# Hyperuniform point sets on projective spaces

Johann S. Brauchart  Institute of Analysis and Number Theory, Graz University
of Technology, Kopernikusgasse 24. 8010 Graz, Austria<EMAIL_ADDRESS>and  Peter J. Grabner‡<EMAIL_ADDRESS>
###### Abstract.

We extend the notion of hyperuniformity to the projective spaces
$\mathbb{RP}^{d-1}$, $\mathbb{CP}^{d-1}$, $\mathbb{HP}^{d-1}$, and
$\mathbb{OP}^{2}$. We show that hyperuniformity implies uniform distribution
and present examples of deterministic point sets as well as point processes
which exhibit hyperuniform behaviour.

‡ This author is supported by the Austrian Science Fund FWF project I 6750

##  1\. Introduction

The quantitative characterisation of density fluctuation in many-particle
systems led to the introduction of the concept of “hyperuniformity” (S.
Torquato; cf. [37]) also called “superhomogeneity” (J. Lebowitz, cf. [18]).

The main feature of hyperuniformity of an infinite discrete point
configuration $X$ in $\mathbb{R}^{d}$ is the fact that local density
fluctuations are of smaller order than for a random (“Poissonian”) point
configuration. In terms of the structure factor

$S(\mathbf{k})=\lim_{B\to\mathbb{R}^{d}}\frac{1}{\\#(B\cap
X)}\sum_{\mathbf{x},\mathbf{y}\in B\cap
X}e^{i\langle\mathbf{k},\mathbf{x}-\mathbf{y}\rangle}\quad\text{(thermodynamic
limit)}$

this means that $\lim_{\mathbf{k}\to\mathbf{0}}S(\mathbf{k})=0$; i.e.,
normalised density fluctuations are completely suppressed at very large
length-scales (cf. [37]). This _thermodynamic limit_ is understood in the
sense that the volume $B$ (for instance a ball of radius $R$) tends to the
whole space $\mathbb{R}^{d}$ while $\lim_{B\to\mathbb{R}^{d}}\frac{\\#(B\cap
X)}{\mathrm{vol}(B)}=\rho$, where $\rho$ is the density. Equivalently, the
number variance $\mathbb{V}[N_{R}]$ (understood in the sense of thermodynamic
limit) of particles within a spherical observation window of radius $R$ can be
used. When scaled by the window volume, this ratio tends to zero as
$R\to\infty$; i.e., $\mathbb{V}[N_{R}]$ grows more slowly than the window
volume of order $R^{d}$ in the large-$R$ limit. When

$S(\mathbf{k})\sim\left\|\mathbf{k}\right\|^{\alpha}\qquad\text{as
$\left\|\mathbf{k}\right\|\to\infty$,}$

where $\alpha>0$, the number variance has the following large-$R$ scaling:

$\mathbb{V}[N_{R}]\sim\begin{cases}R^{d-1}&\alpha>1,\\\ R^{d-1}\log
R&\alpha=1,\\\ R^{d-\alpha}&0<\alpha<1.\end{cases}$

This asymptotic behaviour provides a way to distinguish between different
classes of hyperuniformity: Class I (strongest), II, and III (weakest). For a
thorough discussion and survey regarding, in particular, hyperuniform states
of matter, we refer to [36]. More recent contributions consider
hyperuniformity in biology and geophysics [17, 26, 40], cryptography [20], and
mathematics [7] to name a few. A nice layman’s introduction to hyperuniformity
can be found in [39].

In an answer to a question of S. Torquato, we together with W. Kusner [11]
were able to extend the concept of hyperuniformity to the compact unit sphere
$\mathbb{S}^{d}$ in the Euclidean space $\mathbb{R}^{d+1}$ by considering a
fixed sequence of $N$-point distributions and analysing the asymptotic
behaviour of the number variance with respect to certain spherical caps as
$N\to\infty$. It turns out that there are three regimes of growth of the
sperical caps as test sets that are of interest and only the one dubbed
“hyperuniform with respect to threshold order” can be seen as a direct
analogue to the classical notion of hyperuniformity; see Definition 2 below.
In [11] it is shown that hyperuniformity in each of the three regimes implies
uniform distribution. Furthermore, it is proven that so-called Quasi-Monte
Carlo design sequences (cf. [13]) with strength at least $\frac{d+1}{2}$,
certain energy minimising point set sequences (see, e.g., [10, 9]), and
especially sequences of spherical designs of optimal growth order have
hyperuniform behaviour. The paper [12] studies hyperuniformity on the sphere
for samples of point processes on the sphere. The spherical ensemble (cf.
[25]), the harmonic ensemble introduced in [6], and the jittered sampling
process (shown to be a determinantal) exhibit hyperuniform behaviour.

An essential requirement for our study of the hyperuniformity phenomenon in
the compact setting is that the space is homogeneous. Having considered the
sphere, it seems natural to turn to doubly homogeneous spaces next, which is
the aim of this paper. We point out that another compact setting, flat tori,
is studied in [34].

The paper is organised as follows. Section 2 introduces compact doubly
homogeneous spaces, briefly discusses their harmonic analysis, and defines
hyperuniformity on these spaces. Like in the spherical case [12, 11], there
are three regimes. We arrive at the result that hyperuniformity, indeed,
implies uniform distribution in each case. Section 3 discusses examples:
maximisers of the sum of distances, jittered sampling, and the harmonic
ensemble.

##  2\. Hyperuniformity on compact doubly homogeneous spaces

The main purpose of this paper is to extend the notion of hyperuniformity to
doubly homogeneous manifolds. Thus we first give a concise description of
these spaces and collect the relevant facts that will be used later.

###  2.1. Harmonic analysis on compact doubly homogeneous spaces

Let us recall the definition of doubly homogeneous spaces. A metric space
$(X,\vartheta)$ is called doubly homogeneous, if there exists a group $G$
acting isometrically on $X$ so that

$\forall x_{1},x_{2},y_{1},y_{2}\in
X:\vartheta(x_{1},x_{2})=\vartheta(y_{1},y_{2})\Rightarrow\exists g\in
G:y_{1}=gx_{1}\wedge y_{2}=gx_{2}.$

The compact doubly homogeneous Riemannian manifolds have been characterised in
[38] (see also [22]):

$\displaystyle\mathbb{S}^{d}$
$\displaystyle\cong\mathrm{SO}(d+1)/\mathrm{SO}(d)$
$\displaystyle\mathbb{RP}^{d-1}$
$\displaystyle\cong\mathrm{O}(d)/(\mathrm{O}(d-1)\times\mathrm{O}(1))$
$\displaystyle\mathbb{CP}^{d-1}$
$\displaystyle\cong\mathrm{U}(d)/(\mathrm{U}(d-1)\times\mathrm{U}(1))$
$\displaystyle\mathbb{HP}^{d-1}$
$\displaystyle\cong\mathrm{Sp}(d)/(\mathrm{Sp}(d-1)\times\mathrm{Sp}(1))$
$\displaystyle\mathbb{OP}^{2}$
$\displaystyle\cong\mathrm{F}_{4}/\mathrm{Spin}(9).$

We also refer to [1] for a concise introduction. The case of the sphere has
been studied in [12, 11]; thus we mainly focus on the projective spaces
$\mathbb{FP}^{d-1}$ in this paper, where we use $\mathbb{F}$ to represent the
underlying scalar “field” ($\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$,
$\mathbb{O}$). The dimension of the space as a real manifold is then given by

$D=(d-1)\dim_{\mathbb{R}}(\mathbb{F}),$

where

$\dim_{\mathbb{R}}(\mathbb{C})=2,\quad\dim_{\mathbb{R}}(\mathbb{H})=4,\quad\dim_{\mathbb{R}}(\mathbb{O})=8.$

Furthermore, we associate the parameters

$\alpha=\frac{d-1}{2}\dim_{\mathbb{R}}(\mathbb{F})-1\quad\text{and
}\beta=\frac{1}{2}\dim_{\mathbb{R}}(\mathbb{F})-1$

to the spaces $\mathbb{FP}^{d-1}$. We normalise the geodesic metric
$\vartheta$ to take values in $[0,\frac{\pi}{2}]$, and equip the space with
the normalised surface measure $\sigma$.

The measure of the geodesic ball
$B(x,r)=\\{y\in\mathbb{FP}^{d-1}\mid\vartheta(x,y)<r\\}$ is then given by

$\sigma(B(x,r))=\int_{0}^{r}A(\vartheta)\,d\vartheta,$

where

$A(r)=\frac{2\Gamma(\alpha+\beta+2)}{\Gamma(\alpha+1)\Gamma(\beta+1)}\sin(r)^{2\alpha+1}\cos(r)^{2\beta+1}$

denotes the surface area of the geodesic sphere
$\mathbb{S}(a,r)=\\{x\in\mathbb{FP}^{d-1}\mid\vartheta(a,x)=r\\}$. From now
on, we use the notation

$C_{\alpha,\beta}=\frac{2\Gamma(\alpha+\beta+2)}{\Gamma(\alpha+1)\Gamma(\beta+1)}$

for the normalising constant.

The Laplace operator associated to the underlying metric tensor is then given
by

$\triangle f=-\frac{1}{A(r)}\frac{\partial}{\partial
r}\left(A(r)\frac{\partial f}{\partial
r}\right)+\triangle_{\mathbb{S}(a,r)}f,$

where $\triangle_{\mathbb{S}(a,r)}$ denotes the Laplace operator on the
geodesic sphere. In the following we will restrict our attention to zonal
functions depending only on $r$; for these functions
$\triangle_{\mathbb{S}(a,r)}f$ vanishes. Furthermore, we notice that we use
the geometers convention that the Laplacian has non-negative eigenvalues.

The zonal eigenfunctions of $\triangle$ are then given by the functions

$P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,a))),$

the Jacobi-polynomials with parameters $\alpha$ and $\beta$. The corresponding
eigenvalues are given by

$\lambda_{n}=4n(n+\alpha+\beta+1).$

The dimension of the corresponding space of eigenfunctions $V_{n}$ is then
given by

$m_{n}=\frac{2n+\alpha+\beta+1}{\alpha+\beta+1}\frac{(\alpha+\beta+1)_{n}(\alpha+1)_{n}}{n!(\beta+1)_{n}},$

where $(x)_{n}=x(x+1)\cdots(x+n-1)$ denotes the rising factorial (Pochhammer
symbol).

$X$ | $\alpha$ | $\beta$ | $\lambda_{k}$ | $m_{k}=\dim(V_{k})$
---|---|---|---|---
$\mathbb{RP}^{d-1}$ | $\frac{d-3}{2}$ | $-\frac{1}{2}$ | $2k(2k+d-2)$ | $\frac{4k+d-2}{d-2}\binom{2k+d-3}{d-3}$
$\mathbb{CP}^{d-1}$ | $d-2$ | $0$ | $4k(k+d-1)$ | $\frac{2k+d-1}{d-1}\binom{d+k-2}{d-2}^{2}$
$\mathbb{HP}^{d-1}$ | $2d-3$ | $1$ | $4k(k+2d-1)$ | $\frac{2k+2d-1}{(2d-1)(2d-2)}\binom{k+2d-2}{2d-2}\binom{k+2d-3}{2d-3}$
$\mathbb{OP}^{2}$ | $7$ | $3$ | $4k(k+11)$ | $\frac{2k+11}{1320}\binom{k+10}{7}\binom{k+7}{7}$
Table 1.  The eigenvalues and dimensions of the eigenspaces of the Laplace
operator on the projective spaces $\mathbb{FP}^{d-1}$

The spaces of eigenfunctions $V_{n}$ are invariant under the group $G$ acting
on $\mathbb{FP}^{d-1}$. The corresponding representation turns out to be
irreducible. Let $Y_{n,k}(x)$ ($k=1,\ldots,m_{n}$) denote an orthonormal basis
of $V_{n}$. Then the addition theorem

(1)
$\sum_{k=1}^{m_{n}}Y_{n,k}(x)Y_{n,k}(y)=\frac{m_{n}}{P_{n}^{(\alpha,\beta)}(1)}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))$

is an immediate consequence of the irreducibility of $V_{n}$. Furthermore, we
recall the formula for integrals of zonal functions

(2)
$\begin{split}&\int_{\mathbb{FP}^{d-1}}f(\cos(2\vartheta(x,y)))\,d\sigma(x)\\\
=&C_{\alpha,\beta}\int_{0}^{\frac{\pi}{2}}f(\cos(2\vartheta))\sin(\vartheta)^{2\alpha+1}\cos(\vartheta)^{2\beta+1}\,d\vartheta,\end{split}$

see for instance [21].

Every function $f(\cos(2t))\in
L^{2}([0,\frac{\pi}{2}],\sin(t)^{2\alpha+1}\cos(t)^{2\beta+1}\,dt)$ can be
expanded in terms of the orthogonal system $P_{n}^{(\alpha,\beta)}(\cos(2t))$:

(3)
$f(\cos(2t))=\sum_{n=0}^{\infty}\widehat{f}(n)P_{n}^{(\alpha,\beta)}(\cos(2t)),$

where the coefficients are given by

$\widehat{f}(n)=\frac{C_{\alpha,\beta}m_{n}}{\left(P_{n}^{(\alpha,\beta)}(1)\right)^{2}}\int\limits_{0}^{\frac{\pi}{2}}f(\cos(2t))P_{n}^{(\alpha,\beta)}(\cos(2t))\sin(t)^{2\alpha+1}\cos(t)^{2\beta+1}\,dt.$

The convergence in (3) is _a priori_ only in the $L^{2}$-sense; for continuous
functions $f$ satisfying

(4) $\sum_{j,k=1}^{N}c_{j}\overline{c_{k}}f(\cos(2\vartheta(x_{j},x_{k})))\geq
0$

for all choices of $N$, $x_{1},\ldots,x_{N}\in X$, and
$c_{1},\ldots,c_{N}\in\mathbb{C}$, Mercer’s theorem ensures absolute and
uniform convergence of the series (3) (see for instance [16]). Functions
satisfying (4) are called _positive definite_. It follows from [8] that
positive definite functions are characterised by the non-negativity of the
coefficients $\widehat{f}(n)$.

###  2.2. Hyperuniformity on $\mathbb{FP}^{d-1}$

We use similar ideas as in [12, 11] to define hyperuniform sequences of point
sets in the setting of the spaces $\mathbb{FP}^{d-1}$. As in the original
euclidean setting the concept is based on the _number variance_.

######  Definition 1.

Let $(X_{N})_{N\in\mathbb{N}}$ a sequence of point sets in $\mathbb{FP}^{d-1}$
with $\lim_{N\to\infty}\\#X_{N}=\infty$. The number variance of this sequence
is then given by

(5) $\begin{split}&\mathbb{V}(X_{N},r)=\mathbb{V}_{x}\\#(X_{N}\cap B(x,r))\\\
=&\int_{\mathbb{FP}^{d-1}}\left(\sum_{y\in
X_{N}}\mathbbm{1}_{B(x,r)}(y)-\\#X_{N}\sigma(B(x,r))\right)^{2}\,d\sigma(x).\end{split}$

In a probabilistic setting the number variance is the variance of the number
of points in a geodesic ball of radius $r$. The interest in this quantity
stems from the fact that it can be used as a measure for the quality of a
point distribution in the deterministic as well as the probabilistic setting.

######  Remark 1.

The number variance is intimately related to the $L^{2}$-discrepancy

$\mathrm{D}_{L^{2}}(X_{N})=\frac{1}{\\#X_{N}}\left(\int_{0}^{\frac{\pi}{2}}\mathbb{V}(X_{N},r)\,dr\right)^{\frac{1}{2}}.$

This notion is a classical measure for the deviation of the discrete
distribution $\frac{1}{\\#X_{N}}\sum_{x\in X_{N}}\delta_{x}$ from uniform
distribution. $L^{2}$-discrepancy has been studied to find lower bounds for
this deviation; the theory of _irregularities of distribution_ has developed
techniques to derive lower bounds for various notions of discrepancy. For a
comprehensive introduction and a very good overview over results in this
context we refer to [3].

######  Remark 2.

Stolarsky’s celebrated invariance principle [35] relates the
$L^{2}$-discrepancy of a point set $X_{N}$ on the sphere $\mathbb{S}^{d}$ to
the sum of mutual chordal distances

(6) $\mathrm{D}_{L^{2}}(X_{N})^{2}+c_{d}\frac{1}{(\\#X_{N})^{2}}\sum_{x,y\in
X_{N}}\|x-y\|=c_{d}\iint_{\mathbb{S}^{d}\times\mathbb{S}^{d}}\|x-y\|\,d\sigma(x)\,d\sigma(y)$

for a suitable constant $c_{d}$. This shows that the $L^{2}$-discrepancy is
minimised for sets maximising the sum of mutual distances.

This invariance principle was generalised to the projective spaces
$\mathbb{FP}^{d-1}$ by Skriganov [33]. Furthermore, it is observed there that
the invariance principle is a special case of a much more general relation.

As in the classical notion of hyperuniformity we will be interested in the
dependence of the number variance on the radius $r$. Three regimes turn out to
be of interest.

######  Definition 2.

A sequence of point sets $(X_{N})_{N\in\mathbb{N}}$ is called

  1. (1)

hyperuniform for large balls, if for every $r\in(0,\frac{\pi}{2})$

$\mathbb{V}(X_{N},r)=o\left(\\#X_{N}\right),$

  2. (2)

hyperuniform for small balls, if for every sequence of
$(r_{N})_{N\in\mathbb{N}}$ of radii satisfying

$\displaystyle\lim_{N\to\infty}r_{N}=0$
$\displaystyle\lim_{N\to\infty}\sigma(B(\cdot,r_{N}))\\#X_{N}=\infty$

the following holds

$\mathbb{V}(X_{N},r)=o\left(\\#X_{N}\sigma(B(\cdot,r_{N}))\right),$

  3. (3)

hyperuniform for balls at threshold order, if

$\limsup_{N\to\infty}\mathbb{V}(X_{N},r(\\#X_{N})^{-\frac{1}{D}})=\mathcal{O}(r^{D-1})$

for $r\to\infty$.

We notice that hyperuniformity at threshold order is obtained by rescaling and
adapting the classical notion to the compact situation. The motivation for the
order $r^{D-1}$ on the right hand side comes from the order of the surface
area of the geodesic sphere; i.i.d. random points would achieve the order
$r^{D}$. Furthermore, it is clear that a smaller order of magnitude than
$(\\#X_{N})^{-\frac{1}{D}}$ for the radii would not be sensible, because then
the expected number of points in a ball would tend to $0$. The other two
notions of hyperuniformity have no obvious counterpart in the classical
setting. All three notions have in common that the number variance is required
to have smaller order of magnitude than the variance of i.i.d. random point
sets of the same cardinality.

###  2.3. Uniform distribution

Uniform distribution is a notion that formalises the discretisation of
measures.

######  Definition 3.

A sequence of of point sets $(X_{N})_{N\in\mathbb{N}}$ in $\mathbb{FP}^{d-1}$
with $\lim_{N\to\infty}\\#X_{N}=\infty$ is called uniformly distributed, if

$\lim_{N\to\infty}\frac{1}{\\#X_{N}}\sum_{x\in
X_{N}}\mathbbm{1}_{B(x,r)}=\sigma(B(x,r))$

for all $x\in\mathbb{FP}^{d-1}$ and $r\in[0,\frac{\pi}{2}]$.

For a comprehensive introduction to uniform distribution we refer to [15, 28].

From the general theory (see [28]) it is known that a sequence of point sets
is uniformly distributed, if and only if

$\lim_{N\to\infty}\frac{1}{\\#X_{N}}\sum_{x\in
X_{N}}f(x)=\int_{\mathbb{FP}^{d-1}}f(x)\,d\sigma(x)$

for all continuous functions $f:\mathbb{FP}^{d-1}\to\mathbb{C}$. From this it
follows immediately (using the denseness of the Laplace eigenfunctions and the
addition theorem (1)) that $(X_{N})_{N\in\mathbb{N}}$ is uniformly
distributed, if and only if

(7) $\lim_{N\to\infty}\frac{1}{(\\#X_{N})^{2}}\sum_{x,y\in
X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))=0$

for all $n\geq 1$.

We now want to relate hyperuniformity to uniform distribution. Since uniform
distribution is a property modelled after the law of large numbers, which
holds especially for i.i.d. random points, and hyperuniformity is modelled as
a property “better than i.i.d. random points”, we can only expect that
hyperuniformity implies uniform distribution.

In order to show that in all three regimes, we first derive an expression for
the number variance of $X_{N}$ in terms of Jacobi polynomials. We observe that
$\mathbbm{1}_{B(x,r)}(y)=\mathbbm{1}_{[0,r)}(\vartheta(x,y))$ is a zonal
function. Thus we can expand it as a series in terms of Jacobi polynomials in
$\cos(2\vartheta(x,y))$. Using the formula

$\int_{0}^{r}P_{n}^{(\alpha,\beta)}(\cos(2t))\sin(t)^{2\alpha+1}\cos(t)^{2\beta+1}\,dt\\\
=\frac{1}{2n}\sin(r)^{2\alpha+2}\cos(r)^{2\beta+2}P_{n-1}^{(\alpha+1,\beta+1)}(\cos(2r))$

for $n\geq 1$ (see [29]), we obtain

$\mathbbm{1}_{B(x,r)}(y)=\sigma(B(x,r))+\sum_{n=1}^{\infty}\frac{C_{\alpha,\beta}m_{n}}{P_{n}^{\alpha,\beta}(1)}a_{n}(r)P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y))).$

with

$a_{n}(r)=\frac{1}{2nP_{n}^{\alpha,\beta}(1)}\sin(r)^{2\alpha+2}\cos(r)^{2\beta+2}P_{n-1}^{(\alpha+1,\beta+1)}(\cos(2r)).$

This equation is valid in the $L^{2}$-sense. Applying Parseval’s identity we
obtain

(8)
$\mathbb{V}(X_{n},r)=\sum_{n=1}^{\infty}\frac{C_{\alpha,\beta}^{2}m_{n}}{P_{n}^{\alpha,\beta}(1)}a_{n}(r)^{2}\sum_{x,y\in
X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y))).$

This series is absolutely and uniformly (as a function of $x$ and $y$)
convergent by Mercer’s theorem. Notice that the function

(9)
$\begin{split}g_{r}(x,y)&=\\!\\!\\!\\!\int\limits_{\mathbb{FP}^{d-1}}\\!\\!\\!\left(\mathbbm{1}_{B(x,r)}(z)-\sigma(B(x,r))\right)\left(\mathbbm{1}_{B(y,r)}(z)-\sigma(B(y,r))\right)\,d\sigma(z)\\\
&=\sigma(B(x,r)\cap B(y,r))-\sigma(B(\cdot,r))^{2}\end{split}$

is positive definite and that

(10) $\mathbb{V}(X_{N},r)=\sum_{x,y\in X_{N}}g_{r}(x,y).$

####  2.3.1. Hyperuniformity for large balls implies uniform distribution

Assume now that $(X_{N})_{N\in\mathbb{N}}$ is hyperuniform for large balls.
For fixed $n$ choose a distance $0<r<\frac{\pi}{2}$ such that
$P_{n}^{(\alpha,\beta)}(\cos(2r))\neq 0$. Then equation (8) yields

$0=\lim_{N\to\infty}\frac{\mathbb{V}(X_{N},r)}{\\#X_{N}}\geq\\\
\frac{C_{\alpha,\beta}^{2}m_{n}}{P_{n}^{\alpha,\beta}(1)}a_{n}(r)^{2}\lim_{N\to\infty}\frac{1}{\\#X_{N}}\sum_{x,y\in
X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y))).$

This proves

######  Theorem 1.

Let $(X_{N})_{N\in\mathbb{N}}$ be hyperuniform for large balls. Then
$(X_{N})_{N\in\mathbb{N}}$ is uniformly distributed. More precisely,

(11) $\lim_{N\to\infty}\frac{1}{\\#X_{N}}\sum_{x,y\in
X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))=0$

holds for all $n\geq 1$.

######  Remark 3.

Notice that (11) gives a better rate of convergence as compared to the
criterion (7) for uniform distribution. This reflects the heuristic
expectation that hyperuniformity should give an improved quality of uniform
distribution.

####  2.3.2. Hyperuniformity for small balls implies uniform distribution

Assume that $(X_{N})_{N\in\mathbb{N}}$ is hyperuniform for small balls.
Furthermore, we observe that for fixed $n\geq 1$ the coeffient of
$P_{n}^{(\alpha,\beta)}$ in (8) is of order $r_{N}^{2D}$ for $N\to\infty$
(notice that $4\alpha+4=2D$). Arguing as above we obtain that

$\lim_{N\to\infty}\frac{\mathbb{V}(X_{N},r_{N})}{\\#X_{N}\sigma(B(\cdot,r_{N}))}=0$

implies

$\lim_{N\to\infty}\frac{r_{N}^{D}}{\\#X_{N}}\sum_{x,y\in
X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))=0.$

Since $r_{N}$ can be chosen to tend to $0$ arbitrarily slowly, this implies

$\limsup_{N\to\infty}\frac{1}{\\#X_{N}}\sum_{x,y\in
X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))<\infty.$

Thus we have shown

######  Theorem 2.

Let $(X_{N})_{N\in\mathbb{N}}$ be hyperuniform for small balls. Then
$(X_{N})_{N\in\mathbb{N}}$ is uniformly distributed. More precisely,

(12) $\limsup_{N\to\infty}\frac{1}{\\#X_{N}}\sum_{x,y\in
X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))<\infty$

holds for all $n\geq 1$.

######  Remark 4.

Notice that we have an improvement in the quality of uniform distribution in
comparison with (7) by a power $(\\#X_{N})^{1-\varepsilon}$ for
$\varepsilon>0$.

####  2.3.3. Hyperuniformity for balls at threshold order implies uniform
distribution

Assume that $(X_{N})_{N\in\mathbb{N}}$ is hyperuniform for balls at threshold
order. For fixed $n\geq 1$ we notice that

$a_{n}(r(\\#X_{N})^{-\frac{1}{D}})^{2}\sim\frac{P_{n-1}^{(\alpha+1,\beta+1)}(1)^{2}}{(2nP_{n}^{(\alpha,\beta)}(1))^{2}}\frac{r^{2D}}{(\\#X_{N})^{2}}.$

Combining this with (8) gives

$r^{2D}\frac{C_{\alpha,\beta}^{2}m_{n}P_{n-1}^{(\alpha+1,\beta+1)}(1)^{2}}{(2n)^{2}(P_{n}^{\alpha,\beta}(1))^{3}}\limsup_{N\to\infty}\frac{1}{(\\#X_{N})^{2}}\sum_{x,y\in
X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))\\\
\leq\limsup_{N\to\infty}\mathbb{V}(X_{N},r(\\#X_{N})^{-\frac{1}{D}})=\mathcal{O}(r^{D-1}).$

This can only hold, if

$\limsup_{N\to\infty}\frac{1}{(\\#X_{N})^{2}}\sum_{x,y\in
X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))=0.$

Summing up, we have proved

######  Theorem 3.

Let $(X_{N})_{N\in\mathbb{N}}$ be hyperuniform for balls at threshold order.
Then $(X_{N})_{N\in\mathbb{N}}$ is uniformly distributed.

######  Remark 5.

In this case there seems to be no improvement in the order of convergence of
the sum (7).

##  3\. Examples

###  3.1. Maximisers of the sum of distances

As pointed out in Remark 2 Skriganov [33] generalised Stolarsky’s invariance
principle [35] to the projective spaces $\mathbb{FP}^{d-1}$. As a consequence
the $L^{2}$-discrepancy is minimised amongst all sets of the same cardinality
by configurations maximising the sum of mutual “chordal” distances

(13) $\sum_{x,y\in X_{N}}\sin(\vartheta(x,y)).$

It was shown in [32] that there exist positive constants $K_{d}$ and $k_{d}$
such that

(14) $\begin{split}-&K_{d}(\\#X_{N})^{1-\frac{1}{D}}\leq\\\ &\sum_{x,y\in
X_{N}}\sin(\vartheta(x,y))-(\\#X_{N})^{2}\iint\limits_{(\mathbb{FP}^{d-1})^{2}}\sin(\vartheta(x,y))\,d\sigma(x)\,d\sigma(y)\\\
&\leq-k_{d}(\\#X_{N})^{1-\frac{1}{D}}\end{split}$

for a point set $X_{N}$ maximising (13).

Integrating (8) against $\sin(2r)\,dr$ and using

$\displaystyle\int_{0}^{\frac{\pi}{2}}\sin(r)^{4\alpha+4}\cos(r)^{4\beta+4}P_{n-1}^{(\alpha+1,\beta+1)}(\cos(2r))^{2}\sin(2r)\,dr=$
$\displaystyle\frac{\Gamma(2\alpha+3)\Gamma(2\beta+3)}{\alpha+\beta+2}2^{2n-2}(n+\alpha+\beta+1)\times$
$\displaystyle\frac{(\frac{1}{2})_{n-1}(\alpha+2)_{n-1}(\beta+2)_{n-1}(\alpha+\beta+2)_{n-1}}{((n-1)!)^{2}\Gamma(2(n+\alpha+\beta+3))}$

(see [33]) we obtain

(15)
$\begin{split}&\mathrm{D}_{L^{2}}(X_{N})^{2}=C_{\alpha,\beta}^{2}\Gamma(2\alpha+3)\Gamma(2\beta+3)\times\\\
&\sum_{n=1}^{\infty}2^{2n-2}(2n+\alpha+\beta+1)\frac{(\frac{1}{2})_{n-1}(\alpha+\beta+1)_{n}(\alpha+\beta+2)_{n}}{(\alpha+1)_{n}\Gamma(2(n+\alpha+\beta+3))}\times\\\
&\sum_{x,y\in X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))\end{split}$

(see also [33, 31]). We notice that the coefficients in the sum are of the
order $n^{-\alpha-2}$.

We will use the inequality

(16)
$\begin{split}&\sin(\vartheta)^{a+\frac{1}{2}}\cos(\vartheta)^{b+\frac{1}{2}}|P_{n}^{(a,b)}(\cos(2\vartheta))|\\\
&\leq\frac{\Gamma(a+1)}{\sqrt{\pi}}\binom{n+a}{n}\left(n+\frac{1}{2}(a+b+1)\right)^{-a-\frac{1}{2}}=\mathcal{O}\left(n^{-\frac{1}{2}}\right)\end{split}$

valid for $a\geq b>-1$ (see [14]). Inserting $a=\alpha+1$ and $b=\beta+1$ in
(16) we estimate the number variance by

(17) $\mathbb{V}(X_{N},r)\leq
C\sin(r)^{2\alpha+1}\cos(r)^{2\beta+1}\sum_{n=1}^{\infty}n^{-\alpha-2}\sum_{x,y\in
X_{N}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y))).$

Putting (15) and (17) together and observing that $2\alpha+1=D-1$ we obtain

(18)
$\mathbb{V}(X_{N},r)=\mathcal{O}\left(\sin(r)^{D-1}\mathrm{D}_{L^{2}}(X_{N})\right).$

For point sets with optimal $L^{2}$-discrepancy (or equivalently, maximal sum
of distances) we obtain

$\mathbb{V}(X_{N},r)=\mathcal{O}\left(\sin(r)^{D-1}(\\#X_{N})^{1-\frac{1}{D}}\right).$

This inequality immediately implies the theorem.

######  Theorem 4.

Let $(X_{N})_{N\in\mathbb{N}}$ be a sequence of maximisers of the sum of
mutual distances (13). Then this sequence is hyperuniform in all all three
regimes.

###  3.2. Jittered sampling

A very simple and obvious probabilistic model for a point set $X_{N}$ is given
by jittered sampling. We start with a partition of the space
$\mathbb{FP}^{d-1}$ into $N$ sets $A_{j}$ ($j=1,\ldots,N$) of equal area and
diameter $\leq CN^{-\frac{1}{D}}$, where the constant only depends on the
space. It is known from [19] that such a partition exists.

We define a point process $\mathscr{X}_{N}^{J}$, called the _jittered sampling
process_ , by choosing $N$ points $x_{j}\in A_{j}$ ($j=1,\ldots,N$), where
each point is chosen with respect to the surface measure restricted to
$A_{j}$. The number variance of this process is then given by

(19) $\begin{split}&V_{N}=\mathbb{V}(\mathscr{X}_{N}^{J},r)\\\
&=\\!\\!\\!\int\limits_{\mathbb{FP}^{d-1}}\int\limits_{A_{1}}\\!\\!\cdots\\!\\!\int\limits_{A_{N}}\\!\\!\left(\sum_{j=1}^{N}\mathbbm{1}_{B(x,r)}(x_{j})-N\sigma(B(x,r))\right)^{2}\\!\\!d\sigma_{1}(x_{1})\cdots
d\sigma_{N}(x_{N})\,d\sigma(x),\end{split}$

where $\sigma_{j}(A)=N\sigma(A_{j}\cap A)$ denotes the surface measure
restricted to $A_{j}$.

Expanding the square in (19) and using the fact that
$\mathbbm{1}_{B(y,r)}(x)=\mathbbm{1}_{B(x,r)}(y)$ we obtain

$\displaystyle V_{N}$ $\displaystyle=\sum_{i\neq
j}\int_{A_{i}}\int_{A_{j}}\sigma(B(x_{i},r)\cap
B(x_{j},r))\,d\sigma_{i}(x_{i})\,d\sigma_{j}(x_{j})$
$\displaystyle+N\sigma(B(\cdot,r))-N^{2}\sigma(B(\cdot,r))^{2}$
$\displaystyle=N^{2}\int_{\mathbb{FP}^{d-1}}\int_{\mathbb{FP}^{d-1}}\sigma(B(x,r)\cap
B(y,r))\,d\sigma(x)\,d\sigma(y)$
$\displaystyle-\sum_{i=1}^{N}\int_{A_{i}}\int_{A_{i}}\sigma(B(x_{i},r)\cap
B(x_{j},r))\,d\sigma_{i}(x_{i})\,d\sigma_{j}(x_{j})$
$\displaystyle+N\sigma(B(\cdot,r))-N^{2}\sigma(B(\cdot,r))^{2}$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{N}\int_{A_{i}}\int_{A_{i}}\sigma(B(x_{i},r)\triangle
B(y_{i},r))\,d\sigma_{i}(x_{i})\,d\sigma_{i}(y_{i}),$

where we have used

$\int_{\mathbb{FP}^{d-1}}\int_{\mathbb{FP}^{d-1}}\sigma(B(x,r)\cap
B(y,r))\,d\sigma(x)\,d\sigma(y)=\sigma(B(\cdot,r))^{2}$

and used the notation $A\triangle B=A\cup B\setminus(A\cap B)$ for the
symmetric difference. A similar computation has been used in [12].

The measure of the symmetric difference of two balls can be bounded

(20) $\sigma(B(x,r)\triangle B(y,r))\leq
C^{\prime}\vartheta(x,y)\mathrm{surface}(\partial B(\cdot,r))$

for some constant $C^{\prime}>0$. Since the diameter of each part $A_{j}$ of
the partition is bounded by $CN^{-\frac{1}{D}}$ we obtain the bound

$V_{N}\leq CC^{\prime}N^{1-\frac{1}{D}}\mathrm{surface}(\partial
B(\cdot,r))=\mathcal{O}(r^{D-1}N^{1-\frac{1}{D}}),$

from which we derive the theorem.

######  Theorem 5.

The jittered sampling process $\mathscr{X}_{N}^{J}$ is hyperuniform in all
three regimes.

###  3.3. The harmonic ensemble

Determinantal point processes (see [25]) have become a very useful tool to
provide probabilistic models, which behave better than i.i.d. models due to
built mutual repulsion of the points. Especially, in the context of finding
well distributed point sets such processes have proven to be a very useful
tool to establish good bounds for various quality measures of point
distributions (see, for instance [1, 6, 4, 24, 5, 30, 23]). Since the notion
of hyperuniformity has been motivated as a property of point sets, which
behave “better than i.i.d. random”, it is natural to test samples of
determinantal processes for this property.

The rotation invariant processes on the spaces $\mathbb{FP}^{d-1}$ have been
characterised in [1]. The most natural amongst them is the process induced by
the projection kernel on the spaces of eigenfunctions for the $N+1$ smallest
eigenvalues of the Laplace operator. This kernel is given by

(21)
$\begin{split}K_{N}(x,y)&=\sum_{n=0}^{N}\sum_{k=1}^{m_{n}}Y_{n,k}(x)Y_{n,k}(y)=\sum_{n=0}^{N}\frac{m_{n}}{P_{n}^{(\alpha,\beta)}(1)}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))\\\
&=\frac{(\alpha+\beta+2)_{N}}{(\beta+1)_{N}}P_{N}^{(\alpha+1,\beta)}(\cos(2\vartheta(x,y)))\end{split}$

(see, for instance [2]). This process produces

$K_{N}(x,x)=\frac{(\alpha+\beta+2)_{N}(\alpha+2)_{N}}{N!(\beta+1)_{N}}\sim\frac{\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)\Gamma(\alpha+2)}N^{2\alpha+2}$

points; we use the notation $\mathscr{X}_{N}^{H}$ for this process and call it
the _harmonic process_ like the corresponding process on the sphere introduced
in [6]. For more details on determinantal point processes we refer to [25]
again.

We notice that the number variance of $\mathscr{X}_{N}^{H}$ equals the
expectation of the sum (10) under law of the process $\mathscr{X}_{N}^{H}$,

(22) $\mathbb{V}(\mathscr{X}_{N}^{H},r)=\mathbb{E}\sum_{x,y}g_{r}(x,y).$

Such expectations can be expressed easily by the general theory of
determinantal processes

$\mathbb{E}\sum_{x,y}g_{r}(x,y)=\mathbb{E}\sum_{x\neq
y}g_{r}(x,y)+\mathbb{E}\sum_{x}g_{r}(x,x)\\\
=\iint\limits_{(\mathbb{FP}^{d-1})^{2}}g_{r}(x,y)\left(K_{N}(x,x)K_{N}(y,y)-K_{N}(x,y)K_{N}(y,x)\right)\,d\sigma(x)\,d\sigma(y)\\\
+K_{N}(\cdot,\cdot)g_{r}(\cdot,\cdot).$

Since all functions occurring in this last equation are zonal, we can use (2)
to further simplify this equation

$\mathbb{V}(\mathscr{X}_{N}^{H},r)=\\\
C_{\alpha,\beta}\int_{0}^{\frac{\pi}{2}}g_{r}(\cos(2\vartheta))\left(K_{N}(1)^{2}-K_{N}(\cos(2\vartheta))^{2}\right)\sin(\vartheta)^{2\alpha+1}\cos(\vartheta)^{2\beta+1}\,d\vartheta\\\
+K_{N}(1)g_{r}(1);$

here we have made use of the convention to write
$f(\cos(2\vartheta(x,y)))=f(x,y)$ for zonal functions. Using the facts that

$\displaystyle
C_{\alpha,\beta}\int_{0}^{\frac{\pi}{2}}K_{N}(\cos(2\vartheta))^{2}\sin(\vartheta)^{2\alpha+1}\cos(\vartheta)^{2\beta+1}\,d\vartheta=K_{N}(1)$
$\displaystyle
C_{\alpha,\beta}\int_{0}^{\frac{\pi}{2}}g_{r}(\cos(2\vartheta))\sin(\vartheta)^{2\alpha+1}\cos(\vartheta)^{2\beta+1}\,d\vartheta=0$

we obtain

$\mathbb{V}(\mathscr{X}_{N}^{H},r)=\\\
C_{\alpha,\beta}\int_{0}^{\frac{\pi}{2}}\left(g_{r}(1)-g_{r}(\cos(2\vartheta))\right)K_{N}(\cos(2\vartheta))^{2}\sin(\vartheta)^{2\alpha+1}\cos(\vartheta)^{2\beta+1}\,d\vartheta.$

We now remark that

$g_{r}(1)-g_{r}(\cos(2\vartheta(x,y)))=\frac{1}{2}\sigma(B(x,r)\triangle
B(y,r))\\\ \leq C^{\prime}\vartheta(x,y)\mathrm{surface}(\partial
B(\cdot,r))\leq C^{\prime\prime}\sin(\vartheta)r^{D-1}$

using (20). Thus we have

$\mathbb{V}(\mathscr{X}_{N}^{H},r)=\mathcal{O}\left(r^{D-1}\int_{0}^{\frac{\pi}{2}}K_{N}(\cos(2\vartheta))^{2}\sin(\vartheta)^{2\alpha+2}\cos(\vartheta)^{2\beta+1}\,d\vartheta\right),$

and it remains to estimate the integral

$\frac{(\alpha+\beta+2)_{N}^{2}}{(\beta+1)_{N}^{2}}\int_{0}^{\frac{\pi}{2}}P_{N}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\sin(\vartheta)^{2\alpha+2}\cos(\vartheta)^{2\beta+1}\,d\vartheta;$

notice that this is an integral over the square of a Jacobi polynomial against
the “wrong” weight function. We use the estimate (16) for $a=\alpha+1$ and
$b=\beta$ in the interval $[\frac{c}{N},\frac{\pi}{2}]$ and the trivial bound
$|P_{N}^{(\alpha+1,\beta)}(\cos(2\vartheta))|\leq P_{N}^{(\alpha+1,\beta)}(1)$
in the interval $[0,\frac{C}{N}]$ to obtain

$\int_{0}^{\frac{\pi}{2}}P_{N}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\sin(\vartheta)^{2\alpha+2}\cos(\vartheta)^{2\beta+1}\,d\vartheta\\\
=\mathcal{O}\left(\int_{0}^{\frac{C}{N}}N^{2\alpha+2}\vartheta^{2\alpha+2}\,d\vartheta\right)+\mathcal{O}\left(\int_{\frac{C}{N}}^{\frac{\pi}{2}}\frac{1}{N\sin(\vartheta)}\,d\vartheta\right)\\\
=\mathcal{O}\left(\frac{1}{N}\right)+\mathcal{O}\left(\frac{\log(N)}{N}\right)=\mathcal{O}\left(\frac{\log(N)}{N}\right).$

Summing up, we have obtained

$\mathbb{V}(\mathscr{X}_{N}^{H},r)=\mathcal{O}\left(r^{D-1}\frac{(\alpha+\beta+2)_{N}^{2}}{(\beta+1)_{N}^{2}}\frac{\log(N)}{N}\right)=\mathcal{O}\left(r^{D-1}\frac{K_{N}(1)\log(N)}{N}\right).$

Thus we have shown the following theorem.

######  Theorem 6.

The harmonic process $\mathscr{X}_{N}^{H}$ is hyperuniform for large and small
balls. For balls at threshold order the weaker relation

$\mathbb{V}(\mathscr{X}_{N}^{H},rK_{N}(1)^{-\frac{1}{D}})=\mathcal{O}\left(r^{D-1}\log(N)\right)$

holds.

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