t}k_{2}$ then

$g.o=k_{1}a_{t}.o=(\cosh(t),\sinh(t)u)$

for some $u\in\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$. The sphere
$\mathbb{S}^{n-1}$ is a homogeneous space for $K_{n}$, and a standard argument
using the Gram-Schmidt orthogonalization process produces a continuous section
$\kappa:\mathbb{S}^{n-1}\rightarrow K_{n}$, so
$\varsigma(\cosh(t),\sinh(t)u)=\kappa(u)a_{t}$ is a continuous section from
$\mathbb{H}^{n}$ into $G_{n}$.

####  6.1.3. Invariant reference measures

If we denote by $m_{K_{n}}$ the unique Haar probability measure on $K_{n}$
then a Haar measure on $G_{n}$ written in Cartan coordinates is

$\displaystyle
m_{G_{n}}(\varphi)=\int_{K_{n}}\int_{0}^{\infty}\int_{K_{n}}\varphi(k_{1}a_{t}k_{2})\,dm_{K_{n}}(k_{1})\sinh(t)^{n-1}dt\,dm_{K_{n}}(k_{2})\,,\quad\varphi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n})\,,$

and a $G_{n}$-invariant reference measure on $\mathbb{H}^{n}$ is

$\displaystyle
m_{\mathbb{H}^{n}}(f)=\int_{0}^{\infty}\int_{\mathbb{S}^{n-1}}f(\cosh(t),\sinh(t)u)\,d\sigma_{n-1}(u)\sinh(t)^{n-1}dt\,,\quad
f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{H}^{n})\,,$

where $\sigma_{n-1}$ is the canonical surface measure on $\mathbb{S}^{n-1}$.
The relation between the two measures is

$\displaystyle
m_{\mathbb{H}^{n}}(f)=\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}\int_{G_{n}}f(g.o)dm_{G_{n}}(g)\,.$

In particular, the volume growth of centered Cartan balls in $\mathbb{H}^{n}$
is

$\displaystyle
m_{\mathbb{H}^{n}}(B_{R}.o)=\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}\int_{0}^{R}\sinh(t)^{n-1}dt\asymp_{n}\cosh(R)^{n-1}$
(6.1)

for all sufficiently large $R>0$.

####  6.1.4. The algebra of bi-$K_{n}$-invariant functions

As in the case of isotropic random measures on Euclidean spaces we will
restrict our attention to linear statistics and autocorrelation measures on
the subspace of radial functions in
$\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{H}^{n})$, which we identify with
the space $\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n},K_{n})$ of
bi-$K_{n}$-invariant functions using the section $\varsigma$. A key feature in
defining the spherical functions and the spherical transform is that the
latter space defines a commutative algebra with respect to convolution.

######  Lemma 6.1.

Let $\eta,\nu$ be bi-$K_{n}$-invariant Radon measures on $G_{n}$ and assume
that $\nu$ is finite. Then

$\eta*\nu=\nu*\eta\,.$

In particular, the subalgebra
$\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n},K_{n})\subset\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n})$
is a commutative involutive subalgebra.

With this property, we say that $(G_{n},K_{n})$ is a _Gelfand pair_.

###  6.2. Spherical functions on $\mathbb{H}^{n}$

A _complex hyperbolic wave_ will for us be a function
$\xi_{\lambda,u}:\mathbb{H}^{n}\rightarrow\mathbb{C}$ of the form
$\xi_{\lambda,u}(x)=[x,(1,u)]^{-i\lambda-\frac{n-1}{2}}$, where
$u\in\mathbb{S}^{n-1}$ and $\lambda\in\mathbb{C}$. A _spherical function_ for
hyperbolic space $\mathbb{H}^{n}$ will, similarly to the Bessel functions
appearing in the Euclidean setting, be a function
$\omega_{\lambda}:G_{n}\rightarrow\mathbb{C}$ given by

$\displaystyle\omega_{\lambda}(g)=\int_{K_{n}}\xi_{\lambda,u}(k^{-1}g.o)dm_{K_{n}}(k)=\frac{\Gamma(\frac{n}{2})}{2\pi^{\frac{n}{2}}}\int_{\mathbb{S}^{n-1}}[g.o,(1,v)]^{-i\lambda-\frac{n-1}{2}}d\sigma_{n-1}(v)$

for any choice of $u\in\mathbb{S}^{n-1}$. To emphasize the dimension $n$, we
write $\omega_{\lambda}^{(n)}$ in place of $\omega_{\lambda}$. These functions
are continuous and bi-$K_{n}$-invariant, and can be rewritten as an integral
over $\mathbb{R}$ by a formula that we state next. For the statement, we
remind ourselves of the Beta function

$\displaystyle\mathrm{B}(z,w)=\frac{\Gamma(z)\Gamma(w)}{\Gamma(z+w)}=\int_{0}^{1}s^{z-1}(1-s)^{w-1}ds\,,\quad
z,w\in\mathbb{C}\backslash\mathbb{Z}_{\leq 0}\,.$

######  Lemma 6.2.

The spherical function $\omega_{\lambda}^{(n)}$, $\lambda\in\mathbb{C}$, can
be written as

$\displaystyle\omega_{\lambda}^{(n)}(g)=a_{n}\sinh(\ell(g))^{2-n}\int_{0}^{\ell(g)}(\cosh(\ell(g))-\cosh(s))^{\frac{n-3}{2}}\cos(\lambda
s)ds\,,\quad
a_{n}=\frac{2^{\frac{n-1}{2}}}{\mathrm{B}(\frac{n-1}{2},\frac{1}{2})}\,.$

###### Proof.

From the Cartan decomposition, every $g\in G_{n}$ can be written as
$g=k_{1}a_{\ell(g)}k_{2}$ for some $k_{1},k_{2}\in K_{n}$, where
$\ell(g)=d(g.o,o)$. Using that spherical functions are bi-$K_{n}$-invariant
and introducing spherical coordinates on $\mathbb{S}^{n-1}$ yields

$\displaystyle\omega_{\lambda}^{(n)}(g)$
$\displaystyle=\omega_{\lambda}^{(n)}(a_{\ell(g)})=\frac{\Gamma(\frac{n}{2})}{2\pi^{\frac{n}{2}}}\int_{\mathbb{S}^{n-1}}[a_{\ell(g)}.o,(1,v)]^{-i\lambda-\frac{n-1}{2}}d\sigma_{n-1}(v)$
$\displaystyle=\frac{\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}\Gamma(\frac{n-1}{2})}\int_{0}^{\pi}(\cosh(\ell(g))-\sinh(\ell(g))\cos(\alpha))^{-i\lambda-\frac{n-1}{2}}\sin(\alpha)^{n-2}d\alpha\,.$

Making the substitution
$\mathrm{e}^{s}=\cosh(\ell(g))-\sinh(\ell(g))\cos(\alpha)$ we get

$\displaystyle\omega_{\lambda}^{(n)}(g)$
$\displaystyle=\frac{\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}\Gamma(\frac{n-1}{2})\sinh(\ell(g))}\int_{-\ell(g)}^{\ell(g)}\mathrm{e}^{-(i\lambda+\frac{n-3}{2})s}\Big{(}1-\Big{(}\frac{\cosh(\ell(g))-\mathrm{e}^{s}}{\sinh(\ell(g))}\Big{)}^{2}\Big{)}^{\frac{n-3}{2}}ds$
$\displaystyle=\frac{\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}\Gamma(\frac{n-1}{2})\sinh(\ell(g))^{n-2}}\int_{-\ell(g)}^{\ell(g)}\mathrm{e}^{-i\lambda
s}\Big{(}\mathrm{e}^{-s}\big{(}\sinh^{2}(\ell(g))-(\cosh(\ell(g))-\mathrm{e}^{s})^{2}\big{)}\Big{)}^{\frac{n-3}{2}}ds\,.$

Note that

$\displaystyle\mathrm{e}^{-s}\big{(}\sinh^{2}(\ell(g))-(\cosh(\ell(g))-\mathrm{e}^{s})^{2}\big{)}$
$\displaystyle=\mathrm{e}^{-s}\big{(}\sinh^{2}(\ell(g))-\cosh^{2}(\ell(g))+2\cosh(\ell(g))\mathrm{e}^{s}-\mathrm{e}^{2s}\big{)}$
$\displaystyle=2(\cosh(\ell(g))-\cosh(s))\,,$

so the spherical function is

$\displaystyle\omega_{\lambda}^{(n)}(g)$
$\displaystyle=\frac{2^{\frac{n-3}{2}}\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}\Gamma(\frac{n-1}{2})\sinh(\ell(g))^{n-2}}\int_{-\ell(g)}^{\ell(g)}(\cosh(\ell(g))-\cosh(s))^{\frac{n-3}{2}}\mathrm{e}^{i\lambda
s}ds$
$\displaystyle=\frac{2^{\frac{n-1}{2}}\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}\Gamma(\frac{n-1}{2})\sinh(\ell(g))^{n-2}}\int_{0}^{\ell(g)}(\cosh(\ell(g))-\cosh(s))^{\frac{n-3}{2}}\cos(\lambda
s)ds\,.$

∎

The following result is a recollection of some standard useful properties of
the spherical functions.

######  Lemma 6.3.

The spherical functions $\omega_{\lambda}^{(n)}$, $\lambda\in\mathbb{C}$,
satisfy the following:

  1. (1)

$\omega_{\lambda}^{(n)}(e)=1$ and
$\omega_{-\lambda}^{(n)}=\omega_{\lambda}^{(n)}$ for all
$\lambda\in\mathbb{C}$.

  2. (2)

$\omega_{\lambda}^{(n)}$ is bounded if and only if
$|\mathrm{Im}(\lambda)|\leq\frac{n-1}{2}$. Moreover, if
$|\mathrm{Im}(\lambda)|<\frac{n-1}{2}$ we have that

$|\omega_{\lambda}^{(n)}(g)|<1$

for all $g\neq e$ in $G_{n}$.

  3. (3)

If $g\neq e$ and $\lambda\geq 0$ then
$|\omega_{\lambda}^{(n)}(g)|\leq\omega_{0}^{(n)}(g)$.

###### Proof.

For (1) we have that

$\displaystyle\omega_{\lambda}^{(n)}(e)=\frac{\Gamma(\frac{n}{2})}{2\pi^{\frac{n}{2}}}\int_{\mathbb{S}^{n-1}}[o,(1,v)]^{-i\lambda-\frac{n-1}{2}}d\sigma_{n-1}(v)=1\,,$

and by Lemma 6.2 it is easy to see that
$\omega_{-\lambda}^{(n)}=\omega_{\lambda}^{(n)}$ for all
$\lambda\in\mathbb{C}$ using that $\cos(-\lambda s)=\cos(\lambda s)$.

For (2), the first claim is obvious and for the second one we first assume
that $|\mathrm{Im}(\lambda)|\leq\frac{n-1}{2}$ and use Lemma 6.2 to find that

$\displaystyle|\omega_{\lambda}^{(n)}(g)|$
$\displaystyle\leq\frac{2^{\frac{n-1}{2}}\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}\Gamma(\frac{n-1}{2})\sinh(\ell(g))^{n-2}}\int_{0}^{\ell(g)}(\cosh(\ell(g))-\cosh(s))^{\frac{n-3}{2}}\cosh(|\mathrm{Im}(\lambda)|s)ds$
$\displaystyle\leq\frac{2^{\frac{n-1}{2}}\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}\Gamma(\frac{n-1}{2})\sinh(\ell(g))^{n-2}}\int_{0}^{\ell(g)}(\cosh(\ell(g))-\cosh(s))^{\frac{n-3}{2}}\cosh\Big{(}\frac{n-1}{2}s\Big{)}ds$
$\displaystyle=\omega_{i\frac{n-1}{2}}^{(n)}(g)=1$

for all $g\in G_{n}$. Moreover, it is not hard to see that the last inequality
is strict if $|\mathrm{Im}(\lambda)|<\frac{n-1}{2}$ and $g\neq e$, so the
second statement also holds.

(3) follows from a similar bound as in the proof of (2), using that
$|\cos(\lambda s)|\leq 1$ for all $s\geq 0$. ∎

###  6.3. A trigonometric expansion of spherical functions

The following Proposition can be viewed as a alternate formulation of the
classical asymptotic expansions of spherical functions due to Harish-Chandra,
see [11, Theorem 5.5, p.430]. For us however, we are in need of an expansion
along the line $\lambda\geq 0$, which differs slightly from the referenced
statement. We emphasize that our proof will rely entirely on elementary
arguments.

######  Proposition 6.4.

Let $\lambda>0$. Then there is an $r_{o}>0$ and bounded continuous functions

$\alpha_{n}(\cdot,\lambda),\beta_{n}(\cdot,\lambda):[r_{o},+\infty)\rightarrow\mathbb{R}$

such that

$\displaystyle\omega_{\lambda}^{(n)}(a_{r})=a_{n}\sinh(r)^{-\frac{n-1}{2}}\big{(}\alpha_{n}(r,\lambda)\cos(\lambda
r)+\beta_{n}(r,\lambda)\sin(\lambda r)\big{)}\,,$

with

$\displaystyle\lim_{r\rightarrow+\infty}\alpha_{n}(r,\lambda)=\mathrm{Re}\Big{(}\mathrm{B}\Big{(}i\lambda,\frac{n-1}{2}\Big{)}\Big{)}\quad\mbox{
and
}\quad\lim_{r\rightarrow+\infty}\beta_{n}(r,\lambda)=\mathrm{Im}\Big{(}\mathrm{B}\Big{(}i\lambda,\frac{n-1}{2}\Big{)}\Big{)}\,.$

####  6.3.1. Some identities for the Beta function

For the proof of Proposition 6.4 in even dimensions, it will be useful to
consider the the power series expansion

$\displaystyle(1-x)^{-1/2}=\sum_{\ell=0}^{\infty}E_{\ell}x^{\ell}\,,\quad
E_{\ell}=\frac{(2\ell)!}{4^{\ell}(\ell!)^{2}}$ (6.2)

for $0\leq x<1$. Using Stirling approximation, one gets the asymptotic
$E_{\ell}\sim\sqrt{\frac{2}{\pi\ell}}$ as $\ell\rightarrow+\infty$.

######  Lemma 6.5.

Let $z\in\mathbb{C}\backslash\\{0\\}$ such that $\mathrm{Re}(z)\geq 0$ and
$m\in\mathbb{Z}_{\geq 0}$. Then

$\displaystyle\mathrm{B}(z,m+1)=\sum_{k=0}^{m}\binom{m}{k}\frac{(-1)^{k}}{k+z}\quad\mbox{
and
}\quad\mathrm{B}(z,m+1/2)=\sum_{\ell=0}^{\infty}E_{\ell}\,\mathrm{B}(z+\ell,m+1)\,.$

###### Proof.

The first identity follows from binomial expansion,

$\displaystyle\mathrm{B}(z,m+1)=\int_{0}^{1}x^{z-1}(1-x)^{m}dx=\sum_{k=0}^{m}\binom{m}{k}(-1)^{k}\int_{0}^{1}x^{z+k-1}dx=\sum_{k=0}^{m}\binom{m}{k}\frac{(-1)^{k}}{k+z}\,.$

For the second identity, we first note that

$\displaystyle|\mathrm{B}(z+\ell,m+1)|\leq\mathrm{B}(\mathrm{Re}(z)+\ell,m+1)\leq\mathrm{B}(\ell,m+1)=\frac{m!(\ell-1)!}{(\ell+m)!}\,,$

so

$\displaystyle\sum_{\ell=1}^{\infty}E_{\ell}|\mathrm{B}(z+\ell,m+1)|$
$\displaystyle\leq\sum_{\ell=1}^{\infty}E_{\ell}\mathrm{B}(\ell,m+1)$
$\displaystyle=m!\sum_{\ell=1}^{\infty}E_{\ell}\frac{(\ell-1)!}{(\ell+m)!}\leq
m!\sum_{\ell=1}^{\infty}\frac{E_{\ell}}{\ell^{m+1}}<+\infty\,.$

By dominated convergence, we get

$\displaystyle\sum_{\ell=1}^{\infty}E_{\ell}\,\mathrm{B}(z+\ell,m+1)$
$\displaystyle=\int_{0}^{1}x^{z-1}\Big{(}\sum_{\ell=1}^{\infty}E_{\ell}x^{\ell}\Big{)}(1-x)^{m}dx$
$\displaystyle=\int_{0}^{1}x^{z-1}\big{(}(1-x)^{-1/2}-1\big{)}(1-x)^{m}dx$
$\displaystyle=\mathrm{B}(z,m+1/2)-\mathrm{B}(z,m+1)\,.$

Rearranging this, we finally have that

$\displaystyle\mathrm{B}(z,m+1/2)=\sum_{\ell=0}^{\infty}E_{\ell}\,\mathrm{B}(z+\ell,m+1)\,.$

∎

####  6.3.2. Some formulas for intermediate coefficients

######  Lemma 6.6.

For every $q\in\mathbb{N}$ and all $r\geq 0$,

$\displaystyle\int_{0}^{r}\cosh(s)^{q}ds\leq
2^{q+\frac{1}{2}}+\frac{2\cosh(r)^{q}}{q}\,.$

###### Proof.

Setting $u=\cosh(s)$ we split our integral as

$\displaystyle\int_{0}^{r}\cosh(s)^{q}ds=\int_{1}^{\cosh(r)}\frac{u^{q}}{\sqrt{u^{2}-1}}du=\int_{1}^{2}\frac{u^{q}}{\sqrt{u^{2}-1}}du+\int_{2}^{\cosh(r)}\frac{u^{q}}{\sqrt{u^{2}-1}}du\,.$

For the first integral we make the estimate

$\displaystyle\int_{1}^{2}\frac{u^{q}}{\sqrt{u^{2}-1}}du$
$\displaystyle=\int_{1}^{2}\frac{u^{q}}{\sqrt{(u+1)(u-1)}}du\leq
2^{q-\frac{1}{2}}\int_{1}^{2}\frac{du}{\sqrt{u-1}}=2^{q+\frac{1}{2}}\,.$

For the second integral, we have that

$\displaystyle\int_{2}^{\cosh(r)}\frac{u^{q}}{\sqrt{u^{2}-1}}du$
$\displaystyle=\int_{2}^{\cosh(r)}\frac{u^{q}}{\sqrt{(u+1)(u-1)}}du\leq\int_{2}^{\cosh(r)}\frac{u^{q}}{u-1}du$
$\displaystyle=\int_{2}^{\cosh(r)}\Big{(}1+\frac{1}{u-1}\Big{)}u^{q-1}du\leq
2\int_{2}^{\cosh(r)}u^{q-1}du\leq\frac{2\cosh(r)^{q}}{q}\,.$

Gathering the two estimates we get

$\displaystyle\int_{0}^{r}\cosh(s)^{q}ds=\int_{1}^{2}\frac{u^{q}}{\sqrt{u^{2}-1}}du+\int_{2}^{\cosh(r)}\frac{u^{q}}{\sqrt{u^{2}-1}}du\leq
2^{q+\frac{1}{2}}+\frac{2\cosh(r)^{q}}{q}\,.$

∎

######  Lemma 6.7.

For every $k\in\mathbb{Z}_{\geq 0}$, $r\geq 0$, $\lambda>0$,

$\displaystyle\int_{0}^{r}\cosh(s)^{k}\cos(\lambda
s)ds=a_{k}(r,\lambda)\cos(\lambda r)+b_{k}(r,\lambda)\sin(\lambda r)\,,$ (6.3)

where

$\displaystyle
a_{k}(r,\lambda)=\frac{1}{2^{k}}\sum_{j=0}^{k}\binom{k}{j}\frac{(2j-k)\sinh((2j-k)r)}{(2j-k)^{2}+\lambda^{2}}\,,\,\,b_{k}(r,\lambda)=\frac{\lambda}{2^{k}}\sum_{j=0}^{k}\binom{k}{j}\frac{\cosh((2j-k)r)}{(2j-k)^{2}+\lambda^{2}}\,.$

Moreover, if $r>\mathrm{arctanh}(\frac{1}{2})$ these coefficients satisfy

$\displaystyle a_{k}(r,\lambda)\leq\int_{0}^{r}\cosh(s)^{k}ds\,,\quad
b_{k}(r,\lambda)\leq\frac{1}{\lambda}+2\lambda\int_{0}^{r}\cosh(s)^{k}ds\,,$

and

$\displaystyle\lim_{r\rightarrow+\infty}\frac{a_{k}(r,\lambda)}{\cosh(r)^{k}}=\frac{k}{k^{2}+\lambda^{2}}\,,\quad\lim_{r\rightarrow+\infty}\frac{b_{k}(r,\lambda)}{\cosh(r)^{k}}=\frac{\lambda}{k^{2}+\lambda^{2}}\,.$

###### Proof.

A binomial expansion of $\cosh(s)^{k}$ yields

$\displaystyle\int_{0}^{r}\cosh(s)^{k}\cos(\lambda
s)ds=\frac{1}{2}\int_{-r}^{r}\cosh(s)^{k}\mathrm{e}^{i\lambda
s}ds=\frac{1}{2^{k+1}}\sum_{j=0}^{k}\binom{k}{j}\int_{-r}^{r}\mathrm{e}^{(2j-k+i\lambda)s}ds\,.$

The latter integral is

$\displaystyle\int_{-r}^{r}\mathrm{e}^{(2j-k+i\lambda)s}ds$
$\displaystyle=2\,\mathrm{Re}\Big{(}\frac{\sinh((2j-k+i\lambda)r)}{2j-k+i\lambda}\Big{)}$
$\displaystyle=2\,\frac{(2j-k)\sinh((2j-k)r)\cos(\lambda
r)+\lambda\cosh((2k-j)r)\sin(\lambda r)}{(2j-k)^{2}+\lambda^{2}}$

which means that

$\displaystyle\int_{0}^{r}\cosh(s)^{k}\cos(\lambda s)ds=$
$\displaystyle\Big{(}\frac{1}{2^{k}}\sum_{j=0}^{k}\binom{k}{j}\frac{(2j-k)\sinh((2j-k)r)}{(2j-k)^{2}+\lambda^{2}}\Big{)}\cos(\lambda
r)$
$\displaystyle+\Big{(}\frac{\lambda}{2^{k}}\sum_{j=0}^{k}\binom{k}{j}\frac{\cosh((2j-k)r)}{(2j-k)^{2}+\lambda^{2}}\Big{)}\sin(\lambda
r)$

as desired.

For the upper bound of $a_{k}(r,\lambda)$, we have

$\displaystyle a_{k}(r,\lambda)$
$\displaystyle=\frac{1}{2^{k}}\sum_{j=0}^{k}\binom{k}{j}\frac{(2j-k)\sinh((2j-k)r)}{(2j-k)^{2}+\lambda^{2}}$
$\displaystyle=\frac{1}{2^{k}}\sum_{j=0}^{k}\binom{k}{j}\frac{(2j-k)^{2}}{(2j-k)^{2}+\lambda^{2}}\frac{\sinh((2j-k)r)}{2j-k}$
$\displaystyle\leq\frac{1}{2^{k}}\sum_{j=0}^{k}\binom{k}{j}\frac{\sinh((2j-k)r)}{2j-k}=\int_{0}^{r}\cosh(s)^{k}ds\,.$

For $b_{k}(r,\lambda)$, we isolate the worst case of $2j-k=0$ by

$\displaystyle b_{k}(r,\lambda)$
$\displaystyle=\frac{\lambda}{2^{k}}\sum_{j=0}^{k}\binom{k}{j}\frac{\cosh((2j-k)r)}{(2j-k)^{2}+\lambda^{2}}\leq\frac{1}{\lambda}+\frac{\lambda}{2^{k}}\sum_{2j-k\neq
0}\binom{k}{j}\frac{\cosh((2j-k)r)}{(2j-k)^{2}+\lambda^{2}}\,.$

In the above inequality, we used that $\binom{k}{j}\leq 2^{k}$ for all
$j=0,...,k$ when isolating the term corresponding to $2j-k=0$. Next we observe
that

$\frac{1}{2|2j-k|}\leq\frac{1}{2}\leq\tanh(r)\leq\tanh(|2j-k|r)$

for all $j$ such that $2j-k\neq 0$ and all $r\geq\mathrm{arctanh}(1/2)$, so
that

$\displaystyle\cosh((2j-k)r)\leq 2(2j-k)\sinh((2j-k)r)$

for the same $j$ and $r$. Thus

$\displaystyle\frac{1}{2^{k}}\sum_{2j-k\neq
0}\binom{k}{j}\frac{\cosh((2j-k)r)}{(2j-k)^{2}+\lambda^{2}}\leq\frac{2}{2^{k}}\sum_{j=0}^{k}\binom{k}{j}\frac{(2j-k)\sinh((2j-k)r)}{(2j-k)^{2}+\lambda^{2}}$
$\displaystyle\leq 2a_{k}(r,\lambda)\leq 2\int_{0}^{r}\cosh(s)^{k}ds\,,$

and so

$\displaystyle b_{k}(r,\lambda)$
$\displaystyle\leq\frac{1}{\lambda}+\frac{\lambda}{2^{k}}\sum_{2j-k\neq
0}\binom{k}{j}\frac{\cosh((2j-k)r)}{(2j-k)^{2}+\lambda^{2}}\leq\frac{1}{\lambda}+2\lambda\int_{0}^{r}\cosh(s)^{k}ds$

for all $r>\mathrm{arctanh}(1/2)$. Lastly, the limits in the statement follow
easily. ∎

######  Lemma 6.8.

Suppose that $m\in\mathbb{N}$, $\ell\in\mathbb{Z}_{\geq 0}$. Then

$\displaystyle\int_{0}^{r}\Big{(}\frac{\cosh(s)}{\cosh(r)}\Big{)}^{\ell}\Big{(}1-\frac{\cosh(s)}{\cosh(r)}\Big{)}^{m-1}\cos(\lambda
s)ds=A_{m,\ell}(r,\lambda)\cos(\lambda r)+B_{m,\ell}(r,\lambda)\sin(\lambda
r)\,,$

where

$\displaystyle
A_{m,\ell}(r,\lambda)=\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{(-1)^{k}a_{k+\ell}(r,\lambda)}{\cosh(r)^{k+\ell}}\,,\quad
B_{m,\ell}(r,\lambda)=\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{(-1)^{k}b_{k+\ell}(r,\lambda)}{\cosh(r)^{k+\ell}}\,.$

Moreover, for all $r>\mathrm{arccosh}(2)$ and all $\ell\geq 1$,

$\displaystyle|A_{m,\ell}(r,\lambda)|\leq\frac{2^{m+1}}{\ell}\quad\mbox{ and
}\quad|B_{m,\ell}(r,\lambda)|\leq\big{(}\frac{1}{\lambda}+8\lambda\big{)}\frac{2^{m-1}}{\ell}$

and for all $\ell\geq 0$,

$\displaystyle\lim_{r\rightarrow+\infty}A_{m,\ell}(r,\lambda)$
$\displaystyle=\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{(-1)^{k}(k+\ell)}{(k+\ell)^{2}+\lambda^{2}}=\mathrm{Re}(\mathrm{B}(i\lambda,m))\,,$
$\displaystyle\lim_{r\rightarrow+\infty}B_{m,\ell}(r,\lambda)$
$\displaystyle=\lambda\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{(-1)^{k}}{(k+\ell)^{2}+\lambda^{2}}=\mathrm{Im}(\mathrm{B}(i\lambda,m))\,.$

###### Proof of Lemma 6.8.

Using the binomial Theorem we can write

$\displaystyle\int_{0}^{r}\Big{(}\frac{\cosh(s)}{\cosh(r)}\Big{)}^{\ell}$
$\displaystyle\Big{(}1-\frac{\cosh(s)}{\cosh(r)}\Big{)}^{m-1}\cos(\lambda
s)ds=$
$\displaystyle=\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{(-1)^{k}}{\cosh(r)^{k+\ell}}\int_{0}^{r}\cosh(s)^{k+\ell}\cos(\lambda
s)ds\,,$

which by Lemma 6.7 is

$\displaystyle\sum_{k=0}^{m-1}$
$\displaystyle\binom{m-1}{k}\frac{(-1)^{k}}{\cosh(r)^{k+\ell}}(a_{k+\ell}(r,\lambda)\cos(\lambda
r)+b_{k+\ell}(r,\lambda)\sin(\lambda r))=$
$\displaystyle=\Big{(}\sum_{k=0}^{m-1}$
$\displaystyle\binom{m-1}{k}\frac{(-1)^{k}a_{k+\ell}(r,\lambda)}{\cosh(r)^{k+\ell}}\Big{)}\cos(\lambda
r)+\Big{(}\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{(-1)^{k}b_{k+\ell}(r,\lambda)}{\cosh(r)^{k+\ell}}\Big{)}\sin(\lambda
r)\,.$

For the upper bounds, we use the upper bounds in Lemma 6.7 for
$a_{k}(r,\lambda)$ and $b_{k}(r,\lambda)$, yielding

$\displaystyle|A_{m,\ell}(r,\lambda)|\leq\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{a_{k+\ell}(r,\lambda)}{\cosh(r)^{k+\ell}}\leq\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{1}{\cosh(r)^{k+\ell}}\int_{0}^{r}\cosh(s)^{k+\ell}ds\,.$

Since $r>\mathrm{arccosh}(2)$, Lemma 6.6 gives us

$\displaystyle\int_{0}^{r}\cosh(s)^{k+\ell}ds\leq\sqrt{2}\Big{(}\frac{2}{\cosh(r)}\Big{)}^{k+\ell}+\frac{2}{k+\ell}\leq\frac{4}{\ell}$

for all $k\geq 0$ and all $\ell\geq 1$, so

$\displaystyle\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{1}{\cosh(r)^{k+\ell}}\int_{0}^{r}\cosh(s)^{k+\ell}ds$
$\displaystyle\leq\frac{4}{\ell}\sum_{k=0}^{m-1}\binom{m-1}{k}=\frac{2^{m+1}}{\ell}\,.$

Similarly,

$\displaystyle|B_{m,\ell}(r,\lambda)|$
$\displaystyle\leq\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{b_{k+\ell}(r,\lambda)}{\cosh(r)^{k+\ell}}$
$\displaystyle\leq\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{1}{\cosh(r)^{k+\ell}}\Big{(}\frac{1}{\lambda}+2\lambda\int_{0}^{r}\cosh(s)^{k+\ell}ds\Big{)}$
$\displaystyle\leq\sum_{k=0}^{m-1}\binom{m-1}{k}\Big{(}\frac{1}{\lambda\cosh(r)^{k+\ell}}+\frac{8\lambda}{\ell}\Big{)}$
$\displaystyle\leq
2^{m-1}\Big{(}\frac{1}{\lambda\cosh(r)^{\ell}}+\frac{8\lambda}{\ell}\Big{)}\leq\Big{(}\frac{1}{\lambda}+8\lambda\Big{)}\frac{2^{m-1}}{\ell}\,.$

The limits are a straightforward consequence of Lemma 6.5 and Lemma 6.7. ∎

######  Corollary 6.9.

Let $r\geq 0$ and $\lambda>0$. Then the series

$\displaystyle\sum_{\ell=0}^{\infty}E_{\ell}A_{m,\ell}(r,\lambda)\quad\mbox{
and }\quad\sum_{\ell=0}^{\infty}E_{\ell}B_{m,\ell}(r,\lambda)$

are absolutely convergent.

###### Proof.

Using the upper bounds provided in Lemma 6.8, we get that

$\displaystyle\sum_{\ell=0}^{\infty}E_{\ell}|A_{m,\ell}(r,\lambda)|\leq
2^{m+1}\sum_{\ell=0}^{\infty}\frac{E_{\ell}}{\ell}\quad\mbox{ and
}\quad\sum_{\ell=0}^{\infty}E_{\ell}|B_{m,\ell}(r,\lambda)|\leq
2^{m-1}\Big{(}\frac{1}{\lambda}+8\lambda\Big{)}\sum_{\ell=0}^{\infty}\frac{E_{\ell}}{\ell}\,.$

From the asymptotic $E_{\ell}\sim\sqrt{\frac{2}{\pi\ell}}$ as
$\ell\rightarrow+\infty$, we see that $\sum_{\ell\geq 0}E_{\ell}/\ell$ is
convergent. ∎

####  6.3.3. A Proof of Proposition 6.4 for odd dimensions

Let $n=2m+1$, $m\in\mathbb{N}$. We use the formula in Lemma 6.2, binomial
expansion and Lemma 6.8 to write the spherical function
$\omega_{\lambda}^{(2m+1)}$ as

$\begin{split}\omega^{(2m+1)}_{\lambda}(a_{r})&=\frac{a_{2m+1}}{\sinh(r)^{2m-1}}\int_{0}^{r}(\cosh(r)-\cosh(s))^{m-1}\cos(\lambda
s)ds\\\
&=\frac{a_{2m+1}}{\sinh(r)^{m}}\coth(r)^{m-1}\int_{0}^{r}\Big{(}1-\frac{\cosh(s)}{\cosh(r)}\Big{)}^{m-1}\cos(\lambda
s)ds\\\
&=\frac{a_{2m+1}}{\sinh(r)^{m}}\coth(r)^{m-1}\big{(}A_{m,0}(r,\lambda)\cos(\lambda
r)+B_{m,0}(r,\lambda)\sin(\lambda r)\big{)}\,.\end{split}$ (6.4)

If we define

$\displaystyle\alpha_{2m+1}(r,\lambda)=\coth(r)^{m-1}A_{m,0}(r,\lambda)\quad\mbox{
and }\quad\beta_{2m+1}(r,\lambda)=\coth(r)^{m-1}B_{m,0}(r,\lambda)$

then

$\omega_{\lambda}^{(2m+1)}(a_{r})=a_{n}\sinh(r)^{-\frac{n-1}{2}}\big{(}\alpha_{2m+1}(r,\lambda)\cos(\lambda
r)+\beta_{2m+1}(r,\lambda)\sin(\lambda r)\big{)}\,,$

and Lemma 6.8 tells us that

$\displaystyle\lim_{r\rightarrow+\infty}\alpha_{2m+1}(r,\lambda)=\mathrm{Re}(\mathrm{B}(i\lambda,m))\quad\mbox{
and
}\quad\lim_{r\rightarrow+\infty}\beta_{2m+1}(r,\lambda)=\mathrm{Im}(\mathrm{B}(i\lambda,m))\,.$

####  6.3.4. A Proof of Proposition 6.4 for even dimensions

Let $n=2m$, $m\in\mathbb{N}$. Using the Taylor expansion

$\displaystyle(1-x)^{-\frac{1}{2}}=\sum_{\ell=0}^{\infty}E_{\ell}x^{\ell}$

for $0\leq x<1$, we rewrite the spherical function $\omega_{\lambda}^{(2m)}$
in the same way as in the odd dimensional case,

$\begin{split}\omega^{(2m)}_{\lambda}(a_{r})&=\frac{a_{2m}}{\sinh(r)^{2m-2}}\int_{0}^{r}(\cosh(r)-\cosh(s))^{m-\frac{3}{2}}\cos(\lambda
s)ds\\\
&=\frac{a_{2m}}{\sinh(r)^{m-\frac{1}{2}}}\coth(r)^{m-\frac{3}{2}}\int_{0}^{r}\Big{(}1-\frac{\cosh(s)}{\cosh(r)}\Big{)}^{m-\frac{3}{2}}\cos(\lambda
s)ds\\\
&=\frac{a_{2m}}{\sinh(r)^{m-\frac{1}{2}}}\coth(r)^{m-\frac{3}{2}}\int_{0}^{r}\sum_{\ell=0}^{\infty}E_{\ell}\Big{(}\frac{\cosh(s)}{\cosh(r)}\Big{)}^{\ell}\Big{(}1-\frac{\cosh(s)}{\cosh(r)}\Big{)}^{m-1}\cos(\lambda
s)ds\,.\end{split}$ (6.5)

By Lemma 6.8 and Corollary 6.9, we know that

$\displaystyle\sum_{\ell=0}^{\infty}E_{\ell}\int_{0}^{r}\Big{(}\frac{\cosh(s)}{\cosh(r)}\Big{)}^{\ell}$
$\displaystyle\Big{(}1-\frac{\cosh(s)}{\cosh(r)}\Big{)}^{m-1}\cos(\lambda
s)ds=$
$\displaystyle=\Big{(}\sum_{\ell=0}^{\infty}E_{\ell}A_{m,\ell}(r,\lambda)\Big{)}\cos(\lambda
r)+\Big{(}\sum_{\ell=0}^{\infty}E_{\ell}B_{m,\ell}(r,\lambda)\Big{)}\sin(\lambda
r)$

is absolutely convergent. We may then apply Fubini’s Theorem to write the
spherical function as

$\displaystyle\omega^{(2m)}_{\lambda}(a_{r})$
$\displaystyle=\frac{a_{2m}}{\sinh(r)^{m-\frac{1}{2}}}\coth(r)^{m-\frac{3}{2}}\sum_{\ell=0}^{\infty}E_{\ell}\int_{0}^{r}\Big{(}\frac{\cosh(s)}{\cosh(r)}\Big{)}^{\ell}\Big{(}1-\frac{\cosh(s)}{\cosh(r)}\Big{)}^{m-1}\cos(\lambda
s)ds$
$\displaystyle=\frac{a_{2m}}{\sinh(r)^{m-\frac{1}{2}}}\coth(r)^{m-\frac{3}{2}}\Big{(}\Big{(}\sum_{\ell=0}^{\infty}E_{\ell}A_{m,\ell}(r,\lambda)\Big{)}\cos(\lambda
r)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\Big{(}\sum_{\ell=0}^{\infty}E_{\ell}B_{m,\ell}(r,\lambda)\Big{)}\sin(\lambda
r)\Big{)}\,.$

Thus we are inclined to define

$\displaystyle\alpha_{2m}(r,\lambda)=\coth(r)^{m-\frac{3}{2}}\sum_{\ell=0}^{\infty}E_{\ell}A_{m,\ell}(r,\lambda)\quad\mbox{
and
}\quad\beta_{2m}(r,\lambda)=\coth(r)^{m-\frac{3}{2}}\sum_{\ell=0}^{\infty}E_{\ell}B_{m,\ell}(r,\lambda)\,.$

Since these series are absolutely convergent for all $r>\mathrm{arccosh}(2)$
by Corollary 6.9, we can compute their term-wise limit and use Lemma 6.5 to
see that

$\displaystyle\lim_{r\rightarrow+\infty}\alpha_{2m}(r,\lambda)=\lim_{r\rightarrow+\infty}\sum_{\ell=0}^{\infty}E_{\ell}A_{m,\ell}(r,\lambda)=\sum_{\ell=0}^{\infty}E_{\ell}\mathrm{Re}(\mathrm{B}(i\lambda+\ell,m))=\mathrm{Re}(\mathrm{B}(i\lambda,m-1/2))$

and

$\displaystyle\lim_{r\rightarrow+\infty}\beta_{2m}(r,\lambda)=\lim_{r\rightarrow+\infty}\sum_{\ell=0}^{\infty}E_{\ell}B_{m,\ell}(r,\lambda)=\sum_{\ell=0}^{\infty}E_{\ell}\mathrm{Im}(\mathrm{B}(i\lambda+\ell,m))=\mathrm{Im}(\mathrm{B}(i\lambda,m-1/2))\,.$

This concludes the proof of Proposition 6.4.

######  Remark 6.10.

The spherical functions $\omega_{\lambda}^{(n)}$ on real hyperbolic space
$\mathbb{H}^{n}$ are special cases of _Jacobi functions_. For real rank one
symmetric spaces, the spherical functions are essentially all Jacobi
functions, admitting integral formulas similar to that of Lemma 6.2, see [4,
Eq. 5.10, p.40]. The explicit integral formulas for $n$-dimensional hyperbolic
space over the real, complex and quarternionic field respectively are given by

$\omega_{\lambda}^{(n)}(g)=\mathrm{Vol}(\mathbb{S}^{n-1})^{-1}\int_{\mathbb{S}^{n-1}}|[a_{\ell(g)}.o,(1,v)]|^{-i\lambda-\rho}d\sigma(v)$

for $t\geq 0$, where $\mathbb{S}^{n-1}$ is the unit sphere in the hyperbolic
space with canonical surface measure $\sigma$ and $\rho$ is the half sum of
positive roots with respect to a fixed root system. For the octonionic case,