$n=2$ and one must make some modifications for the integral formula, but we do
not get into any details about this here. We expect that the proof of
Proposition 6.4 can for the complex and quarternionic cases be carried out in
a similar fashion for such real rank one symmetric spaces.

###  6.4. Bounds and asymptotics of spherical functions

We will now derive the main asymptotic bounds of spherical functions that will
lead us to the proof of Theorem 1.4 in the later sections.

######  Proposition 6.11.

The spherical functions satisfy the following:

  1. (1)

For every $r>0$ and every $\lambda>0$,

$\displaystyle|\omega_{\lambda}^{(n)}(a_{r})|\leq\frac{a_{n}}{\lambda\sinh(r)^{\frac{n-1}{2}}}\,.$

  2. (2)

For every $\lambda>0$ we have

$\displaystyle\lim_{R\rightarrow+\infty}\frac{1}{R}\int_{0}^{R}|\omega^{(n)}_{\lambda}(a_{r})|^{2}\sinh(r)^{n-1}dr=2^{n-2}\frac{|\mathrm{B}(i\lambda,\frac{n-1}{2})|^{2}}{\mathrm{B}(\frac{1}{2},\frac{n-1}{2})^{2}}\,.$

In particular, the convergence is uniform in $\lambda\geq\lambda_{o}$ for
every $\lambda_{o}>0$.

  3. (3)

Moreover, if $\lambda\in[0,\frac{n-1}{2}]$ and $r>1$ then there is a constant
$C=C(n)>0$ such that

$\displaystyle\omega_{i\lambda}^{(n)}(a_{r})\geq C\frac{\sinh(\lambda
r)}{\lambda\sinh(r)^{\frac{n-1}{2}}}\,.$

######  Remark 6.12.

The limit formula in item (2) of the above Proposition has previously been
computed by Strichartz in [17, Eq.4.20, p.84] by viewing the spherical
functions on $\mathbb{H}^{n}$ as generalized Legendre functions. We present
another proof of this using the trigonometric expansion given in Proposition
6.4.

To prove item (2) in Proposition 6.11, we use of the following elementary
Lemma.

######  Lemma 6.13.

Let $R_{o}\geq 0$ and suppose that
$\alpha,\beta:[R_{o},+\infty)\rightarrow\mathbb{R}$ are bounded continuous
functions such that $\alpha(R)\rightarrow a$ and $\beta(R)\rightarrow b$ as
$R\rightarrow+\infty$. Then for every $\lambda>0$,

$\displaystyle\frac{1}{R}\int_{R_{o}}^{R}(\alpha(r)\cos(\lambda
r)+\beta(r)\sin(\lambda r))^{2}dr\longrightarrow\frac{a^{2}+b^{2}}{2}\,,\quad
R\rightarrow+\infty\,.$

###### Proof of Proposition 6.11.

To prove (1) we use that $\cosh(r)-1\leq\sinh(r)$ for all $r\geq 0$ to find
that

$\displaystyle|\omega_{\lambda}^{(n)}(a_{r})|$
$\displaystyle=\frac{a_{n}}{\sinh(r)^{n-2}}\Big{|}\int_{0}^{r}(\cosh(r)-\cosh(s))^{\frac{n-3}{2}}\cos(\lambda
s)ds\Big{|}$ $\displaystyle\leq
a_{n}\frac{(\cosh(r)-1)^{\frac{n-3}{2}}}{\sinh(r)^{n-2}}\Big{|}\int_{0}^{r}\cos(\lambda
s)ds\Big{|}\leq\frac{a_{n}}{\lambda\sinh(r)^{\frac{n-1}{2}}}\,.$

For (2) it is enough to show that

$\displaystyle\lim_{R\rightarrow+\infty}\frac{1}{R}\int_{R_{o}}^{R}|\omega^{(n)}_{\lambda}(a_{r})|^{2}\sinh(r)^{n-1}dr=2^{n-2}\frac{|\mathrm{B}(i\lambda,\frac{n-1}{2})|^{2}}{\mathrm{B}(\frac{1}{2},\frac{n-1}{2})^{2}}$

for some choice of $R_{o}\geq 0$. Choose $R_{o}$ sufficiently large so that we
can apply Proposition 6.4 and Lemma 6.13 to find that

$\displaystyle\frac{1}{R}\int_{R_{o}}^{R}$
$\displaystyle|\omega^{(n)}_{\lambda}(a_{r})|^{2}\sinh(r)^{n-1}dr=\frac{a_{n}^{2}}{R}\int_{R_{o}}^{R}(\alpha_{n}(r,\lambda)\cos(\lambda
r)+\beta_{n}(r,\lambda)\sin(\lambda r))^{2}dr$ $\displaystyle\longrightarrow
a_{n}^{2}\frac{\mathrm{Re}(\mathrm{B}(i\lambda,\frac{n-1}{2}))^{2}+\mathrm{Im}(\mathrm{B}(i\lambda,\frac{n-1}{2}))^{2}}{2}=2^{n-2}\frac{|\mathrm{B}(i\lambda,\frac{n-1}{2})|^{2}}{\mathrm{B}(\frac{1}{2},\frac{n-1}{2})^{2}}$

as $R\rightarrow+\infty$.

For (3), we bound from below by

$\displaystyle\omega_{i\lambda}(a_{r})$
$\displaystyle=\frac{a_{n}}{\sinh(r)^{n-2}}\int_{0}^{r}(\cosh(r)-\cosh(s))^{\frac{n-3}{2}}\cosh(\lambda
s)ds$
$\displaystyle\geq\frac{a_{n}}{\sinh(r)^{n-2}}\int_{0}^{r-1}(\cosh(r)-\cosh(s))^{\frac{n-3}{2}}\cosh(\lambda
s)ds$ $\displaystyle\geq
a_{n}\frac{(\cosh(r)-\cosh(r-1))^{\frac{n-3}{2}}}{\sinh(r)^{n-2}}\int_{0}^{r-1}\cosh(\lambda
s)ds$
$\displaystyle=a_{n}\frac{\cosh(r)^{\frac{n-3}{2}}}{\sinh(r)^{n-2}}\Big{(}1-\frac{\cosh(r-1)}{\cosh(r)}\Big{)}^{\frac{n-3}{2}}\frac{\sinh(\lambda(r-1))}{\lambda}$
$\displaystyle\geq\frac{a_{n}}{\sinh(r)^{\frac{n-1}{2}}}\Big{(}1-\frac{\cosh(r-1)}{\cosh(r)}\Big{)}^{\frac{n-3}{2}}\frac{\sinh(\lambda(r-1))}{\lambda}\,.$

Using that $\lambda\in[0,\frac{n-1}{2}]$ and $r>1$, elementary calculations
yield the lower bounds

$\displaystyle\sinh(\lambda(r-1))\geq\mathrm{e}^{-\frac{n-1}{2}}\sinh(\lambda
r)\quad\mbox{ and }\quad 1-\frac{\cosh(r-1)}{\cosh(r)}\geq
1-\frac{2}{\mathrm{e}}$

so that

$\displaystyle\omega_{i\lambda}(a_{r})\geq C\frac{\sinh(\lambda
r)}{\lambda\sinh(r)^{\frac{n-1}{2}}}\,,\quad
C=a_{n}\mathrm{e}^{-\frac{n-1}{2}}(1-2\mathrm{e}^{-1})^{\frac{n-3}{2}}\,.$

∎

###  6.5. The spherical transform on $\mathbb{H}^{n}$

Given a function $\varphi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n})$, its
_Fourier-Helgason transform_ is the function
$\mathscr{F}\varphi:\mathbb{C}\times\mathbb{S}^{n-1}\rightarrow\mathbb{C}$
given by

$\displaystyle\mathscr{F}\varphi(\lambda,u)=\int_{G_{n}}\varphi(g)\overline{\xi_{\lambda,u}(g)}dm_{G_{n}}(g)\,.$

If $\varphi$ happens to be bi-$K_{n}$-invariant, then

$\displaystyle\mathscr{F}\varphi(\lambda,u)=\int_{G_{n}}\varphi(g)\overline{\omega_{\lambda}(g)}dm_{G_{n}}(g)$

for any choice of $u\in\mathbb{S}^{n-1}$.

######  Definition 6.14 (Hyperbolic spherical transform).

The _spherical transform_ of a function
$\varphi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n},K_{n})$ is the function
$\widehat{\varphi}:\mathbb{C}\rightarrow\mathbb{C}$ given by

$\displaystyle\widehat{\varphi}(\lambda)=\int_{G_{n}}\varphi(g)\overline{\omega_{\lambda}(g)}dm_{G_{n}}(g)=\int_{0}^{\infty}\varphi(a_{t})\overline{\omega_{\lambda}(a_{t})}\sinh(t)^{n-1}dt\,.$

One shows that $\widehat{\varphi}$ is even and analytic on $\mathbb{C}$ for
every $\varphi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n},K_{n})$ and that
$\widehat{\varphi_{1}^{*}*\varphi_{2}}(\lambda)=\overline{\widehat{\varphi}_{1}(\overline{\lambda})}\widehat{\varphi}_{2}(\lambda)$.
In particular, if $\lambda\in\mathbb{R}\cup i\mathbb{R}$ then
$\widehat{\varphi^{*}*\varphi}(\lambda)=|\widehat{\varphi}(\lambda)|^{2}$.

As for the $L^{2}$-theory of the spherical transform, we again have a
Plancherel formula. To state it, we introduce the _Harish-Chandra
$c$-function_

$\displaystyle
c_{n}(\lambda)=\rho_{n}2^{\frac{n-2}{2}}\frac{\mathrm{B}(i\lambda,\frac{n-1}{2})}{\mathrm{B}(\frac{1}{2},\frac{n-1}{2})}=\rho_{n}2^{\frac{n-2}{2}}\frac{\Gamma(\frac{n}{2})\Gamma(i\lambda)}{\Gamma(\frac{n-1}{2})\Gamma(i\lambda+\frac{n-1}{2})}\,,\quad\lambda\in\mathbb{C}\backslash
i\mathbb{Z}_{\geq 0}\,,$

where $\rho_{n}>0$ is a normalizing constant such that the Plancherel formula
below holds. The precise value of $\rho_{n}$ is not crucial to us in proving
the remaining main results. A computation of the $c$-function with an exact
value of $\rho_{n}$ can be found in [19, Section 4, Thm 4.2. p.334], but we
note that the value depends on the normalization of the Haar measure
$m_{G_{n}}$. Note that $c_{n}$ is meromorphic and does not have any zeros in
$\mathbb{C}$. In particular, $|c_{n}|$ is a strictly positive function.

######  Lemma 6.15 (Spherical Plancherel formula).

For every $\varphi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n},K_{n})$,

$\displaystyle\int_{G_{n}}|\varphi(g)|^{2}dm_{G_{n}}(g)=\int_{0}^{\infty}|\widehat{\varphi}(\lambda)|^{2}\frac{d\lambda}{|c_{n}(\lambda)|^{2}}$

Equivalently, the spherical transform extends to a unitary map

$L^{2}(G_{n},K_{n})\rightarrow L^{2}(\mathbb{R}_{\geq
0},|c_{n}(\lambda)|^{-2}d\lambda)\,.$

Before moving on, we will compute the spherical transform of indicator
functions on balls.

######  Example 6.1 (Spherical transform of the indicator function on a
centered Cartan ball).

The computation of the spherical transform for the indicator function
$\chi_{B_{r}}$ is similar to the Euclidean case. Using the formula

$\omega^{(n)}_{\lambda}(t)=a_{n}\sinh(t)^{2-n}\int_{0}^{t}(\cosh(t)-\cosh(s))^{\frac{n-3}{2}}\cos(\lambda
s)ds\,,\quad
a_{n}=\frac{2^{\frac{n-1}{2}}}{\mathrm{B}(\frac{1}{2},\frac{n-1}{2})}$

for the spherical functions, Fubini’s Theorem yields

$\displaystyle\widehat{\chi}_{B_{r}}(\lambda)$
$\displaystyle=\int_{0}^{r}\omega^{(n)}_{\lambda}(t)\sinh(t)^{n-1}dt$
$\displaystyle=a_{n}\int_{0}^{r}\Big{(}\int_{0}^{t}(\cosh(t)-\cosh(s))^{\frac{n-3}{2}}\cos(\lambda
s)ds\Big{)}tdt$
$\displaystyle=a_{n}\int_{0}^{r}\Big{(}\int_{s}^{r}(\cosh(t)-\cosh(s))^{\frac{n-3}{2}}\sinh(t)dt\Big{)}\cos(\lambda
s)ds$
$\displaystyle=\frac{2a_{n}}{n-1}\int_{0}^{r}(\cosh(r)-\cosh(s))^{\frac{n-1}{2}}\cos(\lambda
s)ds\,.$

In terms of spherical functions, we have

$\displaystyle\widehat{\chi}_{B_{r}}(\lambda)=b_{n}\sinh(r)^{n}\omega_{\lambda}^{(n+2)}(r)\,,\quad
b_{n}=\frac{2^{\frac{n+1}{2}}}{n}\,.$ (6.6)

##  7\. Hyperbolic diffraction measures

In this section we will introduce the diffraction measure associated to a
random measure on $\mathbb{H}^{n}$. Furthermore, we provide an asymptotic
upper bound on the diffraction measures of large intervals, similar to the
Euclidean case. The support such diffraction measures will be confined to the
subset

$\displaystyle\Omega_{n}^{+}=(0,+\infty)\cup i[0,(n-1)/2]\subset\mathbb{C}\,.$

The following Lemma ensures existence and uniqueness of the diffraction
measure of a random measure on $\mathbb{H}^{n}$, which we will prove in larger
generality in upcoming work.

######  Lemma 7.1.

Let $\mu$ be a locally square-integrable invariant random measure on
$\mathbb{H}^{n}$ with autocorrelation measure $\eta_{\mu}$ on $G_{n}$. Then
there is a unique positive Radon measure $\widehat{\eta}_{\mu}$ on
$\Omega_{n}^{+}$ such that

$\displaystyle\int_{G_{n}}\varphi(g)d\eta_{\mu}(g)=\int_{\Omega_{n}^{+}}\widehat{\varphi}(\lambda)d\widehat{\eta}_{\mu}(\lambda)\,,\quad\forall\,\varphi\in
C_{c}^{\infty}(G_{n},K_{n})\,.$

Moreover, $\widehat{\eta}_{\mu}(\\{i\frac{n-1}{2}\\})=0$ and the measures
$\eta_{\mu},\widehat{\eta}_{\mu}$ are tempered in the sense that there are
constants $a,b\in\mathbb{Z}_{\geq 0}$ such that

$\displaystyle\int_{G_{n}}(1+\ell(g))^{-a}\omega_{0}^{(n)}(g)^{-\frac{n+1}{2}}d\eta_{\mu}(g)<+\infty\quad\mbox{
and
}\quad\int_{\Omega_{n}^{+}}(1+|\lambda|)^{-b}d\widehat{\eta}_{\mu}(\lambda)<+\infty\,.$

Since the spherical function $\omega_{0}^{(n)}$ decays exponentially, the
following family of functions are integrable with respect to the
autocorrelation measure.

######  Corollary 7.2.

If $\varphi:G_{n}\rightarrow\mathbb{C}$ is a measurable function such that

$|\varphi(g)|\mathrm{e}^{-\alpha d(g.o,o)}\longrightarrow 0$

as $g\rightarrow+\infty$ for every $\alpha>0$, then $\varphi$ is integrable
with respect to $\eta_{\mu}$.

In fact, $\eta_{\mu}$ extends to a continuous linear functional on a certain
_Harish-Chandra $L^{p}$-space_ of smooth functions that are rapidly decaying
with respect to a power of $\omega_{0}$, but for our purposes we will not need
this here.

We define the _principal diffraction measure_ $\widehat{\eta}_{\mu}^{(p)}$ and
the _complementary diffraction measure $\widehat{\eta}_{\mu}^{(c)}$_ to be the
restrictions of $\widehat{\eta}_{\mu}$ to $(0,+\infty)$ and
$i[0,\frac{n-1}{2})$ respectively, so that

$\displaystyle\int_{\Omega_{n}^{+}}\widehat{\varphi}(\lambda)d\widehat{\eta}_{\mu}(\lambda)=\int_{0}^{\infty}\widehat{\varphi}(\lambda)d\widehat{\eta}_{\mu}^{(p)}(\lambda)+\int_{0}^{\frac{n-1}{2}}\widehat{\varphi}(i\lambda)d\widehat{\eta}_{\mu}^{(c)}(\lambda)\,,\quad\varphi\in
C_{c}^{\infty}(G_{n},K_{n})\,.$

Both measures $\widehat{\eta}_{\mu}^{(p)},\widehat{\eta}_{\mu}^{(c)}$ are
positive Radon measures, and $\widehat{\eta}_{\mu}^{(c)}$ is necessarily a
finite measure. As in the Euclidean case, we can extend the defining identity

$\displaystyle\mathrm{Var}_{\mu}(Sf)=\eta_{\mu}(\varphi_{f}^{*}*\varphi_{f})=\int_{0}^{\infty}|\widehat{\varphi}_{f}(\lambda)|^{2}d\widehat{\eta}_{\mu}^{(p)}(\lambda)+\int_{0}^{\frac{n-1}{2}}|\widehat{\varphi}_{f}(i\lambda)|^{2}d\widehat{\eta}_{\mu}^{(c)}(\lambda)$

to all radial functions
$f\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(\mathbb{H}^{n})$, in particular the
indicator functions $f=\chi_{B_{r}}$, $r>0$.

######  Lemma 7.3.

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

$\displaystyle\eta_{\mu}(\varphi^{*}*\varphi)=\int_{\Omega_{n}^{+}}|\widehat{\varphi}(\lambda)|^{2}d\widehat{\eta}_{\mu}(\lambda)\,.$

In particular, $\widehat{\varphi}\in
L^{2}(\Omega_{n}^{+},\widehat{\eta}_{\mu})$.

###### Proof.

Same proof as that of Lemma 4.3 by letting $\beta_{\varepsilon}$ be the
bi-$K_{n}$-invariant rapidly decaying smooth function on $G_{n}$ satisfying
$\widehat{\beta}_{\varepsilon}(\lambda)=\mathrm{e}^{-\varepsilon^{2}(\lambda^{2}+(\frac{n-1}{2})^{2})}$.
∎

######  Example 7.1 (Diffraction of the $m_{\mathbb{H}^{n}}$-Poisson point
process).

Recall that the autocorrelation measure $\eta_{\mathrm{Poi}}$ of the
$m_{\mathbb{H}^{n}}$-Poisson point process is evaluation at the identity $e\in
G_{n}$ when considered as a linear functional on
$\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n},K_{n})$. Therefore, by the
Plancherel formula for the spherical transform,

$\displaystyle\eta_{\mathrm{Poi}}(\varphi^{*}*\varphi)=(\varphi^{*}*\varphi)(e)=\int_{G_{n}}|\varphi(g)|^{2}dm_{G_{n}}(g)=\int_{0}^{\infty}|\widehat{\varphi}(\lambda)|^{2}\frac{d\lambda}{|c_{n}(\lambda)|^{2}}\,.$

Thus the Poisson diffraction measure is

$\displaystyle
d\widehat{\eta}_{\mathrm{Poi}}^{(p)}(\lambda)=|c_{n}(\lambda)|^{-2}d\lambda\quad\mbox{
and }\quad d\widehat{\eta}^{(c)}_{\mathrm{Poi}}(\lambda)=0\,.$

We will next prove the following analogue of Lemma 4.4.

######  Proposition 7.4.

Let $\mu$ be a locally square-integrable invariant random measure on
$\mathbb{H}^{n}$. Then the principal part $\widehat{\eta}_{\mu}^{(p)}$ of the
diffraction measure $\widehat{\eta}_{\mu}$ of $\mu$ satisfies

$\displaystyle\widehat{\eta}_{\mu}^{(p)}([0,L])\ll_{n,\mu}L^{n}$

for all sufficiently large $L>0$.

######  Remark 7.5.

For the $m_{\mathbb{H}^{n}}$-Poisson point process on $\mathbb{H}^{n}$ we have
that $d\widehat{\eta}^{(c)}_{\mathrm{Poi}}(\lambda)=0$ and
$d\widehat{\eta}^{(p)}_{\mathrm{Poi}}(\lambda)=|c_{n}(\lambda)|^{-2}d\lambda$,
and the $c$-function has the asymptotic bounds

$\displaystyle
c_{n}(\lambda)=\rho_{n}2^{\frac{n-2}{2}}\frac{\Gamma(\frac{n}{2})\Gamma(i\lambda)}{\Gamma(\frac{n-1}{2})\Gamma(i\lambda+\frac{n-1}{2})}\asymp\begin{cases}\lambda^{-\frac{n-1}{2}}&\mbox{
if }\lambda\gg 0\\\ \lambda^{-1}&\mbox{ if }\lambda\approx 0\,.\end{cases}$
(7.1)

For sufficiently large $L>0$ we then have

$\displaystyle\widehat{\eta}_{\mu}^{(p)}([0,L])=\int_{0}^{L}\frac{d\lambda}{|c_{n}(\lambda)|^{2}}\asymp\int_{0}^{L}\lambda^{n-1}d\lambda\asymp
L^{n}\,.$

The upper bound in Proposition 7.4 is thus, up to constant, a strict upper
bound in general and which is achieved by the invariant Poisson point process.

In order to prove Proposition 7.4 we make use of the _heat kernel_ $h_{\tau}$
on $G_{n}$ with $\tau>0$. It is the bi-$K_{n}$-invariant function

$\displaystyle
h_{\tau}(g)=\int_{0}^{\infty}\mathrm{e}^{-\tau((\frac{n-1}{2})^{2}+\lambda^{2})}\omega_{\lambda}(g)\frac{d\lambda}{|c_{n}(\lambda)|^{2}}\,.$

Equivalently,
$\widehat{h}_{\tau}(\lambda)=\mathrm{e}^{-\tau((\frac{n-1}{2})^{2}+\lambda^{2})}$.
For the heat kernel in even dimensions $n$, there is no closed formula in
terms of elementary functions, but it will be sufficient for us to consider
the following uniform bounds in [10, Theorem 3.1, p.186].

######  Lemma 7.6 (Davies-Mandouvalos).

The heat kernel satisfies

$\displaystyle
h_{\tau}(a_{r})\asymp\tau^{-\frac{n}{2}}(1+\tau+r)^{\frac{n-3}{2}}(1+r)\mathrm{e}^{-(\frac{n-1}{2})^{2}\tau-\frac{n-1}{2}r-\frac{1}{4\tau}r^{2}}$

uniformly in $r\in[0,+\infty)$ and $\tau\in(0,+\infty)$.

In particular, the heat kernel $h_{\tau}$ is tempered and hence integrable
with respect to the autocorrelation measure $\eta_{\mu}$ of a locally square-
integrable random measure $\mu$ on $\mathbb{H}^{n}$ by Lemma 7.1.

###### Proof of Proposition 7.4.

Denote by $\chi_{[0,L]}$ the indicator function of the interval $[0,L]$ in
$\mathbb{R}_{\geq 0}$ and bound it from above by
$\chi_{[0,L]}(\lambda)\leq\mathrm{e}^{1-\lambda^{2}/L^{2}}$, so that

$\displaystyle\widehat{\eta}_{\mu}^{(p)}([0,L])\leq\int_{0}^{\infty}\mathrm{e}^{1-\frac{\lambda^{2}}{L^{2}}}d\widehat{\eta}_{\mu}^{(p)}(\lambda)$
$\displaystyle\leq\int_{\Omega_{n}^{+}}\mathrm{e}^{1-\frac{\lambda^{2}}{L^{2}}}d\widehat{\eta}_{\mu}(\lambda)$
$\displaystyle=\mathrm{e}^{1+(\frac{n-1}{2L})^{2}}\int_{\Omega_{n}^{+}}\mathrm{e}^{-\frac{1}{L^{2}}((\frac{n-1}{2})^{2}+\lambda^{2})}d\widehat{\eta}_{\mu}(\lambda)\,.$

Note that the latter integral is finite since $\widehat{\eta}_{\mu}^{(p)}$ is
tempered by Lemma 7.1. Since

$\widehat{h}_{\tau}(\lambda)=\mathrm{e}^{-\frac{1}{L^{2}}((\frac{n-1}{2})^{2}+\lambda^{2})}$

then from the definition of the diffraction measure $\widehat{\eta}_{\mu}$ we
see that

$\displaystyle\int_{\Omega_{n}^{+}}\mathrm{e}^{-\frac{1}{L^{2}}((\frac{n-1}{2})^{2}+\lambda^{2})}d\widehat{\eta}_{\mu}(\lambda)=\int_{G_{n}}h_{L^{-2}}(g)d\eta_{\mu}(g)\,,$

so that
$\widehat{\eta}_{\mu}^{(p)}([0,L])\leq\mathrm{e}^{1+(\frac{n-1}{2L})^{2}}\eta_{\mu}(h_{L^{-2}})$.
Consider the Radon measure $\eta_{\mu}^{+}=\eta_{\mu}+\iota_{\mu}m_{G_{n}}$,
which is a positive tempered measure satisfying
$\eta_{\mu}(h_{\tau})\leq\eta_{\mu}^{+}(h_{\tau})$ for all $\tau>0$. By Lemma
7.6, we find in particular that there is a constant $C>0$ such that

$\displaystyle 0<h_{L^{-2}}(a_{r})\leq
CL^{n}(1+L^{-2}+r)^{\frac{n-3}{2}}(1+r)\mathrm{e}^{-\frac{(Lr)^{2}}{4}}\leq
CL^{n}(2+r)^{\frac{n-1}{2}}\mathrm{e}^{-\frac{r^{2}}{4}}$

for all $L>1$ and all $r\in[0,+\infty)$. We moreover bound
$\mathrm{e}^{1+(\frac{n-1}{2L})^{2}}\leq\mathrm{e}^{1+(\frac{n-1}{2})^{2}}$
for $L>1$, so that the final bound becomes

$\displaystyle\widehat{\eta}_{\mu}^{(p)}([0,L])\leq\mathrm{e}^{1+(\frac{n-1}{2})^{2}}CL^{n}\int_{G_{n}}(2+\ell(g))^{\frac{n-1}{2}}\mathrm{e}^{-\frac{\ell(g)^{2}}{4}}d\eta_{\mu}^{+}(g)\,.$

Since $\eta_{\mu}^{+}$ is tempered, the latter integral is finite by Corollary
7.2 and thus $\widehat{\eta}_{\mu}^{(p)}([0,L])\ll_{n,\mu}L^{n}$. ∎

######  Remark 7.7.

It should be noted that, there are uniform estimates of the heat kernel by
Anker and Ostellari in the Main Theorem of [3], and using those the above
proof can be approached in the same way for general non-compact symmetric
spaces $X=G/K$ with $G$ a non-compact connected real semisimple Lie group with
finite center and maximal compact subgroup $K<G$.

##  8\. A hyperbolic analouge of Beck’s theorem

We prove Theorem 1.4 using the following hyperbolic analogue of Theorem 5.1.

######  Theorem 8.1.

Let $\mu$ be an invariant locally square-integrable random measure on
$\mathbb{H}^{n}$ with diffraction measure $\widehat{\eta}_{\mu}$. The
following holds:

  1. (1)

if $\widehat{\eta}_{\mu}^{(c)}([0,\frac{n-1}{2}))>0$, then for every $R_{o}>1$
there is a constant $C=C(n,R_{o})>0$ such that

$\displaystyle\frac{\mathrm{NV}_{\mu}(R)}{\cosh(R)^{n-1}}\geq
C\int_{0}^{\frac{n-1}{2}}\frac{\sinh^{2}(R\lambda)}{\lambda^{2}}d\widehat{\eta}_{\mu}^{(c)}(\lambda)\,,\quad\forall\,R\geq
R_{o}\,.$

  2. (2)

if $\widehat{\eta}_{\mu}^{(p)}((0,+\infty))>0$ then for every $\lambda_{o}>0$
there is $R_{o}=R_{o}(\lambda_{o})>0$ and a constant $C(n,\lambda_{o})>0$ such
that

$\displaystyle\frac{1}{R}\int_{0}^{R}\frac{\mathrm{NV}_{\mu}(r)}{\cosh(r)^{n-1}}dr\geq
C\int_{\lambda_{o}}^{\infty}|c_{n+2}(\lambda)|^{2}d\widehat{\eta}_{\mu}^{(p)}(\lambda)\,,\quad\forall\,R\geq
R_{o}\,.$

###### Proof.

By definition of the diffraction measure $\widehat{\eta}_{\mu}$ and Equation
(6.6) in Example 6.1,

$\displaystyle\frac{\mathrm{NV}_{\mu}(r)}{\cosh(r)^{n-1}}$
$\displaystyle=\int_{0}^{\infty}\frac{|\widehat{\chi}_{B_{r}}(\lambda)|^{2}}{\cosh(r)^{n-1}}d\widehat{\eta}_{\mu}^{(p)}(\lambda)+\int_{0}^{\frac{n-1}{2}}\frac{|\widehat{\chi}_{B_{r}}(i\lambda)|^{2}}{\cosh(r)^{n-1}}d\widehat{\eta}_{\mu}^{(c)}(\lambda)$
$\displaystyle=\frac{2^{n+1}}{n^{2}}\tanh(r)^{n-1}\sinh(r)^{n+1}\Big{(}\int_{0}^{\infty}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}d\widehat{\eta}_{\mu}^{(p)}(\lambda)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad+\int_{0}^{\frac{n-1}{2}}|\omega_{i\lambda}^{(n+2)}(a_{r})|^{2}d\widehat{\eta}_{\mu}^{(c)}(\lambda)\Big{)}\,.$

To prove (1), we use item (3) in Proposition 6.11 to find a constant
$C_{o}=C_{o}(n)>0$ for $R_{o}>1$ such that

$\displaystyle\omega_{i\lambda}^{(n+2)}(a_{R})\geq C_{o}\frac{\sinh(\lambda
R)}{\lambda\sinh(R)^{\frac{n+1}{2}}}\,\quad\forall\,R\geq R_{o}\,.$

Then

$\displaystyle\frac{\mathrm{NV}_{\mu}(R)}{\cosh(R)^{n-1}}$
$\displaystyle\geq\frac{2^{n+1}}{n^{2}}\tanh(R)^{n-1}\sinh(R)^{n+1}\int_{0}^{\frac{n-1}{2}}|\omega_{i\lambda}^{(n+2)}(a_{R})|^{2}d\widehat{\eta}_{\mu}^{(c)}(\lambda)$
$\displaystyle\geq\frac{2^{n+1}}{n^{2}}C_{o}\tanh(R_{o})^{n-1}\int_{0}^{\frac{n-1}{2}}\frac{\sinh^{2}(R\lambda)}{\lambda^{2}}d\widehat{\eta}_{\mu}^{(c)}(\lambda)\,.$

If we let $C(n,R_{o})=2^{n+1}n^{-2}C_{o}(n)\tanh(R_{o})^{n-1}$ then we are
done.

For (2), we assume without loss of generality that
$\widehat{\eta}_{\mu}^{(c)}=0$. An application of Fubini yields

$\displaystyle\frac{1}{R}\int_{0}^{R}\frac{\mathrm{NV}_{\mu}(r)}{\cosh(r)^{n-1}}dr$
$\displaystyle\geq\frac{2^{n+1}}{n^{2}}\int_{\lambda_{o}}^{\infty}\Big{(}\frac{1}{R}\int_{0}^{R}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}\tanh(r)^{n-1}\sinh(r)^{n+1}dr\Big{)}d\widehat{\eta}_{\mu}^{(p)}(\lambda)\,.$

We claim that

$\displaystyle\lim_{R\rightarrow+\infty}\frac{1}{R}\int_{0}^{R}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}\tanh(r)^{n-1}\sinh(r)^{n+1}dr=\rho_{n+2}^{-2}|c_{n+2}(\lambda)|^{2}$

uniformly for all $\lambda\geq\lambda_{o}$, which will finish the proof by
dominated convergence. To compute this limit, write

$\displaystyle\int_{0}^{R}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}\tanh(r)^{n-1}$
$\displaystyle\sinh(r)^{n+1}dr=\int_{0}^{R}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}\sinh(r)^{n+1}dr$
$\displaystyle\quad+\int_{0}^{R}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}(\tanh(r)^{n-1}-1)\sinh(r)^{n+1}dr\,.$

By item (1) in Proposition 6.11,
$\omega_{\lambda}^{(n+2)}(a_{r})\leq\lambda^{-1}\sinh(r)^{-\frac{n+1}{2}}$,
and since $1-\tanh(r)^{n-1}\leq n\mathrm{e}^{-r}$ we get

$\displaystyle\frac{1}{R}\int_{0}^{R}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}|\tanh(r)^{n-1}-1|\sinh(r)^{n+1}dr\leq\frac{n}{R\lambda^{2}}\int_{0}^{R}\mathrm{e}^{-r}dr\leq\frac{n}{R\lambda^{2}}\underset{R\rightarrow+\infty}{\longrightarrow}0\,.$

By item (2) in Proposition 6.11 we finally have that

$\displaystyle\frac{1}{R}\int_{0}^{R}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}\tanh(r)^{n-1}\sinh(r)^{n+1}dr=$
$\displaystyle\,\frac{1}{R}\int_{0}^{R}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}\sinh(r)^{n+1}dr$
$\displaystyle\underset{R\rightarrow+\infty}{\longrightarrow}$
$\displaystyle\,2^{n}\frac{|\mathrm{B}(i\lambda,\frac{n+1}{2})|^{2}}{\mathrm{B}(\frac{1}{2},\frac{n+1}{2})^{2}}=\rho_{n+2}^{-2}|c_{n+2}(\lambda)|^{2}\,.$

∎

######  Remark 8.2.

The lower bounds in Theorem 8.1 are finite for fixed $R>0$. For the
complementary diffraction bound the statement is trivial, and for the
principal diffraction bound we write

$\displaystyle\int_{\lambda_{o}}^{\infty}|c_{n+2}(\lambda)|^{2}d\widehat{\eta}_{\mu}^{(p)}(\lambda)\ll_{n}\int_{\lambda_{o}}^{\infty}\widehat{\eta}_{\mu}^{(p)}([0,u])\frac{d}{du}|c_{n+2}(u)|^{2}du\,.$

Using the standard asymptotic

$\displaystyle\frac{\Gamma(iu)}{\Gamma(iu+\frac{n-1}{2})}\sim(iu)^{-\frac{n-1}{2}}$

as $u\rightarrow+\infty$ one can see that
$\frac{d}{du}|c_{n+2}(u)|^{2}\ll_{n,\lambda_{o}}u^{-(n+2)}$ for large
$u\geq\lambda_{o}$, and since $\widehat{\eta}_{\mu}^{(p)}$ satisfies
$\widehat{\eta}_{\mu}^{(p)}([0,L])\ll_{n,\mu}L^{n}$ for sufficiently large
$L>0$ by Lemma 7.4, then

$\displaystyle\int_{\lambda_{o}}^{\infty}|c_{n+2}(\lambda)|^{2}d\widehat{\eta}_{\mu}^{(p)}(\lambda)\ll_{n,\lambda_{o}}\int_{\lambda_{o}}^{\infty}\frac{\widehat{\eta}_{\mu}^{(p)}([0,u])}{u^{n+2}}du\ll_{n,\mu,\lambda_{o}}\int_{\lambda_{o}}^{\infty}\frac{du}{u^{2}}=\lambda_{o}^{-1}<+\infty\,.$

Thus the right hand side in item (2) of Theorem 8.1 is indeed finite.

###### Proof of Theorem 1.4.

The first statement of the Theorem follows from item (2) of Theorem 8.1,

$\displaystyle\limsup_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{\cosh(R)^{n-1}}\geq\limsup_{R\rightarrow+\infty}\frac{1}{R}\int_{0}^{R}\frac{\mathrm{NV}_{\mu}(r)}{\cosh(r)^{n-1}}dr\geq
C\int_{0}^{\infty}|c_{n+2}(\lambda)|^{2}d\widehat{\eta}_{\mu}^{(p)}(\lambda)\,.$

Since $|c_{n+2}(\lambda)|^{2}>0$ for all $\lambda>0$ then the right hand side
is positive, possibly infinite.

Item (1) of Theorem 1.4 follows from item (1) of Theorem 8.1, since

$\displaystyle\liminf_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{R^{2}\cosh(R)^{n-1}}\geq\liminf_{R\rightarrow+\infty}\frac{C}{R^{2}}\int_{0}^{\frac{n-1}{2}}\frac{\sinh^{2}(R\lambda)}{\lambda^{2}}d\widehat{\eta}^{(c)}_{\mu}(\lambda)\geq
C\widehat{\eta}^{(c)}_{\mu}(\\{0\\})>0\,.$

Lastly, for any $\delta\in(0,1]$ with
$\widehat{\eta}_{\mu}^{(c)}([\delta\frac{n-1}{2},\frac{n-1}{2}))>0$ we have
that

$\displaystyle\liminf_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{\cosh(R)^{(n-1)(1+\delta)}}$
$\displaystyle\geq\liminf_{R\rightarrow+\infty}\frac{C}{\cosh(R)^{(n-1)\delta}}\int_{\delta\frac{n-1}{2}}^{\frac{n-1}{2}}\frac{\sinh^{2}(R\lambda)}{\lambda^{2}}d\widehat{\eta}_{\mu}^{(c)}(\lambda)$
$\displaystyle\geq
C\widehat{\eta}_{\mu}^{(c)}\Big{(}\Big{[}\delta\frac{n-1}{2},\frac{n-1}{2}\Big{)}\Big{)}\liminf_{R\rightarrow\infty}\frac{\sinh^{2}(R\delta\frac{n-1}{2})}{\delta^{2}\cosh(R)^{(n-1)\delta}}$
$\displaystyle=C2^{(n-1)\delta-2}\delta^{-2}\widehat{\eta}_{\mu}^{(c)}\Big{(}\Big{[}\delta\frac{n-1}{2},\frac{n-1}{2}\Big{)}\Big{)}>0\,.$

Thus item (2) of Theorem 1.4 follows. ∎

##  9\. Spectral hyperuniformity and stealth

We define spectral hyperuniformity of an invariant locally square-integrable
random measure on $\mathbb{H}^{n}$ in Subsection 9.1 and define the relative
notion of stealth in Subsection 9.2.

###  9.1. Spectral hyperuniformity

By Theorem 1.4, we know that random measures $\mu$ on $\mathbb{H}^{n}$ are
never geometrically hyperuniform in the sense that

$\displaystyle\limsup_{R\rightarrow+\infty}\frac{\mathrm{NV}_{\mu}(R)}{\cosh(R)^{n-1}}>0\,,$
(9.1)

and if $\mu$ admits non-trivial complementary spectrum, then this upper limit
is infinite. A useful definition of spectral hyperuniformity would then
necessarily have to differ from this geometric definition. Having understood
the large scale behaviour of diffraction measures, one is inclined to study
the small scale/local behaviour of the diffraction measure in order to obtain
information on the large scale behaviour of the number variance. We take the
stance that one should interpret spectral hyperuniformity as comparing the
diffraction measure of a given random measure to the diffraction measure of
the invariant Poisson point process around $\lambda=0$. For the definition we
shall use that

$\displaystyle\widehat{\eta}_{\mathrm{Poi}}((0,\varepsilon])=\int_{0}^{\varepsilon}\frac{d\lambda}{|c_{n}(\lambda)|^{2}}\asymp\int_{0}^{\varepsilon}\lambda^{2}d\lambda\asymp\varepsilon^{3}$

for sufficiently small $\varepsilon>0$. Note that the exponent $3$ is the same
in all dimensions $n$.

######  Definition 9.1 (Hyperbolic spectral hyperuniformity).

A random measure $\mu$ on $\mathbb{H}^{n}$ is _spectrally hyperuniform_ if
$\widehat{\eta}^{(c)}_{\mu}=0$ and

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

######  Remark 9.2.

There are many examples of point processes $\mu$ on $\mathbb{H}^{n}$ such that
$\widehat{\eta}_{\mu}^{(c)}\neq 0$, even for many lattices, including
cocompact ones. These will be hyperfluctuating in the sense of Theorem 1.4 and
not spectrally hyperuniform according to our definition.

Having defined spectral hyperuniformity, we have a point of reference for
defining more rigid properties of random measures on $\mathbb{H}^{n}$.

###  9.2. Stealthy random measures

######  Definition 9.3 (Stealthy random measure on $\mathbb{H}^{n}$).

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

It is clear from this definition that every stealthy random measure on
$\mathbb{H}^{n}$ is spectrally hyperuniform. As mentioned in the introduction,
in [13, Section 1.2., Eq.B] Jenni provides a lattice
$\Gamma<G_{2}\cong\mathrm{SL}_{2}(\mathbb{R})$ such that the random lattice
orbit $\mu_{\Gamma}$ on $\mathbb{H}^{2}$ is a stealthy point process, in
particular spectrally hyperuniform.

An almost immediate consequence of Theorem 8.1 is that we can compute the
average asymptotic number variance of a given stealthy random measure in terms
of the diffraction measure.

######  Corollary 9.4.

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

$\displaystyle\lim_{R\rightarrow+\infty}\frac{1}{R}\int_{0}^{R}\frac{\mathrm{NV}_{\mu}(r)}{\cosh(r)^{n-1}}dr=\frac{2^{n+1}}{n^{2}\rho_{n+2}^{2}}\int_{0}^{\infty}|c_{n+2}(\lambda)|^{2}d\widehat{\eta}_{\mu}^{(p)}(\lambda)\,,$

which is strictly positive.

###### Proof.

If $\mu$ is stealthy there is a $\lambda_{o}>0$ such that the diffraction
measure $\widehat{\eta}_{\mu}$ is supported on
$[\lambda_{o},+\infty)\subset\Omega_{n}^{+}$. Thus

$\displaystyle\frac{1}{R}\int_{0}^{R}\frac{\mathrm{NV}_{\mu}(r)}{\cosh(r)^{n-1}}dr$
$\displaystyle=\frac{2^{n+1}}{n^{2}}\int_{\lambda_{o}}^{\infty}\Big{(}\frac{1}{R}\int_{0}^{R}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}\tanh(r)^{n-1}\sinh(r)^{n+1}dr\Big{)}d\widehat{\eta}_{\mu}^{(p)}(\lambda)\,.$

As shown in the proof of item (2) of Theorem 8.1, we have

$\displaystyle\lim_{R\rightarrow+\infty}\frac{1}{R}\int_{0}^{R}|\omega_{\lambda}^{(n+2)}(a_{r})|^{2}\tanh(r)^{n-1}\sinh(r)^{n+1}dr=\rho_{n+2}^{-2}|c_{n+2}(\lambda)|^{2}$

uniformly in $\lambda\geq\lambda_{o}$, so by dominated convergence we’re done.
∎

##  10\. Random perturbed lattice orbits in hyperbolic spaces

In Subsection 10.1 we provide examples of non-compact perturbations of random
lattice orbits that are locally square-integrable. In Subsection 10.2 we
compute the diffraction measure of random perturbed lattice orbits that are
locally square-integrable and lastly we show that random perturbed lattice
orbits in $\mathbb{H}^{n}$ are not spectrally hyperuniform in Subsection 10.3.

###  10.1. A criterion for local square-integrability

We first extend Lemma 2.6 to include right-$K_{n}$-invariant perturbations
$\nu\in\mathrm{Prob}(G_{n})$ that are absolutely continuous with respect to
the Haar measure $m_{G_{n}}$ with sufficent exponential decay of the Radon-
Nikodym derivative at infinity. To do this we make use of the fact that the
critical exponent of a lattice $\Gamma<G_{n}$ is that of the volume growth.
The following Lemma is a special case of Theorem C in [2], which includes
higher rank analogues as well.

######  Lemma 10.1.

Let $\Gamma<G_{n}$ be a lattice. Then for every $\varepsilon>0$,

$\displaystyle\sum_{\gamma\in\Gamma}\mathrm{e}^{-(n-1+\varepsilon)d(\gamma.o,o)}<+\infty\,.$

######  Lemma 10.2.

Let $\beta:G_{n}\rightarrow[0,+\infty)$ be a right-$K_{n}$-invariant
measurable function with $\int\beta\,dm_{G_{n}}=1$ and assume that there are
$C>0$ and $\varepsilon>0$ such that

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

for all $g\in G_{n}$. Then for $d\nu(g)=\beta(g)dm_{G_{n}}(g)$ and any lattice
$\Gamma<G_{n}$, the random perturbed lattice orbit $\mu_{\Gamma,\nu}$ in
$\mathbb{H}^{n}$ is locally square-integrable.

By the same argument in the proof in Lemma 2.6, it suffices to bound the
expression in Equation 2.2, that is,

$\displaystyle
I_{\Gamma,\nu}:=\int_{G_{n}/\Gamma}\Big{(}\sum_{\gamma\in\Gamma}\nu(\gamma^{-1}g^{-1}Q)\Big{)}^{2}dm_{G_{n}/\Gamma}(g\Gamma)\,.$

######  Lemma 10.3.

Let $Q\subset G_{n}$ be a pre-compact measurable set and let
$\nu\in\mathrm{Prob}(G_{n})$. Then

$\displaystyle
I_{\Gamma,\nu}=\frac{1}{\mathrm{covol}(\Gamma)}\sum_{\gamma\in\Gamma}\int_{G_{n}}\int_{G_{n}}m_{G_{n}}(Qh_{1}^{-1}\gamma
h_{2}\cap Q)d\nu(h_{1})d\nu(h_{2})\,.$

###### Proof.

By Fubini,

$\displaystyle\int_{G_{n}/\Gamma}\Big{(}$
$\displaystyle\sum_{\gamma\in\Gamma}\nu(\gamma^{-1}g^{-1}Q)\Big{)}^{2}dm_{G_{n}/\Gamma}(g\Gamma)=$
$\displaystyle=\int_{G_{n}}\int_{G_{n}}\Big{(}\int_{G_{n}/\Gamma}|g\Gamma
h_{1}\cap Q||g\Gamma h_{2}\cap
Q|dm_{G_{n}/\Gamma}(g\Gamma)\Big{)}d\nu(h_{1})d\nu(h_{2})\,.$

From the computation for the second moment of a random lattice in example 2.5,
we have that

$\displaystyle\int_{G_{n}/\Gamma}|g\Gamma h_{1}\cap Q||g\Gamma h_{2}\cap
Q|dm_{G_{n}/\Gamma}(g\Gamma)$
$\displaystyle=\frac{1}{\mathrm{covol}(\Gamma)}\sum_{\gamma\in\Gamma}(\chi^{*}_{Qh_{1}^{-1}}*\chi_{Qh_{2}^{-1}})(\gamma)$
$\displaystyle=\frac{1}{\mathrm{covol}(\Gamma)}\sum_{\gamma\in\Gamma}m_{G_{n}}(Qh_{1}^{-1}\gamma
h_{2}\cap Q)\,,$

and Fubini then gives us that

$\displaystyle\int_{G_{n}/\Gamma}\Big{(}\sum_{\gamma\in\Gamma}$
$\displaystyle\nu(\gamma^{-1}g^{-1}Q)\Big{)}^{2}dm_{G_{n}/\Gamma}(g\Gamma)=$
$\displaystyle\frac{1}{\mathrm{covol}(\Gamma)}\sum_{\gamma\in\Gamma}\int_{G_{n}}\int_{G_{n}}m_{G_{n}}(Qh_{1}^{-1}\gamma
h_{2}\cap Q)d\nu(h_{1})d\nu(h_{2})\,.$

∎

###### Proof of Lemma 10.2.

For convenience of notation we set $\alpha_{\varepsilon}=n-1+\varepsilon$. Let
$Q=\pi^{-1}(B)=\varsigma(B)K$, so that $Q\subset G_{n}$ is pre-compact,
measurable and $Q.o=B$. By Lemma 10.3, it suffices to show that

$\displaystyle\sum_{\gamma\in\Gamma}\int_{G_{n}}\int_{G_{n}}m_{G_{n}}(Qh_{1}^{-1}\gamma
h_{2}\cap Q)\beta(h_{1})\beta(h_{2})dm_{G_{n}}(h_{1})dm_{G_{n}}(h_{2})$

is finite. We first rewrite this expression as

$\displaystyle\int_{G_{n}}\int_{G_{n}}$ $\displaystyle
m_{G_{n}}(Qh_{1}^{-1}\gamma h_{2}\cap
Q)\beta(h_{1})\beta(h_{2})dm_{G_{n}}(h_{1})dm_{G_{n}}(h_{2})=$
$\displaystyle=\int_{G_{n}}\Big{(}\int_{G_{n}}m_{G_{n}}(Qh_{2}\cap
Q)\beta(\gamma^{-1}h_{1}h_{2})dm_{G_{n}}(h_{2})\Big{)}\beta(h_{1})dm_{G_{n}}(h_{1})\,.$

If $m_{G_{n}}(Qh_{2}\cap Q)>0$ then $h_{2}\in Q^{-1}Q$, and using the assumed
upper bound on $\beta$ we get that

$\displaystyle\int_{G_{n}}m_{G_{n}}(Qh_{2}\cap Q)$
$\displaystyle\beta(\gamma^{-1}h_{1}h_{2})dm_{G_{n}}(h_{2})\leq$
$\displaystyle\leq C\int_{G_{n}}m_{G_{n}}(Qh_{2}\cap
Q)\mathrm{e}^{-2\alpha_{\varepsilon}d(h_{2}.o,h_{1}^{-1}\gamma.o)}dm_{G_{n}}(h_{2})$
$\displaystyle\leq
Cm_{G_{n}}(Q)\mathrm{e}^{-2\alpha_{\varepsilon}d(Q^{-1}Q.o,h_{1}^{-1}\gamma.o)}\,,$

where $d(Q^{-1}Q.o,h_{1}^{-1}\gamma.o)$ denotes the minimal distance between
$h_{1}^{-1}\gamma.o$ and $Q^{-1}Q.o$ in $\mathbb{H}^{n}$. Since $Q^{-1}Q$ is
pre-compact, we use the reverse triangle inequality to find that

$\mathrm{e}^{-2\alpha_{\varepsilon}d(Q^{-1}Q.o,h_{1}^{-1}\gamma.o)}\leq\sup_{q\in
Q^{-1}Q}\mathrm{e}^{2\alpha_{\varepsilon}d(q.o,o)}\mathrm{e}^{-2\alpha_{\varepsilon}d(h_{1}.o,\gamma.o)}$

for all $h_{1}\in G,\gamma\in\Gamma$. Thus it remains to prove finiteness of

$\displaystyle\sum_{\gamma\in\Gamma}\int_{G_{n}}\mathrm{e}^{-2\alpha_{\varepsilon}d(h_{1}.o,\gamma.o)}$
$\displaystyle\beta(h_{1})dm_{G_{n}}(h_{1})\leq
C\sum_{\gamma\in\Gamma}\int_{G_{n}}\mathrm{e}^{-2\alpha_{\varepsilon}(d(h_{1}.o,\gamma.o)+d(h_{1}.o,o))}dm_{G_{n}}(h_{1})\,.$

To see this this we use the triangle inequality to find that

$\displaystyle\mathrm{e}^{-2\alpha_{\varepsilon}(d(h_{1}.o,\gamma.o)+d(h_{1}.o,o))}$
$\displaystyle=\mathrm{e}^{-\alpha_{\varepsilon}(d(h_{1}.o,\gamma.o)+d(h_{1}.o,o))}\mathrm{e}^{-\alpha_{\varepsilon}(d(h_{1}.o,\gamma.o)+d(h_{1}.o,o))}$
$\displaystyle\leq\mathrm{e}^{-\alpha_{\varepsilon}d(\gamma.o,o)}\mathrm{e}^{-\alpha_{\varepsilon}d(h_{1}.o,o)}\,.$

Finally,

$\displaystyle\sum_{\gamma\in\Gamma}\int_{G_{n}}\mathrm{e}^{-2\alpha_{\varepsilon}(d(h_{1}.o,\gamma.o)+d(h_{1}.o,o))}dm_{G_{n}}(h_{1})\leq\sum_{\gamma\in\Gamma}\mathrm{e}^{-\alpha_{\varepsilon}d(\gamma.o,o)}\int_{G_{n}}\mathrm{e}^{-\alpha_{\varepsilon}d(h_{1}.o,o)}dm_{G_{n}}(h_{1})\,,$

which is finite by Lemma 10.1. ∎

###  10.2. A formula for the diffraction measure

Let $\Gamma<G_{n}$ be a lattice and $\nu\in\mathrm{Prob}(G_{n})$ be
right-$K_{n}$-invariant and absolutely continuous with respect to the Haar
measure satisfying the conditions in Lemma 10.2, so that the random perturbed
lattice $\mu_{\Gamma,\nu}$ is locally square-integrable. Then from the
computations in example 2.6, the autocorrelation measure of the random
perturbed lattice orbit $\mu_{\Gamma,\nu}$ is

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

as a linear functional on $\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n},K_{n})$,
where $\Gamma_{o}=\mathrm{Stab}_{\Gamma}(o)=\Gamma\cap K_{n}$. If we let
$\widehat{\eta}_{\Gamma}$ denote the diffraction measure of the random lattice
orbit $\mu_{\Gamma}$ and $\widehat{\eta}_{\Gamma,\nu}$ that of
$\mu_{\Gamma,\nu}$, then

$\displaystyle\int_{\Omega_{n}^{+}}\widehat{\varphi}(\lambda)d\widehat{\eta}_{\Gamma,\nu}(\lambda)=|\Gamma_{o}|^{2}\int_{\Omega_{n}^{+}}\widehat{\varphi}_{\nu}(\lambda)d\widehat{\eta}_{\Gamma}(\lambda)+\frac{1}{\mathrm{covol}(\Gamma)}\int_{0}^{\infty}\big{(}\widehat{\varphi}(\lambda)-\widehat{\varphi}_{\nu}(\lambda)\big{)}\frac{d\lambda}{|c_{n}(\lambda)|^{2}}$

for all $\varphi\in\mathscr{L}^{\infty}_{\mathrm{bnd}}(G_{n},K_{n})$, where
$\varphi_{\nu}=\nu*\varphi*\check{\nu}$. If $\nu$ happens to be
bi-$K_{n}$-invariant, then by Lemma 6.1 we can write

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

Considering the spherical transform of $\nu$ given by

$\widehat{\nu}(\lambda)=\int_{G_{n}}\omega_{\lambda}(g)d\nu(g)$

we can write the diffraction measure $\widehat{\eta}_{\Gamma,\nu}$ as

$\displaystyle
d\widehat{\eta}_{\Gamma,\nu}(\lambda)=|\Gamma_{o}|^{2}|\widehat{\nu}(\lambda)|^{2}d\widehat{\eta}_{\Gamma}(\lambda)+\frac{1}{\mathrm{covol}(\Gamma)}(1-|\widehat{\nu}(\lambda)|^{2})\frac{d\lambda}{|c_{n}(\lambda)|^{2}}\,.$
(10.1)

###  10.3. Random perturbed lattice orbits in $\mathbb{H}^{n}$ are not
spectrally hyperuniform

Let $\Gamma<G_{n}$ be a lattice and $\nu\in\mathrm{Prob}(G_{n})$ a
bi-$K_{n}$-invariant probability measure that is absolutely continuous with
respect to the Haar measure $m_{G_{n}}$, satisfying the exponential decay
condition in Lemma 10.2.

###### Proof of Proposition 1.7.

From the formula for the diffraction measure $\widehat{\eta}_{\Gamma,\nu}$ in
Equation 10.1 we cut off the first term to get the lower bound

$\displaystyle\limsup_{\varepsilon\rightarrow
0^{+}}\frac{\widehat{\eta}_{\Gamma,\nu}((0,\varepsilon])}{\varepsilon^{3}}$
$\displaystyle\geq\frac{1}{\mathrm{covol}(\Gamma)}\limsup_{\varepsilon\rightarrow
0^{+}}\frac{1}{\varepsilon^{3}}\int_{0}^{\varepsilon}(1-|\widehat{\nu}(\lambda)|^{2})\frac{d\lambda}{|c_{n}(\lambda)|^{2}}\,.$

By item (3) of Lemma 6.3 we have that

$\displaystyle\widehat{\nu}(\lambda)=\int_{G_{n}}\omega_{\lambda}(g)d\nu(g)\leq\int_{G_{n}}\omega_{0}(g)d\nu(g)=\widehat{\nu}(0)\,,$

so

$\displaystyle\limsup_{\varepsilon\rightarrow
0^{+}}\frac{\widehat{\eta}_{\Gamma,\nu}((0,\varepsilon])}{\varepsilon^{3}}$
$\displaystyle\geq\frac{1}{\mathrm{covol}(\Gamma)}(1-|\widehat{\nu}(0)|^{2})\limsup_{\varepsilon\rightarrow
0^{+}}\frac{1}{\varepsilon^{3}}\int_{0}^{\varepsilon}\frac{d\lambda}{|c_{n}(\lambda)|^{2}}\,.$

Since $|c_{n}(\lambda)|^{-2}\asymp\lambda^{2}$ when $\lambda\rightarrow 0^{+}$
then there is a constant $c_{o}>0$ such that $|c(\lambda)|^{-2}\geq
c_{o}\lambda^{2}$ for all sufficiently small $\lambda\geq 0$, so that

$\limsup_{\varepsilon\rightarrow
0^{+}}\frac{1}{\varepsilon^{3}}\int_{0}^{\varepsilon}\frac{d\lambda}{|c_{n}(\lambda)|^{2}}\geq\frac{c_{o}}{3}\,.$

Moreover, by item (2) in Lemma 6.3, $\omega_{0}(g)<1$ for all $g\neq e$, and
since $m_{G_{n}}(K_{n})=0$ then $\nu(K_{n})=0$ by absolute continuity, so

$\displaystyle\widehat{\nu}(0)=\int_{G_{n}}\omega_{0}(g)d\nu(g)<\nu(G_{n})=1\,.$

Gathering these lower bounds,

$\displaystyle\limsup_{\varepsilon\rightarrow
0^{+}}\frac{\widehat{\eta}_{\Gamma,\nu}((0,\varepsilon])}{\varepsilon^{3}}\geq\frac{c_{o}}{3\,\mathrm{covol}(\Gamma)}(1-|\widehat{\nu}(0)|^{2})>0\,.$

∎

######  Remark 10.4.

Although we have not introduced diffraction measures for random measures on
other homogeneous spaces, we do so in upcoming work for general commutative
metric spaces $X=G/K$ with $K$ compact. This includes semisimple Lie groups
$G$ with finite center and maximal compact subgroup $K$, and there the same
argument as in the above proof holds, so no perturbation of a stealthy lattice
yields a hyperuniform point process.

As a consequence of Proposition 1.7, we prove that the number variance of a
perturbed lattice orbit grows as fast as the volume along _all_ subsequences
of radii.

######  Corollary 10.5.

Let $\Gamma<G_{n}$ be a lattice and $\nu\in\mathrm{Prob}(G_{n})$ be
bi-$K_{n}$-invariant as in Lemma 10.2. Then

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

###### Proof.

By the formula for the diffraction measure of $\mu_{\Gamma,\nu}$ in equation
10.1 we can bound the number variance form below using the spherical
Plancherel formula as

$\displaystyle\mathrm{NV}_{\mu_{\Gamma,\nu}}(R)$
$\displaystyle\geq\frac{1}{\mathrm{covol}(\Gamma)}\int_{0}^{\infty}|\widehat{\chi}_{B_{R}}(\lambda)|^{2}(1-|\widehat{\nu}(\lambda)|^{2})\frac{d\lambda}{|c_{n}(\lambda)|^{2}}$
$\displaystyle\geq\frac{1}{\mathrm{covol}(\Gamma)}(1-|\widehat{\nu}(0)|^{2})\int_{0}^{\infty}|\widehat{\chi}_{B_{R}}(\lambda)|^{2}\frac{d\lambda}{|c_{n}(\lambda)|^{2}}$
$\displaystyle=\frac{1}{\mathrm{covol}(\Gamma)}(1-|\widehat{\nu}(0)|^{2})\int_{G}|\chi_{B_{R}}(g)|^{2}dm_{G_{n}}(g)$
$\displaystyle=\frac{\Gamma(\frac{n}{2})}{\mathrm{covol}(\Gamma)2\pi^{\frac{n}{2}}}(1-|\widehat{\nu}(0)|^{2})\mathrm{Vol}_{\mathbb{H}^{n}}(B_{R})\,.$

Since $\widehat{\nu}(0)<1$ the right hand side is strictly positive. ∎

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Department of Mathematics, Chalmers and University of Gothenburg, Gothenburg,
Sweden
Email address<EMAIL_ADDRESS>
Department of Mathematics, Chalmers and University of Gothenburg, Gothenburg,
Sweden
Email address<EMAIL_ADDRESS>


aainstitutetext: Dipartimento di Fisica e Astronomia, Università di Bologna,
via Irnerio 46, 40126 Bologna, Italybbinstitutetext: INFN, Sezione di Bologna,
viale Berti Pichat 6/2, 40127 Bologna, Italyccinstitutetext: School of Physics
and Astronomy, University of Nottingham, Nottingham NG7 2RD,
UKddinstitutetext: Nottingham Centre of Gravity, University of Nottingham,
Nottingham NG7 2RD, UK

#  Quintessence and the Swampland:
The parametrically controlled regime of moduli space

Michele Cicoli c,d Francesc Cunillera c,d Antonio Padilla a,b Francisco G.
Pedro<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We provide evidence that slow roll is not possible in any parametrically
controlled regime of the moduli space of string theory. This is proven in full
generality in the asymptotic limit of the moduli space of type II and
heterotic Calabi-Yau compactifications for the dilaton and any number of
Kähler moduli. Our results suggest that in order to build quintessence into
string theory one must work in the interior of moduli space where numerical,
even if not parametric, control could still be achieved.

##  1 Introduction

Cosmological observations hep-th/9812133 ; astro-ph/9805201 ; 1807.06209
suggest that our universe has experienced at least two phases of accelerated
expansion — one at early times, usually referred to as inflation, and one in
the most recent epoch, known as dark energy. Establishing the microscopic
origin of inflation and/or dark energy, is a significant challenge for string
theory phenomenology. Indeed, it has been conjectured that the canonical form
of accelerated expansion - that of a de Sitter vacuum driven by a positive
cosmological constant deSitter:1917zz  \- cannot be obtained from string
theory 1806.08362 ; 1807.05193 ; 1810.05506 . Here our interest lies in the
late time alternative to de Sitter, some times known as quintessence
Peebles:1987ek ; Ratra:1987rm ; astro-ph/9708069 . This is where we have a
scalar field, or fields, in slow roll, yielding a dynamical theory of dark
energy that is almost de Sitter, but not quite. Such a scenario might be
appealing in string theory, partly because of the difficulties in modelling
late time de Sitter, and partly because they offer the possibility of a
solution to the coincidence problem 2105.03426 .

However, this short note is the first of two papers identifying significant
challenges to quintessence model building in string theory compactifications.
Here we will prove that quintessence is not possible in the runaway region
close to the boundary of moduli space, because one cannot satisfy the slow
roll constraints. Related conclusions were drawn in 0711.2512 ; 1810.09406 ;
Ibe:2018ffn ; 2005.10168 ; Andriot:2020vlg . Here we build on these works by
simultaneously analysing the Kähler-dilaton sectors, by highlighting the rôle
of supersymmetry and by considering corrections to moduli potentials which at
tree-level feature a no-scale cancellation. Further difficulties with
quintessence model building in string theory were also discussed in 1909.08625
. In the following we discuss the absence of a slow roll region in the runaway
of type II and heterotic supergravities. For a comprehensive review of the
pros-and-cons of the different theories regarding other model building
aspects, see 1808.08967 .

To proceed with our proof, let us approach the boundary of moduli space in a
string theory compactification on some Calabi-Yau threefold, $\mathcal{X}$. In
the strict boundary limit, we should think of the dilaton or the volume moduli
going to infinity. As is well known, such a limit is expected to bring down an
infinite tower of light states, spoiling the effective field theory
description hep-th/0605264 ; 1610.00010 ; 1802.08264 ; 1803.04989 . In
particular, the asymptotic limits of the volume moduli bring down either
Kaluza-Klein modes, consistent with a decompactification of the internal
space, or winding modes, consistent with decompactification of the dual. For
the asymptotic limits of the dilaton, the tower of light states is less clear
— it has been suggested that this can introduce a tower of tensionless strings
and domain walls 1904.05379 .

Since the physics at the boundary is clearly not phenomenologically viable,
let us relax the limit and consider what happens when the dilaton or some
Kähler modulus becomes arbitrarily large but finite. In this case, the
effective theory corresponds to tree-level supergravity close to the boundary,
where the moduli fields are in a runaway regime. We are interested in theories
that might admit no four-dimensional vacuum, or potentially a supersymmetric
Minkowski or AdS vacuum somewhere in the bulk of moduli space. The
supersymmetric requirement serves to align our discussion with the usual
swampland lore of non-supersymmetric vacua being unstable 1610.01533 . It
follows from the arguments given in 1810.05506  that de Sitter vacua cannot
arise in any parametrically controlled regime of the moduli space, and our
only hope to achieve phenomenologically viable models of dark energy is to
consider runaway quintessence. For example, type IIB flux compactifications
feature at tree-level supersymmetric Minkowski vacua where the Kähler moduli
are flat directions due to the underlying no-scale structure Burgess:2020qsc .
Leading order $\alpha^{\prime}$ corrections, or a non-supersymmetric
stabilisation of the dilaton and the complex structure moduli, can then
generate this type of de Sitter runaway for large volume.

In the following we will assume that there exists some mechanism to fix the
complex structure moduli at tree-level in a controlled manner, while the
remaining moduli fields correspond to the axio-dilaton $S=s+i\alpha$ and the
Kähler moduli $T^{a}=\tau^{a}+i\theta^{a}$, which are identified with either
two-cycles or four-cycles, depending on which is most convenient. In the
runaway regime, close to the boundary, the dynamics of these moduli is
governed by the following Kähler potential in its tree-level sum separable
form

$K=-p\ln(\mathcal{V})-\ln(S+\bar{S})+K_{0}\ ,$ (1)

and a superpotential $W$ to be specified for each supergravity. We have
denoted the internal volume by $\mathcal{V}(\tau^{a})$. For any Calabi-Yau
threefold $\mathcal{X}$, the volume is a homogeneous function of degree $3/2$
in the Kähler moduli for four-cycles, or equivalently, of degree $3$ in the
Kähler moduli for two-cycles. Furthermore, for type IIB supergravity, we have
$p=2$, and work with four-cycles, while for type IIA and heterotic
supergravities, we have $p=1$, and work with two-cycles. As already stated,
the complex structure contribution to the Kähler potential, $K_{0}$, is
considered to be fixed. For further details, see hep-th/0507153 ;
Cicoli:2013rwa .

The Lagrangian for the scalar moduli is given by

$\mathcal{L}=K_{S\bar{S}}dS\wedge\star d\bar{S}+K_{a\bar{b}}dT^{a}\wedge\star
dT^{\bar{b}}-V\ ,$ (2)

where $K_{I\bar{J}}=\partial_{I}\partial_{\bar{J}}K$ is the Kähler metric for
moduli space, which given equation (1) is block diagonal. The scalar potential
is obtained by computing

$V=e^{K}(K^{I\bar{J}}D_{I}WD_{\bar{J}}\bar{W}-3|W|^{2})\ ,$ (3)

where $D_{I}W=\partial_{I}W+W\partial_{I}K$ is the Kähler covariant
derivative, and $I$ runs over all the moduli of the theory.

At the perturbative level, the axions, $\theta^{a}=\text{Im}\ (T^{a})$ and
$\alpha$, do not contribute to the scalar potential and cannot play the rôle
of runaway quintessence. Focusing instead on the saxions/moduli,
$\tau^{a}=\text{Re}\ (T^{a})$ and $s$, we find that the relevant part of the
Lagrangian is now given by

$\mathcal{L}=\frac{1}{4s^{2}}(\partial
s)^{2}+K_{a\bar{b}}\partial\tau^{a}\partial\tau^{b}-V(s,\tau^{a})\ ,$ (4)

A necessary condition for phenomenologically viable quintessence is the
existence of a slow roll regime, defined by the condition
$\epsilon:=-\dot{H}/H^{2}<1$ where $H$ is the Hubble parameter and dot denotes
differentiation with respect to proper cosmological time. For a generic
multiscalar theory described by a Lagrangian
$\mathcal{L}=\frac{1}{2}Z_{IJ}\partial\phi^{I}\partial\phi^{J}-V(\phi^{I})$
one can show in the limit of vanishing acceleration, $\ddot{\phi}^{I}=0$, the
dynamics of the system is described by

$H^{2}\approx\frac{1}{3}V\qquad\text{and}\qquad
3H\dot{\phi}^{I}+\Gamma^{I}_{JK}\dot{\phi}^{J}\dot{\phi}^{K}\approx-Z^{IJ}\partial_{\phi^{J}}V,$
(5)

where $Z^{IJ}$ is the inverse of the field space metric and $\Gamma^{I}_{JK}$
is the corresponding metric connection. These equations admit two distinct
classes of slow roll solutions, whose existence and stability has been
analysed in the context of dark energy in Cicoli:2020cfj . In this note we
will focus on the cases when
$3H\dot{\phi}^{I}\gg\Gamma^{I}_{JK}\dot{\phi}^{J}\dot{\phi}^{K}\rightarrow 0$,
for which

$\dot{H}\approx\frac{\left(\partial_{\phi^{I}}V\right)\dot{\phi}^{I}}{6H}\approx-\frac{(\partial_{\phi^{I}}V)Z^{IJ}(\partial_{\phi^{J}}V)}{18H^{2}}\,,$
(6)

and so

$\epsilon=-\frac{\dot{H}}{H^{2}}\approx\frac{1}{2}\left(\partial_{\phi^{I}}\ln
V\right){Z^{IJ}}\left(\partial_{\phi^{J}}\ln V\right)\,.$ (7)

In this regime, the requirement of quasi-de Sitter expansion implies the need
for flat (multifield) scalar potentials. The other class of accelerating
solutions, that we will not analyse here, forego the flatness of the potential
in exchange for large field space curvature, see Cicoli:2020cfj ;
Cicoli:2020noz .111Strictly speaking the requirement is that
$\Gamma^{I}_{JK}\dot{\phi}^{J}\dot{\phi}^{K}\gg 3H\dot{\phi}^{I}$ for one of
the scalars, which can be achieved even in flat field space if, for instance,
one uses polar coordinates as in Boyle:2001du . While promising, this avenue
is not without challenges from a string model building perspective
Brinkman:2022 .

For the supergravity compactifications under consideration (4), the first slow
roll parameter is given by $\epsilon=\epsilon_{s}+\epsilon_{\mathcal{V}}$,
where the dilaton contribution is

$\epsilon_{s}=s^{2}\left(\partial_{s}\ln V\right)^{2}\ ,$ (8)

and the Kähler contribution is

$\epsilon_{\mathcal{V}}=\frac{1}{4}(\partial_{\tau^{a}}\ln
V)K^{a\bar{b}}(\partial_{\tau^{b}}\ln V)\ ,$ (9)

Throughout this paper, we will make regular use of Euler’s theorem and its
corollary: if $P_{n}$ is a homogeneous function of degree $n$ in $\tau^{a}$,
then

$\tau^{a}\partial_{\tau^{a}}\ln P_{n}=n\
,\qquad\tau^{a}\partial_{\tau^{a}}\partial_{\tau^{b}}\ln
P_{n}=-\partial_{\tau^{b}}\ln P_{n}\ .$

In particular, this allows us to infer the following conditions on the
derivatives of the Kähler potential

$K^{a\bar{b}}\partial_{\bar{T}^{b}}K=-2\tau^{a}\
,\qquad(\partial_{T^{a}}K)K^{a\bar{b}}(\partial_{\bar{T}^{b}}K)=3\ ,$ (10)

where the indices run over the Kähler moduli only. The latter result holds as
long as we work with four-cycle Kähler moduli in type IIB supergravity, and
two-cycle Kähler moduli in type IIA and heterotic.

Let us now derive the detailed form of the tree-level scalar potential, and
the corresponding slow roll parameter, for the type II and heterotic
supergravities, in turn. We will see that the slow roll condition $\epsilon<1$
can never be satisfied in the runaway regions close to the boundary of moduli
space.

##  2 The type IIB runaway

Let us begin with type IIB supergravity (for further details, see hep-
th/0507153 ). In this instance, at tree-level, the superpotential is linear in
the dilaton $W=h_{0}S+f_{0}$, with $h_{0}$ and $f_{0}$ set by the three-form
fluxes, respectively $H_{3}$ and $F_{3}$, that stabilise the complex structure
moduli supersymmetrically. Furthermore, the Kähler moduli
$T^{a}=\tau^{a}+i\theta^{a}$ that enter the Kähler potential through the
volume will be identified with four-cycles, so that the volume is a
homogeneous function of degree $\frac{3}{2}$ in the corresponding saxions,
$\tau^{a}$. Using the form of the F-term scalar potential given by equation
(3) and the Kähler metric (1) with $p=2$, we obtain a scalar potential

$V={e^{K_{0}}\over 2s\mathcal{V}^{2}}|h_{0}\bar{S}-f_{0}|^{2}\ ,$ (11)

where $\bar{S}=s-i\alpha$. If $h_{0}\neq 0$, the imaginary part of the axio-
dilaton can be stabilised at the point where $\partial_{\alpha}V=0$, or
equivalently, $\alpha=-\text{Im}\ \left(\frac{f_{0}}{h_{0}}\right)$, so that
the scalar potential becomes

$V(s,\mathcal{V})={e^{K_{0}}\over
2s\mathcal{V}^{2}}|h_{0}|^{2}\left(s-\text{Re}\
\left(\frac{f_{0}}{h_{0}}\right)\right)^{2}\ .$ (12)

It follows that

$\epsilon_{s}=\left(\frac{s+\text{Re}\
\left(\frac{f_{0}}{h_{0}}\right)}{s-\text{Re}\
\left(\frac{f_{0}}{h_{0}}\right)}\right)^{2},\qquad\epsilon_{\mathcal{V}}=3\
,$ (13)

defined whenever $V\neq 0$. We immediately infer that
$\epsilon=\epsilon_{s}+\epsilon_{\mathcal{V}}\geq 3$, ruling out slow roll in
the runaway regime. Notice that vacua with $h_{0}=0$ feature exactly
$\epsilon_{s}=1\,\Rightarrow\,\epsilon=4$.

For type IIB supergravity, if $h_{0}\neq 0$, it is possible to further
stabilise the dilaton at the location of the supersymmetric minimum,
$s=\text{Re}\ \left(\frac{f_{0}}{h_{0}}\right)$, where the leading order
potential for the Kähler moduli vanishes. When this happens, we should
consider additional perturbative and non-perturbative corrections to the
potential. In order to be compatible with a robust large volume expansion,
these go as Burgess:2020qsc ; AbdusSalam:2020ywo ; Cicoli:2021rub

$V={\mathcal{A}\over\mathcal{V}^{2+p}}+{\mathcal{B}\,e^{-f}\over\mathcal{V}^{2+q}}+\frac{\mathcal{C}}{\mathcal{V}^{2+r}g^{n}}\ldots\
,$ (14)

where $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ will in general depend on
the stabilised dilaton and complex structure moduli, while $f>0$ and $g>0$ are
homogeneous functions of degree one in the $\tau^{a}$. In particular, $f$
corresponds to the dominant saxion in the non-perturbative expansion. In order
for us to trust the large volume expansion, we require each of these terms to
scale away more quickly than the leading order $1/\mathcal{V}^{2}$ term, which
happened to vanish once the dilaton was stabilised at its supersymmetric
minimum. This implies that $p>0$, $q\geq 0$, and $r+\frac{2n}{3}>0$,
suggesting that the volume direction is made even steeper and the volume
modulus will not slow roll in this case either. To see this explicitly, we
compute the slow roll parameter in the asymptotic regime where only one of the
terms dominates, in accordance with our definition of the boundary of moduli
space. If $V\sim{\mathcal{A}\over\mathcal{V}^{2+p}}$, we find that
$\epsilon\sim 3\left(1+\frac{p}{2}\right)^{2}>3$. In contrast, if
$V\sim{\mathcal{B}\,e^{-f}\over\mathcal{V}^{2+q}}$, we find that

$\epsilon=3\left(1+\frac{q}{2}\right)^{2}+f(2+q)+\frac{1}{4}K^{a\bar{b}}f_{a}f_{b}\geq
3\ ,$ (15)

where the inequality follows from the fact that $f,q\geq 0$ and $K^{a\bar{b}}$
being a symmetric matrix with all positive eigenvalues. Finally, if
$V\sim\frac{\mathcal{C}}{\mathcal{V}^{2+r}g^{n}}$ we find that

$\epsilon=3\left(1+\frac{r}{2}\right)^{2}+n(2+r)+\frac{n^{2}}{4}K^{a\bar{b}}(\ln
g)_{a}(\ln g)_{b}\geq 3\ .$ (16)

In each case we see that slow roll is impossible in this asymptotic region,
where a single one of the corrections in equation (14) dominates. It is
interesting to note that leading order supersymmetry, imposed for the sake of
the stability of the compactification, only aggravates the problem, by
cancelling off the $1/\mathcal{V}^{2}$ term that would otherwise dominate the
potential at the boundary of moduli space and replacing it with a steeper
term.

Alternatively, we could consider a non-supersymmetric stabilisation of the
complex structure sector. This would induce a correction to (12) of the form
$\lambda/s\mathcal{V}^{2}$, where $\lambda$ is a positive constant
proportional to the F-terms of the complex structure moduli. The vacuum
expectation value of the dilaton is then shifted to a non-supersymmetric value
leaving a $1/\mathcal{V}^{2}$ runaway for the volume mode, which is too steep
to give rise to slow roll. Note further that if we move into the bulk of
moduli space, we might hope to stabilise the volume at some fixed value and
achieve slow roll along some other saxion direction, through the last term in
(14). This is indeed possible, although such a scenario runs into further
difficulties associated with a light gravitino Cicoli:2022 . It is also
possible that interference between two terms in equation (14) can give rise to
a (short) field range for the volume where $\epsilon<1$. This avenue is likely
to involve considerable tuning and, by definition, lies in the bulk of moduli
space and therefore will not be analysed in the present paper.

##  3 The type IIA runaway

We now turn our attention to the type IIA runaway (for further details, see
hep-th/0507153 ). The superpotential now contains RR fluxes
$\left(e_{0},e_{a},q^{a}\right)$, the (3,0)-component of the $H_{3}$ flux,
$h_{0}$, and the Romans mass, $m$ Romans:1985tz :

$W=e_{0}+e_{a}T^{a}+{1\over 2}\mathcal{K}_{abc}q^{a}T^{b}T^{c}+{m\over
6}\mathcal{K}_{abc}T^{a}T^{b}T^{c}-ih_{0}S\ .$ (17)

Here the Kähler moduli $T^{a}=v^{a}+i\theta^{a}$ are identified with two-
cycles and the volume is a homogeneous function of degree $3$ in the
corresponding saxions, $v^{a}$. In particular,
$\mathcal{V}=\frac{1}{6}\mathcal{K}_{abc}v^{a}v^{b}v^{c}$, where
$\mathcal{K}_{abc}$ are the triple intersection numbers for the Calabi-Yau.
Without additional fluxes or corrections to the tree-level action, the complex
structure sector remains flat. To continue the discussion on the slow roll
regime, we must therefore assume that some other mechanism under computational
control exists to stabilise the complex structure sector, so that, at tree-
level, it only enters through some constant in the Kähler potential,
$K_{0}=\int_{\mathcal{X}}\Omega\wedge\bar{\Omega}$.

The tree-level scalar potential for the type IIA theory is then given by

$V={e^{K_{0}}\over
2s\mathcal{V}}\left[4|h_{0}|^{2}s^{2}+K^{a\bar{b}}\rho_{a}\bar{\rho}_{b}+4s\text{Im}\
(W\bar{h}_{0})-4\tau^{a}\text{Re}\ (W\bar{\rho}_{a})+|W|^{2}\right]\ ,$ (18)

where

$\rho_{a}=\partial_{T^{a}}W=e_{a}+\mathcal{K}_{abc}q^{b}T^{c}+\frac{m}{2}\mathcal{K}_{abc}T^{b}T^{c}\
.$ (19)

We can always absorb the vacuum expectation value of the axions into a
redefinition of the fluxes. Therefore, without loss of generality, we set the
axions to vanish. With the axions gone, the superpotential can be written as

$W=e_{0}+e_{a}v^{a}+\mathcal{V}(q^{a}\omega_{a}+m)-ih_{0}s\ ,$ (20)

and

$\rho_{a}=e_{a}+\mathcal{V}\left[q^{b}(\omega_{ab}+\omega_{a}\omega_{b})+m\omega_{a}\right]\
,$ (21)

where $\omega_{a}=\partial_{v^{a}}\ln\mathcal{V}$ and
$\omega_{ab}=\partial_{v^{a}}\partial_{v^{b}}\ln\mathcal{V}$. Keeping only the
leading order Kähler terms at large volume, we obtain a scalar potential that
goes as

$V={e^{K_{0}}\over 2s\mathcal{V}}\left[|h_{0}|^{2}s^{2}+14\text{Im}\
(m\bar{h}_{0})\mathcal{V}s+|m|^{2}\mathcal{V}^{2}\right]+\ldots$ (22)

We now compute the slow roll parameter. Since the scalar potential is a
function of $\mathcal{V}/s$ to leading order in this asymptotic regime, we
have that $\epsilon_{\mathcal{V}}=3\epsilon_{s}$. Furthermore,

$\epsilon_{s}=\left[\frac{|h_{0}|^{2}-|m|^{2}\frac{\mathcal{V}^{2}}{s^{2}}}{|h_{0}|^{2}+14\text{Im}\
(m\bar{h}_{0})\frac{\mathcal{V}}{s}+|m|^{2}\frac{\mathcal{V}^{2}}{s^{2}}}\right]^{2}\
.$ (23)

We remark that, in order to keep $\alpha^{\prime}$ corrections from becoming
important, we require222In units of $\alpha^{\prime}=1$.

$\mathcal{V}_{\text{S}}\gg 1\ ,$ (24)

where $\mathcal{V}_{\text{S}}$ is the volume of $\mathcal{X}$ in the string
frame. After expressing the Einstein frame volume in string frame,
$\mathcal{V}=s^{3/2}\mathcal{V}_{\text{S}}$, we see that
$\mathcal{V}/s=\sqrt{s}\mathcal{V}_{\text{S}}\gg 1$ to be consistent with (24)
and weak coupling. Thus, in the runaway regime, the ratio $\mathcal{V}/s$ has
to be very large and it follows that $\epsilon_{s}\approx 1$, and so
$\epsilon=\epsilon_{s}+\epsilon_{\mathcal{V}}\approx 4$, ruling out slow roll.

##  4 The heterotic runaway

We finish with the heterotic runaway, where the tree-level superpotential does
not depend on the dilaton or the Kähler moduli (for further details see
Cicoli:2013rwa ), so that $W=W_{\rm cs}$. The complex structure moduli are
assumed to lie in their supersymmetric vacuum by the vanishing of the
corresponding F-terms. As a result, the scalar potential becomes a runaway in
the dilaton and the volume modulus

$V=e^{K_{0}}{\left|W_{\text{cs}}\right|^{2}\over 2s\mathcal{V}}\ .$ (25)

Computing the slow roll parameters, we find that $\epsilon_{s}=1$ and
$\epsilon_{\mathcal{V}}=3$, and so $\epsilon=4$. Clearly slow roll cannot be
achieved in the heterotic runaway.

##  5 Conclusions

We have found that the tree-level type II and heterotic supergravities cannot
contain a slow roll region in a parametrically controlled regime. The result
is proven for any number of Kähler moduli. It is clear that obtaining a slow
roll region requires breaking the form of the Kähler potential in (1). This
can be done by introducing corrections to the tree-level action. In the sequel
to this paper Cicoli:2022 , we will step away from the runaway, by considering
perturbative and non-perturbative corrections to this leading order behaviour
in detail. There we find that phenomenologically viable quintessence requires
non-supersymmetric vacua. This suggests that proponents of the swampland
should now object to quintessence as vigorously as they object to de Sitter,
placing them on a collision course with observations.

###### Acknowledgements.

We would like to thank Ivonne Zavala, Susha Parameswaran and Arthur Hebecker
for useful discussions. AP was supported by an STFC consolidated grant number
ST/T000732/1 and FC by a University of Nottingham studentship.

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Some remarks regarding l-elements defined in algebras obtained by the Cayley-
Dickson process

Cristina FLAUT and Diana SAVIN

Abstract. In this paper, we define a special class of elements in the algebras
obtained by the Cayley-Dickson process, called $l-$elements. We find
conditions such that these elements to be invertible. These conditions can be
very useful for finding new identities, identities which can help us in the
study of the properties of these algebras.

Key Words: quaternion algebras; octonion algebras; Cayley-Dickson algebras;
special sequences.

2000 AMS Subject Classification: 15A24, 15A06, 16G30, 11R52, 11B39, 11R54.

1\. Introduction

In the last years, many papers were devoted to the study of some special
sequences or special elements, especially in their connections with particular
cases of algebras obtained by the Cayley-Dickson (quaternions, octonions).
Some of them are: [Ba, Pr; 09], [Ca; 16], [Ci, Ip; 16], [Fa, Pl; 07(1)], [Fa,
Pl; 07(2)], [Fl; 09], [Fl, Sa; 15(1)], [Fl, St; 09], [Ha, 15], [Mo; 17],
[Si,18].

Since the algebras obtained by the Cayley-Dickson process, denoted $A_{t}$,
are poor in properties when their dimension increase, losing commutativity,
associativity and alternativity, in this paper, we will find elements with
supplementary properties, namely a special set of invertible elements in
$A_{t},$ which can help us in the study of these algebras. Algebras obtained
by the Cayley-Dickson process will be presented in the following.

Let $K$ be a commutative field with $char\left(K\right)\neq 2.~{}$Let $A$ be a
finite dimensional unitary algebra over a field $\ K.$ We define a scalar
involution

$\,\,\,\overline{\phantom{x}}:A\rightarrow A,a\rightarrow\overline{a},$

to be a linear map with the following properties

$\overline{ab}=\overline{b}\overline{a},\,\overline{\overline{a}}=a,$

and

$a+\overline{a},a\overline{a}\in K\cdot 1\ \text{for all }a,b\in A.\text{ }$

An element $\,\overline{a}\in A$ is called the conjugate of the element $a\in
A,$ the linear form

$\,\,\mathbf{t}:A\rightarrow K\,,\,\,\mathbf{t}\left(a\right)=a+\overline{a}$

and the quadratic form

$\mathbf{n}:A\rightarrow K,\,\,\mathbf{n}\left(a\right)=a\overline{a}\ $

are called the trace and the norm of the element $a.$

We consider$\,\,\,\gamma\in K$ a fixed non-zero element. We define the
following algebra multiplication on the vector space $A\oplus A:$

$\left(a_{1},a_{2}\right)\left(b_{1},b_{2}\right)=\left(a_{1}b_{1}+\gamma\overline{b}_{2}a_{2},a_{2}\overline{b_{1}}+b_{2}a_{1}\right).$
(1.1.)

The obtained algebra structure over $A\oplus A,$ denoted by
$\left(A,\gamma\right),$ is called the algebra obtained from $A$ by the
Cayley-Dickson process. We have $\dim\left(A,\gamma\right)=2\dim A$.

Let $x\in\left(A,\gamma\right)$, $x=\left(a_{1},a_{2}\right)$. The map

$\,\,\,\overline{\phantom{x}}:\left(A,\gamma\right)\rightarrow\left(A,\gamma\right)\,,\,\,x\rightarrow\bar{x}\,=\left(\overline{a}_{1},\text{-}a_{2}\right),$

is a scalar involution of the algebra $\left(A,\gamma\right)$, extending the
involution $\overline{\phantom{x}}\,\,\,$of the algebra $A.$ We take

$\,\mathbf{t}\left(x\right)=\mathbf{t}(a_{1})$

and

$\mathbf{n}\left(x\right)=\mathbf{n}\left(a_{1}\right)-\gamma\mathbf{n}(a_{2})$

to be the trace and the norm of the element $x\in$ $\left(A,\gamma\right)$.

If we consider $A=K$ and we apply this process $t$ times, $t\geq 1,\,\,$we
obtain an algebra over $K,\,\,$denoted by

$A_{t}=\left(\frac{\gamma_{1},...,\gamma_{t}}{K}\right).$ (1.2. )

Using induction, in this algebra, the set $\\{1,e_{2},...,e_{n}\\},n=2^{t},$
generates a basis with the properties:

$e_{i}^{2}=\gamma_{i}1,\,\,_{i}\in K,\gamma_{i}\neq 0,\,\,i=2,...,n$ (1.3.)

and

$e_{i}e_{j}=-e_{j}e_{i}=\beta_{ij}e_{k},\,\,\beta_{ij}\in K,\,\,\beta_{ij}\neq
0,i\neq j,i,j=\,\,2,...n,$ (1.4.)

$\ \ \beta_{ij}$ and $e_{k}$ being uniquely determined by $e_{i}$ and $e_{j}.$
We remark that if $x\in A_{t}$ and $n\left(x\right)\neq 0,$ then $x$ is an
invertible element in $A_{t}.$

A finite-dimensional algebra $A$ is a division algebra if and only if $A$ does
not contain zero divisors. For other details regarding algebras obtained by
the Cayley-Dickson process, the reader is referred to [Sc;66].

For $t=2,$ we obtain the generalized quaternion algebras, denoted
$\,\mathbb{H}(\gamma_{1},\gamma_{2}),$ and for $t=3,~{}$we obtain the
generalized octonion algebras, denoted
$\mathbb{O}(\gamma_{1},\gamma_{2},\gamma_{3})$.

For example, for $x\in\mathbb{H}(\gamma_{1},\gamma_{2}),$
$x=x_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3},$ its norm is

$\,\mathbf{n}\left(x\right)=x\overline{x}=x_{0}^{2}-\gamma_{1}x_{1}^{2}-\gamma_{2}x_{2}^{2}+\gamma_{1}\gamma_{2}x_{3}^{2}\in
K.$ (1.5.)

If, for $x\in\mathbb{A},$ where
$\mathbb{A=\\{\,\mathbb{H}(}\gamma_{1},\gamma_{2}\mathbb{),\mathbb{O}(}\gamma_{1},\gamma_{2},\gamma_{3}\mathbb{)\\}},$
the relation $\mathbf{n}\left(x\right)=0$ implies $x=0$, then these algebras
are division algebras. A quaternion and an octonion non-division algebras are
called split algebras. Using the above notations, we remark that the
quaternion algebra
$\mathbb{H}\left(-1,-1\right)=\left(\frac{-1,-1}{\mathbb{R}}\right)$ and the
octonion algebra
$\mathbb{O}\left(-1,-1,-1\right)=\left(\frac{-1,-1,-1}{\mathbb{R}}\right)$ are
division algebras. If the field $K$ is a finite field, then the quaternion and
octonion algebras are always split algebras.

An algebra $A\in\mathbb{A}$ such that
$n\left(xy\right)=n\left(x\right)n\left(y\right),$ for all $x,y\in A,$ is
called a composition algebra. For $t\geq 4,~{}$the algebras $A_{t}$ are not
composition algebras.

A unitary algebra $A\neq K$ such that we have $x^{2}+\alpha_{x}x+\beta_{x}=0,$
for each $x\in A,$ with $\alpha_{x},\beta_{x}\in K,$ is called a quadratic
algebra. An algebra $A$ with a scalar involution is quadratic. From here, we
get that the algebras obtained by the Cayley-Dickson process are quadratic.

An algebra $A$ is called alternative if $x^{2}y=x\left(xy\right)$ and
$xy^{2}=\left(xy\right)y,$ for all $x,y\in A,$ flexible if
$x\left(yx\right)=\left(xy\right)x=xyx,$ for all $x,y\in A$ and power
associative if the subalgebra $<x>$ of $A,$ generated by any element $x\in A,$
is associative.

Algebras $A_{t}$ are noncommutative, for $t\geq 2,$ nonassocitive, for $t\geq
3,$ non-alternative, for $t\geq 4$ and, in general, non division algebras.
They are power associative and flexible for all $t.$

2. $l$-numbers and some of their properties

In [Ho; 61] and [Fl, Sa; 17], were defined and studied the $(a,b,x_{0},x_{1})$
numbers. If we consider $l$ a nonzero natural number, for
$a=l,b=1,x_{0}=0,x_{1}=1,~{}$in [Sa; 18] was considered the following
sequence, where

$p_{n}=l\cdot p_{n-1}+p_{n-2},\;n\geq 2,p_{0}=0,p_{1}=1.$ (2.1.)

Since these numbers are $\left(l,1,0,1\right)$ numbers, we will call them
$l-$numbers. For $l=1,$ it is obtained the Fibonacci numbers and for $l=2,$ it
is obtained the Pell numbers.

Remark 2.1. ([Sa; 18]). Let $\left(p_{n}\right)_{n\geq 0}$ be the sequence
previously defined. Then, the following relations are true:
1)

$p_{n}^{2}+p_{n+1}^{2}=p_{2n+1},\ \left(\forall\right)n\in\mathbb{N}.$

2) For $\alpha=\frac{l+\sqrt{l^{2}+4}}{2}$ and
$\beta=\frac{l-\sqrt{l^{2}+4}}{2},$ we obtain that

$p_{n}=\frac{\alpha^{n}-\beta^{n}}{\alpha-\beta}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{l^{2}+4}},\
\ \left(\forall\right)n\in\mathbb{N},$

called the Binet’s formula for the sequence $\left(p_{n}\right)_{n\geq
0}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proposition 2.2. With the above notations, the following relations hold:

1) If $d\mid n,$ then  $p_{d}\mid p_{n}.$

2) $p_{m+n}=p_{m}p_{n+1}+p_{m-1}p_{n}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof. 1) We use the Binet formula from the above remark. If $d\mid n,n=dc,$

$p_{n}=\frac{\alpha^{dc}-\beta^{dc}}{\sqrt{l^{2}+4}}=\frac{(\alpha^{d})^{c}-(\beta^{d})^{c}}{\sqrt{l^{2}+4}}=$
$=\frac{(\alpha^{d}-\beta^{d})M}{\sqrt{l^{2}+4}},$

therefore $p_{d}\mid p_{n}.$

2) We use induction after $n.$ For $n=0,$ we have $p_{m}=p_{m}.$ Supposing
affirmation true for each $k$$\leq$ $n,$ we will prove for $n+1.$ We have
$p_{m+n+1}=lp_{m+n}+p_{m+n-1}=l\left(p_{m}p_{n+1}+p_{m-1}p_{n}\right)+$
$+p_{m}p_{n}+p_{m-1}p_{n-1}=p_{m}\left(lp_{n+1}+p_{n}\right)+p_{m-1}\left(lp_{n}+p_{n-1}\right)=$
$=p_{m}p_{n+2}+p_{m-1}p_{n+1}.\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

Proposition 2.3. Let $(p_{n})_{n\geq 0}$ be the $l-$sequence defined in
$\left(2.1\right)$. Then, the following relations are true:
i)

$p_{n}+p_{n+4}=\left(l^{2}+2\right)\cdot p_{n+2};$ (2.2.)

ii)

$p_{n}+p_{n+8}=\left[\left(l^{2}+2\right)^{2}-2\right]\cdot p_{n+4};$ (2.3.)

For $k\geq 3,$ we have

$p_{n}+p_{n+2^{k}}=\underbrace{\left[\left(\left(\left(l^{2}+2\right)^{2}-2\right)^{2}...-2\right)^{2}-2\right]}_{k-2\;times\;of\;-2}\cdot
p_{n+2^{k-1}}.$ (2.4.)

Proof. i) With the above notations, we remark that
$\alpha^{2}+\frac{1}{\alpha^{2}}=\alpha^{2}+\beta^{2}=\left(\alpha+\beta\right)^{2}-2\alpha\beta=l^{2}+2.~{}$
Applying Binet’s formula for $l-$ sequence, we have:

$p_{n}+p_{n+4}=\frac{\alpha^{n}-\beta^{n}}{\alpha-\beta}+\frac{\alpha^{n+4}-\beta^{n+4}}{\alpha-\beta}=$
$\frac{\alpha^{n+2}\cdot\left(\alpha^{2}+\frac{1}{\alpha^{2}}\right)-\beta^{n+2}\cdot\left(\beta^{2}+\frac{1}{\beta^{2}}\right)}{\alpha-\beta}=$
$\left(l^{2}+2\right)\cdot\frac{\alpha^{n+2}-\beta^{n+2}}{\alpha-\beta}=\left(l^{2}+2\right)p_{n+2}.$

ii) Using i), we obtain:

$p_{n}+p_{n+8}=\left(p_{n}+p_{n+4}\right)+\left(p_{n+4}+p_{n+8}\right)-2\cdot
p_{n+4}=$ $=\left(l^{2}+2\right)\cdot p_{n+2}+\left(l^{2}+2\right)\cdot
p_{n+6}-2\cdot p_{n+4}=$
$=\left(l^{2}+2\right)\cdot\left(p_{n+2}+p_{n+6}\right)-2\cdot p_{n+4}=$
$=\left[\left(l^{2}+2\right)^{2}-2\right]\cdot p_{n+4}.$

iii) We use induction after $k$$\in$$\mathbb{N},$ $k$$\geq 3$, to prove the
following statement:

$P\left(k\right):\
p_{n}+p_{n+2^{k}}=\underbrace{\left[\left(\left(\left(l^{2}+2\right)^{2}-2\right)^{2}...-2\right)^{2}-2\right]}_{k-2\;times\;of\;-2}\cdot
p_{n+2^{k-1}}.$

In i) we proved $P\left(2\right),$ in ii) we proved $P\left(3\right).$
Supposing that $P\left(k\right)$ is true, it is easy to prove
$P\left(k+1\right),$ in the same way as we passed from $P\left(2\right)$ to
$P\left(3\right).\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

Remark 2.4. We will denote

$M_{k}=\underbrace{\left[\left(\left(\left(l^{2}+2\right)^{2}-2\right)^{2}...-2\right)^{2}-2\right]}_{k-2\;times\;of\;-2},k\geq
3.$ (2.5.)

We remark that $M_{k+1}=M_{k}^{2}-2.$ We put $M_{2}=$ $l^{2}+2.$ It results,
$M_{k}>0,$ for all $k\in\mathbb{N}$, $k\geq 2$.

3\. $l$-elements in algebras obtained by the Cayley-Dickson process

With the above notations, we will denote

$S_{t}=M_{2}M_{3}M_{4}...M_{t-2}M_{t-1}M_{t},t\geq 2.$

We will put $S_{1}=1.$

Let $A_{t}=\left(\frac{\gamma_{1},...,\gamma_{t}}{K}\right)$ be an algebra
obtained by the Cayley-Dickson process and $n$$\in$$\mathbb{N}.$ We define the
$n$-th $l-$element $P_{n}\in A_{t}~{}$to be an element of the form

$P_{n}\text{=}p_{n}\cdot 1+p_{n+1}\cdot e_{1}+p_{n+2}\cdot e_{2}+p_{n+3}\cdot
e_{3}+...+p_{n+2^{t}-1}e_{t-1}.$ (3.1.)

Proposition 3.1. Let $v$ be a real number, $q$ be a prime number,
$K\in\\{\mathbb{Q},\mathbb{R,Z}_{q}\\},~{}$and
$A_{t}=\left(\frac{-1,...,-1,v}{K}\right)$ be an algebra obtained by the
Cayley-Dickson process of dimension $r=2^{t}$, $t\geq 2,$ and

$P_{n}=p_{n}\cdot 1+p_{n+1}\cdot e_{1}+p_{n+2}\cdot e_{2}+p_{n+3}\cdot
e_{3}+...+p_{n+2^{t}-1}e_{t-1},$

be an $l-$element. Therefore, the following statements are true:

1) We have

$\boldsymbol{n}\left(P_{n}\right)=S_{t-1}(p_{2n+2^{t-1}-1}-vp_{2n+2^{t+1}-1-2^{t-1}}).$
(3.2.)

2) If $v\in\mathbb{R}-\left(-1,1\right),$ the $n$-th $l-$element $P_{n}$ is
invertible in $A_{t}=\left(\frac{-1,...,-1,v}{K}\right),$ for all
$n$$\in$$\mathbb{N}$.

3) If $v=-1,$ we have

$\boldsymbol{n}\left(P_{n}\right)=S_{t}p_{2n+2^{t}-1}.$ (3.3.)

Proof. 1) Using Remark 2.1 and Proposition 2.3, we compute the norm. In this
case, we obtain:

$\boldsymbol{n}\left(P_{n}\right)\text{=}p_{n}^{2}\text{+}p_{n+1}^{2}\text{+}p_{n+2}^{2}\text{+}p_{n+3}^{2}\text{+...+}p_{n+2^{t-1}-1}^{2}\text{-}v(p_{n+2^{t-1}}^{2}\text{+...+}p_{n+2^{t}-2}^{2}\text{+}p_{n+2^{t}-1}^{2})\text{=}$
$=p_{2n+1}+p_{2n+5}+p_{2n+9}+p_{2n+13}+...-v(p_{2n+2^{t}+1}+...+p_{2n+2^{t+1}-3})=$
$=\left(l^{2}+2\right)p_{2n+3}+\left(l^{2}+2\right)p_{2n+11}+...-v(...+\left(l^{2}+2\right)p_{2n+2^{t+1}-5})=$
$\text{=}\left(l^{2}\text{+}2\right)\text{\\{[}\left(l^{2}\text{+}2\right)^{2}\text{-}2\text{]}p_{2n\text{+}7}\text{+[}\left(l^{2}\text{+}2\right)^{2}\text{-}2\text{]}p_{2n\text{+}23}\text{+...-}v\text{(...+[}\left(l^{2}\text{+}2\right)^{2}\text{-}2\text{]}p_{2n\text{+}2^{t+1}-9}\text{)\\}=}$
$M_{2}M_{3}[M_{4}p_{2n+15}+...-v(...+M_{4}p_{2n+2^{t+1}-17})]=$ $.....$

$=M_{2}M_{3}M_{4}...M_{t-2}[M_{t-1}p_{2n+2^{t-1}-1}-vM_{t-1}p_{2n+2^{t+1}-1-2^{t-1}})]=$

$=M_{2}M_{3}M_{4}...M_{t-2}M_{t-1}(p_{2n+2^{t-1}-1}-vp_{2n+2^{t+1}-1-2^{t-1}}).$

2) Indeed, since $2n+2^{t-1}-1<2n+2^{t+1}-1-2^{t-1},$ we have that
$p_{2n+2^{t+1}-1-2^{t-1}}>p_{2n+2^{t-1}-1},$ since $(p_{n})_{n\geq 0},$ is an
increasing sequence. From here, we get that
$\boldsymbol{n}\left(P_{n}\right)\neq 0,$ even if $v$ is positive or negative.
Therefore $P_{n}$ is an invertible element in
$A_{t}=\left(\frac{-1,...,-1,v}{K}\right).$

3) Taking $v=-1$ in relation (3.2), we get

$\boldsymbol{n}\left(P_{n}\right)=M_{2}M_{3}M_{4}...M_{t-2}M_{t-1}M_{t}(p_{2n+2^{t-1}-1}+p_{2n+2^{t+1}-1-2^{t-1}})=$
$=M_{2}M_{3}M_{4}...M_{t-2}M_{t-1}M_{t}p_{2n+2^{t}-1}.$

$\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

An idempotent element in a unitary ring, with $0\neq 1,$ is an element $x$,
$x\neq 0$ and $x\neq 1,$ such that $x^{2}=x.\vskip 6.0pt plus 2.0pt minus
2.0pt\vskip 12.0pt plus 4.0pt minus 4.0pt$

Theorem 3.2.  With the above notations, let $q$ be a prime number and let $d$
be the first positive integer such that $q\mid p_{d}.$

1) Therefore $q\mid p_{n}$ if and only if $d\mid n.$

2) If $\gcd\left(q,S_{t}\right)=1,~{}t\geq 2,$ and $d$ is an even number, then
all $l-$elements $P_{n},n\geq d,$ from the algebra
$A_{t}=\left(\frac{-1,...,-1}{\mathbb{Z}_{q}}\right)$ are invertible.

3) If $t\in\\{2,3\\},$ $\gcd\left(q,S_{t}\right)=1,~{}t\geq 2,$ and $d$ is an
odd number such that $d\mid(2^{t}-1),$ then there are no idempotent
$l-$elements in the algebra
$A_{t}=\left(\frac{-1,...,-1}{\mathbb{Z}_{q}}\right).$

Proof.

1) Supposing that $q\mid p_{n},$ let $n=dc+r,0\leq r<d,$ therefore

$p_{n}=p_{dc+r}=p_{r}p_{dc+1}+p_{r-1}p_{dc},$

from Proposition 2.2. Since $q\mid p_{d},$ we have that $q\mid p_{dc}.$ Using
that $q\mid p_{n},$ it results $q\mid p_{r}p_{dc+1}.$ If $q\mid p_{r}$, we
have a contradiction with the choice of the number $d.$ If $q\mid p_{dc+1},$
we have $q\mid p_{dc-1}$ and so on. Then we obtain that $q\mid p_{0},$ false.
Therefore, $r=0$ and $d\mid n.$ Conversely, assuming that $d\mid n,$ we have
$n=dc,~{}c$ a natural number. From here, we have $p_{d}\mid p_{n},$ then
$q\mid p_{n}.$

2) Indeed, since $\boldsymbol{n}\left(P_{n}\right)=S_{t}p_{2n+2^{t}-1}\equiv
0($mod$~{}q),$ we obtain $p_{2n+2^{t}-1}\equiv 0$ (mod $q$). It results
$d/\left(2n+2^{t}-1\right),$ which is false.

3) Supposing that there are $l-$elements $P_{n}$$\in
A_{t}=\left(\frac{-1,...,-1}{\mathbb{Z}_{q}}\right)$ such that
$P_{n}^{2}=P_{n}$ in $A_{t},$ we obtain $P_{n}\cdot\left(P_{n}-1\right)=0.$
Since $t\in\\{2,3\\},$ the algebra $A_{t}$ is a composition algebra. It
results that $P_{n}$ and $P_{n}-1$ are zero divisors in the algebra $A_{t}$
and their norm are zero. These are equivalent with the system

$\left\\{\begin{array}[]{c}\boldsymbol{n}\left(P_{n}\right)=\widehat{0}\\\
\boldsymbol{n}\left(P_{n}-1\right)=\widehat{0}\end{array}\right.$

in $\mathbb{Z}_{q}.$ Using Proposition 3.1, we get

$\left\\{\begin{array}[]{c}S_{t}p_{2n+2^{t}-1}\equiv 0(\text{mod}~{}q)\\\
(p_{n}-1)^{2}+p_{n+1}^{2}+p_{n+2}^{2}+...+p_{n+2^{t}-1}^{2}\equiv
0(\text{mod}~{}q).\end{array}\right.$

Since $\gcd\left(q,S_{t}\right)=1,$ we obtain $p_{2n+2^{t}-1}=0($mod$~{}q),$
that means $q\mid p_{2n+2^{t}-1}.$ It results, from above, that

$2n+2^{t}-1\equiv 0(\text{mod}~{}d)$

which is equivalent with

$n\equiv 0(\text{mod}~{}d)$

and

$S_{t}p_{2n+2^{t}-1}-2p_{n}+1\equiv 0(\text{mod}~{}q).$

From the last relation, we have $\widehat{0}=\widehat{2p_{n}-1}$ in
$\mathbb{Z}_{q}.$ From here, since $n\equiv 0$ (mod $d$), we get
$\widehat{2p_{n}-1}=\widehat{q-1}$ , which is false. We obtain that there are
no idempotent $l-$elements in the algebra
$A_{t}=\left(\frac{-1,...,-1}{\mathbb{Z}_{q}}\right)$.$\Box\vskip 3.0pt plus
1.0pt minus 1.0pt$

Remark 3.3. Statement i) from the above Theorem was proved in [Da, Dr; 70] for
the Fibonacci sequence, that means for $l=1$.

4\. Examples

Example 4.1. For $t=2,~{}$when $\gamma_{1}=-1$ and $\gamma_{2}=q$ is a prime
positive integer, it is known, from ([La; 04]), that the quaternion algebra
$\mathbb{H}_{\mathbb{Q}}\left(-1,q\right)$ splits if and only if $q\equiv 1$
(mod $4$). We wonder how many invertible $l-$elements are in the algebra
$\mathbb{H}_{\mathbb{Q}}\left(-1,q\right)?~{}$ For $q$ a prime positive
integer, $q\equiv 1$ (mod $4$), we consider the quaternion algebra
$A_{t}=\left(\frac{-1,q}{\mathbb{Q}}\right)=H_{\mathbb{Q}}\left(-1,q\right).$
Let $P_{n}$ be the $n$-th $l-$quaternion. Then, we have
$n\left(P_{n}\right)=p_{2n+1}-q\cdot p_{2n+5},$ therefore the $n$-th
$l-$quaternion $P_{n}$ is invertible for all $n$$\in\mathbb{N}.$ Indeed, using
Proposition 3.1, we have:

$\boldsymbol{n}\left(P_{n}\right)=p_{n}^{2}+p_{n+1}^{2}-q\cdot
p_{n+2}^{2}-q\cdot p_{n+3}^{2}=p_{2n+1}-q\cdot p_{2n+5}.$

Remarking that the sequence of $l-$numbers is strictly increasing and $q\geq
1,$ it results that $p_{2n+1}<q\cdot p_{2n+5},$ for all $n$$\in$$\mathbb{N}.$
Therefore $\boldsymbol{n}\left(P_{n}\right)\neq 0,$ for all
$n$$\in$$\mathbb{N}.$ We obtain that $P_{n}$ is invertible, for all
$n$$\in$$\mathbb{N}.\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

Example 4.2. Even if the octonion algebra
$\mathbb{O}\left(-1,-1,-1\right)=\left(\frac{-1,-1,-1}{\mathbb{R}}\right)$ is
division algebra, for $t=4,$ the real sedenion algebra

$\left(\frac{-1,-1,-1,-1}{\mathbb{R}}\right),$

with the basis $\\{1,e_{1},.....,e_{15}\\},$ is not a division algebra. For
example,
$\left(e_{3}+e_{10}\right)\left(e_{6}-e_{15}\right)=0,$(see [Fl; 13] ). The
$l-$element $P_{1}=p_{1}+\overset{15}{\underset{i=1}{\sum}}p_{i+1}\cdot e_{i}$
has the norm $n\left(P_{1}\right)=M_{2}M_{3}M_{4}p_{17}\neq 0,$ therefore
$P_{1}$ is an invertible element.

Let $q$ be a prime positive integer, $q\geq 3,$ and
$\left(\mathbb{Z}_{q},+,\cdot\right)$ be the finite field. It is know that the
quaternion algebra $\mathbb{H}_{\mathbb{Z}_{q}}\left(-1,-1\right)$ splits.
Therefore, it has proper zero divisors. In the paper [Sa;17], using many
properties of Fibonacci and Lucas numbers, were determined Fibonacci
quaternions and generalized Fibonacci-Lucas quaternions which are invertible,
respectively zero divisors elements in some quaternion algebras
$\mathbb{H}_{\mathbb{Z}_{q}}\left(-1,-1\right).$ Here, for $q=3,$ $q=5,$
respectively $q=7,$ we get the invertible elements, respectively zero divisors
elements and idempotent elements in the quaternion algebras
$\mathbb{H}_{\mathbb{Z}_{3}}\left(-1,-1\right),$
$\mathbb{H}_{\mathbb{Z}_{5}}\left(-1,-1\right),$
$\mathbb{H}_{\mathbb{Z}_{7}}\left(-1,-1\right).$ A quaternion from
$\mathbb{H}_{\mathbb{Z}_{q}}\left(-1,-1\right)$ is a zero divisor if and only
if its norm is zero.

Example 4.3. Let $q\geq 3$ be a prime positive integer, let
$H_{\mathbb{Z}_{q}}\left(-1,-1\right)$ be the quaternion algebra and let
$P_{n}$ be the $n$-th generalized $l-$quaternion. Then, from Proposition 3.1,
taking $t=2,~{}$we have

$\boldsymbol{n}\left(P_{n}\right)=\left(l^{2}+2\right)p_{2n+3}.$

Example 4.4. All Pell quaternions in the quaternion algebra
$H_{\mathbb{Z}_{3}}\left(-1,-1\right)$ are zero divisors. Indeed, we apply
Proposition 3.1, that means $l=2,$ and we get
$\boldsymbol{n}\left(P_{n}\right)=\widehat{0}$ in $\mathbb{Z}_{3},$ for all
$n$$\in$$\mathbb{N}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Example 4.5. Let $(p_{n})_{n\geq 0}$ be the Pell sequence. Then, we have:
$p_{n}$ $\vdots$ $5$ if and only if $n\equiv 0$ (mod $3$). Indeed, $d=3$ is
the first number such that $p_{d}$ $\vdots$ $5$ and we apply Theorem 3.2.

Example 4.6. A Pell quaternion is a zero divisor in the quaternion algebra
$H_{\mathbb{Z}_{5}}\left(-1,-1\right)$ if and only if $n\equiv 0$ (mod $3$).
Indeed, this statement results from Theorem 3.2.

Example 4.7. There are no idempotent Pell quaternions in the quaternion
algebra $H_{\mathbb{Z}_{5}}\left(-1,-1\right)$. Indeed, it is a particular
case of Theorem 3.2.

Example 4.8. Let $(p_{n})_{n\geq 0}$ be the Pell sequence. Then, we have:
$p_{n}$ $\vdots$ $7$ if and only if $n\equiv 0$ (mod $6$). Obviously, $d=6$ is
the first number such that $p_{d}$ $\vdots$ $7.$ Therefore, we apply Theorem
3.2.

Example 4.9. There are no Pell quaternion zero divisors in the quaternion
algebra $H_{\mathbb{Z}_{7}}\left(-1,-1\right)$. Therefore all Pell quaternions
$P_{n}$ from the quaternion algebra $H_{\mathbb{Z}_{7}}\left(-1,-1\right)$ are
invertible. It is a consequence of the Theorem 3.2.

Conclusions. In this paper, we found elements, namely $l-$elements, with
supplementary properties which can help us in the study of algebras $A_{t}$
obtained by the Cayley-Dickson process. In Proposition 3.1 and Theorem 3.2, we
found conditions such that these elements are invertible. These conditions can
be very useful in solving equations in algebras $A_{t}$ or to find new
identities, identities which can help us in the study of the properties of
these algebras.

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Cristina FLAUT

Faculty of Mathematics and Computer Science, Ovidius University,

Bd. Mamaia 124, 900527, CONSTANTA, ROMANIA

http://www.univ-ovidius.ro/math/

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Diana SAVIN

Faculty of Mathematics and Computer Science,

Ovidius University,

Bd. Mamaia 124, 900527, CONSTANTA, ROMANIA

http://www.univ-ovidius.ro/math/

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>