# Effective Rate of RIS-aided Networks with Location and Phase Estimation
Uncertainty

Long Kong‡, Steven Kisseleff†, Symeon Chatzinotas†, Björn Ottersten†, and
Melike Erol-Kantarci‡
‡ School of Electrical Engineering and Computer Science, University of Ottawa,
Ontario, Canada
†The Interdisciplinary Centre for Security Reliability and Trust (SnT),
University of Luxembourg, Luxembourg
Emails<EMAIL_ADDRESS>{steven.kisseleff, symeon.chatzinotas,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Reconfigurable Intelligent Surfaces (RIS) are planar structures connected to
electronic circuitry, which can be employed to steer the electromagnetic
signals in a controlled manner. Through this, the signal quality and the
effective data rate can be substantially improved. While the benefits of RIS-
assisted wireless communications have been investigated for various scenarios,
some aspects of the network design, such as coverage, optimal placement of
RIS, etc., often require complex optimization and numerical simulations, since
the achievable effective rate is difficult to predict. This problem becomes
even more difficult in the presence of phase estimation errors or location
uncertainty, which can lead to substantial performance degradation if
neglected. Considering randomly distributed receivers within a ring-shaped
RIS-assisted wireless network, this paper mainly investigates the effective
rate by taking into account the above-mentioned impairments. Furthermore,
exact closed-form expressions for the effective rate are derived in terms of
Meijer’s $G$-function, which (i) reveals that the location and phase
estimation uncertainty should be well considered in the deployment of RIS in
wireless networks; and (ii) facilitates future network design and performance
prediction.

###### Index Terms:

Reconfigurable intelligent surfaces (RISs), effective rate, phase estimation
error, location uncertainty.

##  I Introduction

Reconfigurable intelligent surfaces (RISs) are considered key enablers for
future wireless communications [1]. Accordingly, RISs have been proposed for
various scenarios and applications including 6G [2], internet of things (IoT)
[3], smart cities [4], challenging environments [5], etc. The benefit of RIS
lies in its capability of shaping the wireless propagation environments by
adjusting the signal reflections [6]. Through this, the signal quality and
connectivity can be substantially improved. Furthermore, the energy
consumption at RIS is extremely low, which is a favorable property compared to
traditional relaying.

Various aspects of RIS-assisted networks have been investigated so far, such
as optimal placement of RIS [7], signal routing using multiple RIS [8], hybrid
automatic repeat request for RIS-aided communication systems [9], etc. Due to
the associated system complexity, the methods proposed in these works
typically employ highly complex and computationally intensive optimization
techniques. For the same reason, it is difficult to predict the system
performance for RIS-assisted networks without accurate and thorough numerical
evaluation. On the other hand, if an important performance metric, such as
effective rate, can be directly determined without prior simulations of the
environment merely based on the known network configuration, the network
design and planning can be substantially simplified. This motivates the
analytical derivation of such metrics, which is the focus of this work.

Effective rate connecting both physical layer and link layer, is a popular
performance metric used in the RIS-aided wireless networks. [10] describes the
RIS-assisted links using the cascaded fading channel model and also provides a
derivation of the effective rate with exact and approximated expressions. The
authors in [11] study the effective rate under two extreme assumptions with
regard to the availability of the channel state information. However, the
impact of phase estimation errors and location uncertainty on the effective
rate has not been addressed yet. This paper aims at bridging this gap by
providing analytical expressions for the effective rate with consideration of
the aforementioned impairments.

The contributions of this work are three-fold:

  * •

analytical closed-form expressions for the effective rate are derived in terms
of Meijer’s $G$-function;

  * •

the obtained solution incorporates important impairments, which are likely to
occur in practical scenarios of RIS-assisted networks, i.e. location
uncertainty and imperfect phase estimation;

  * •

extensive simulations validate the accuracy of the derived closed-form
expressions independently from the network configuration, impairments, and
number of RIS elements.

The remainder of this paper is organized as follows. In Section II, the system
and signal propagation models are described. In Section III, the closed-form
expressions for the effective rate under impairments are derived. Section IV
provides insights upon the numerical evaluation of the derived expressions and
the predicted system performance under various conditions. Subsequently,
Section V concludes the paper.

Mathematical Functions and Notations: $j=\sqrt{-1}$, $\mathbf{K}_{n}(\cdot)$
is the modified Bessel function of the second kind [12, Eq. (5.52)].
$\Gamma(L)$ is the gamma function [12, Eq. (8.310.1)], $G_{p,q}^{m,n}[\cdot]$
is the univariate Meijer’s $G$-function111It is worth mentioning that the
univariate Meijer’s $G$-function is a built-in function and available at
MATLAB, i.e., $\text{meijerG}(a,b,c,d,x)$, Mathematica, and Maple. It can be
flexibly transformed into many functions, e.g. the exponential, logarithm,
incomplete gamma functions, etc. More specifically, it has been proved useful
and generic when analyzing outage probability, bit error rate, channel
capacity, secrecy outage probability, etc. Interested readers can find
examples in [13, 14]. [12, Eq. (9.301)]. $H_{p,q}^{m,n}[\cdot]$ is the Fox’s
$H$-function [15, Eq. (8.3.1.1)]. $\mathbb{E}[x]$ is the expectation operator
over a random variable (RV) $x$.

##  II System model and preliminaries

Figure 1: The system model.

We consider a single-input single-output (SISO) communication system assisted
by a single RIS, embedded on the surface of a building, shown in Fig. 1.
Specifically, the source node (S) transmits signal $s$, which is reflected by
RIS toward the destination node (D). The RIS comprises $L$ passive and low-
cost reflecting elements. We assume that the direct links are much weaker than
the reflection-based ones due to obstacles and fading effects [16]. Thus, the
direct links are omitted in our investigation. In addition, the exact location
of D and its distance to RIS might be unknown. As such, D is assumed to be
randomly located within a ring centered at the RIS’s position according to
homogeneous Poisson point processes (HPPP), where the ring outer radius
$R_{2}$ indicates its coverage region, and the inner radius $R_{1}$ depicts
the minimum distance.

The instantaneous received signal at D is given by

$y_{D}=\sqrt{P}\left|\sum\limits_{l=1}^{L}h_{l}u_{l}g_{l}\right|s+\mathfrak{n},$
(1)

where $P$ is the transmit power, $h_{l}=\mathfrak{h}_{l}\exp(-j\phi_{l})$ and
$g_{l}=\mathfrak{g}_{l}\exp(-j\psi_{l})$ are independent and identically
distributed (i.i.d.) Rayleigh RVs, which correspond to the channel
coefficients from S to the $l$-th reflector element and the $l$-th reflector
element to D, respectively. $\mathfrak{h}_{l}$, $\mathfrak{g}_{l}$,
$\phi_{l}$, and $\psi_{l}$ denote the amplitudes and phases of the
corresponding fading channel gains. $u_{l}=w_{l}(\theta_{l})\exp(j\theta_{l})$
is the reflection coefficient produced by the $l$-th element of the RIS,
herein $w_{l}(\theta_{l})=1$ for the ideal phase shifts, $l=1,\cdots,L$. $s$
is the transmitted signal with unit energy. $\mathfrak{n}$ is the additive
white Gaussian noise (AWGN) with zero mean and $\sigma^{2}$ variance.

Based on (1), the signal-to-noise ratio (SNR) observed at D is obtained as

$\displaystyle\gamma_{D}$
$\displaystyle=\frac{P}{d_{SR}^{\delta}d_{RD}^{\delta}\sigma^{2}}\left|\sum\limits_{l=1}^{L}\mathfrak{h}_{l}\mathfrak{g}_{l}\exp\big{(}j(\theta_{l}-\phi_{l}-\psi_{l})\big{)}\right|^{2}$
$\displaystyle=\frac{\rho}{d_{SR}^{\delta}d_{RD}^{\delta}}\left|\sum\limits_{l=1}^{L}\mathfrak{h}_{l}\mathfrak{g}_{l}\exp(j\epsilon_{l})\right|^{2},$
(2)

where $\rho=\frac{P}{\sigma^{2}}$ and $\delta$ is the path loss exponent.
$d_{SR}$ and $d_{RD}$ denote the distances from S to RIS and from RIS to D,
respectively. The phase estimation error $\epsilon_{l}$ is assumed to be an
i.i.d. RV following the uniform distribution. Here, S may employ a cheap off-
the-shelf hardware, which has a very poor phase estimation accuracy, i.e.,
$\epsilon_{l}\sim U(-\pi,\pi)$. For simplicity of notations, we fix $d_{SR}$
in our system setup, and let $\bar{\gamma}=\frac{\rho}{d_{SR}^{\delta}}$.
Next, we would like to determine the probability density of D, which is needed
for the calculation of the effective rate.

######  Proposition 1.

The probability density function (PDF) of $\gamma_{D}$ is given by (7), shown
at the top of next page.

$\displaystyle
f_{D}(\gamma)=\frac{1}{2\delta\bar{\gamma}\Gamma(L)(R_{2}^{2}-R_{1}^{2})}\left[R_{2}^{2+\delta}G_{1,3}^{2,1}\left[{\frac{R_{2}^{\delta}\gamma}{4\bar{\gamma}}\left|{\begin{array}[]{*{20}c}{-\frac{2}{\delta}}\\\
{L-1,0,-1-\frac{2}{\delta}}\\\
\end{array}}\right.}\right]-R_{1}^{2+\delta}G_{1,3}^{2,1}\left[{\frac{R_{1}^{\delta}\gamma}{4\bar{\gamma}}\left|{\begin{array}[]{*{20}c}{-\frac{2}{\delta}}\\\
{L-1,0,-1-\frac{2}{\delta}}\\\ \end{array}}\right.}\right]\right],$ (7)

###### Proof.

Let
$X=\left|\sum\limits_{l=1}^{L}\mathfrak{h}_{l}\mathfrak{g}_{l}\exp(j\epsilon_{l})\right|^{2}$,
using the result in [16, Eq. (43)], the cumulative distribution function (CDF)
of $X$ is given by

$\displaystyle
F_{X}(x)=1-\frac{2^{1-L}}{\Gamma(L)}\left(\sqrt{x}\right)^{L}\mathbf{K}_{L}\left(\sqrt{x}\right).$
(8)

Next, by taking the derivative of (8) with respect to $x$, and then using
$\frac{d}{dz}[z^{L}\mathbf{K}_{L}(z)]=-z^{L}\mathbf{K}_{L-1}(z)$ from [12, Eq.
(8.486.14)], we obtain the PDF of $X$ with

$\displaystyle f_{X}(x)$
$\displaystyle=\frac{1}{2^{L}\Gamma(L)}\left(\sqrt{x}\right)^{L-1}\mathbf{K}_{L-1}\left(\sqrt{x}\right)$
$\displaystyle\mathop{=}^{(a)}\frac{1}{4\Gamma(L)}G_{0,2}^{2,0}\left[{\frac{x}{4}\left|{\begin{array}[]{*{20}c}{-}\\\
{L-1,0}\\\ \end{array}}\right.}\right],$ (11)

where step $(a)$ is developed using the technique [12, Eq. (9.34.3)]. With
[17, Eq. (28)] in mind, the PDF of $d_{RD}$ is given by

$\displaystyle f_{d}(r)=\frac{2r}{R_{2}^{2}-R_{1}^{2}},\quad R_{1}\leq r\leq
R_{2},$ (12)

Correspondingly, the PDF of $Y=d_{RD}^{\delta}$ is

$\displaystyle
f_{Y}(y)=\frac{2y^{\frac{2}{\delta}-1}}{\delta(R_{2}^{2}-R_{1}^{2})},\quad
R_{1}^{\delta}\leq r\leq R_{2}^{\delta}.$ (13)

Subsequently, plugging (II) and (13) into $\gamma_{D}=\bar{\gamma}\frac{X}{Y}$
leads to

$\displaystyle
f_{D}(\gamma)=\frac{1}{\bar{\gamma}}\int_{R_{1}^{\delta}}^{R_{2}^{\delta}}yf_{X}\left(\frac{\gamma
y}{\bar{\gamma}}\right)f_{Y}(y)dy.$ (14)

Then, using [15, Eq. (1.16.4)], we derive (3) and the proof is achieved. ∎

##  III Effective rate

The effective rate is a dual concept connecting both physical layer and link
layer. It is widely used to determine the maximal effective bandwidth that the
system can support for a given delay constraint [18]. Assuming the block
fading channel for our system configuration, the normalized effective rate at
D is analytically given by

$\mathcal{R}=-\frac{1}{A}\log_{2}\underbrace{\left[\mathbb{E}\left((1+\gamma_{M})^{-A}\right)\right]}_{\mathcal{M}}\text{bit/s/Hz},$
(15)

where $A=\frac{\theta TB}{\ln 2}$, where $\theta$, $T$, and $B$ represent the
asymptotic decay rate of the buffer occupancy, the block length, and the
system bandwidth, respectively.

######  Proposition 2.

Considering the existence of phase estimation error and unknown exact
location, the effective rate for the given system configuration is given by

$\displaystyle\mathcal{R}=-\frac{1}{A}\log_{2}\left[\frac{\mathcal{W}}{2\delta\bar{\gamma}\Gamma(L)\Gamma(A)(R_{2}^{2}-R_{1}^{2})}\right],$
(16)

where

$\displaystyle\mathcal{W}=$ $\displaystyle
R_{2}^{2+\delta}G_{2,4}^{3,2}\left[{\frac{R_{2}^{\delta}}{4\bar{\gamma}}\left|{\begin{array}[]{*{20}c}{-\frac{2}{\delta},0}\\\
{L-1,0,A-1,-1-\frac{2}{\delta}}\\\ \end{array}}\right.}\right]$
$\displaystyle-
R_{1}^{2+\delta}G_{2,4}^{3,2}\left[{\frac{R_{1}^{\delta}}{4\bar{\gamma}}\left|{\begin{array}[]{*{20}c}{-\frac{2}{\delta},0}\\\
{L-1,0,A-1,-1-\frac{2}{\delta}}\\\ \end{array}}\right.}\right].$

###### Proof.

The proof starts by re-expressing $\frac{1}{(1+x)^{c}}$ in terms of Meijer’s
$G$-function [15, Chpt. 8.4]

$\displaystyle\frac{1}{(1+x)^{c}}$
$\displaystyle=\frac{1}{\Gamma(c)}G_{1,1}^{1,1}\left[{x\left|{\begin{array}[]{*{20}c}{1-c}\\\
{0}\\\ \end{array}}\right.}\right]$ (19)
$\displaystyle\mathop{=}^{(b)}\frac{1}{\Gamma(c)}H_{1,1}^{1,1}\left[{x\left|{\begin{array}[]{*{20}c}{(1-c,1)}\\\
{(0,1)}\\\ \end{array}}\right.}\right],$ (22)

where step $(b)$ is developed using the property of Fox’s $H$-function [15,
Eq. (8.3.2.21)]. Next, we insert (19) and (7) into $\mathcal{M}$, the
expression $\mathcal{M}$ can be easily stated as

$\displaystyle\mathcal{M}=$
$\displaystyle\frac{1}{2\delta\bar{\gamma}\Gamma(L)\Gamma(A)(R_{2}^{2}-R_{1}^{2})}\int_{0}^{\infty}G_{1,1}^{1,1}\left[{\gamma\left|{\begin{array}[]{*{20}c}{1-A}\\\
{0}\\\ \end{array}}\right.}\right]$ (25)
$\displaystyle\times\left[R_{2}^{2+\delta}G_{1,3}^{2,1}\left[{\frac{R_{2}^{\delta}\gamma}{4\bar{\gamma}}\left|{\begin{array}[]{*{20}c}{-\frac{2}{\delta}}\\\
{L-1,0,-1-\frac{2}{\delta}}\\\ \end{array}}\right.}\right]\right.$ (28)
$\displaystyle\left.\hskip 17.22217pt-
R_{1}^{2+\delta}G_{1,3}^{2,1}\left[{\frac{R_{1}^{\delta}\gamma}{4\bar{\gamma}}\left|{\begin{array}[]{*{20}c}{-\frac{2}{\delta}}\\\
{L-1,0,-1-\frac{2}{\delta}}\\\ \end{array}}\right.}\right]\right],$ (31)

subsequently performing the Mellin transform of the product of two Fox’s
$H$-function [15, Eq.(2.25.1)], the proof is eventually concluded. ∎

######  Remark 1.

Considering the phase estimation error alone, the effective rate for the given
system is given by

$\displaystyle\mathcal{R}=-\frac{1}{A}\log_{2}\left[\frac{1}{4\bar{\gamma}\Gamma(A)\Gamma(L)}G_{1,3}^{3,1}\left[{\frac{1}{4\bar{\gamma}}\left|{\begin{array}[]{*{20}c}{0}\\\
{L-1.0,A-1}\\\ \end{array}}\right.}\right]\right].$ (34)

###### Proof.

Fixing both $d_{SR}$ and $d_{RD}$, the derivation of (34) is based on the PDF
of $\gamma_{D}=\bar{\gamma}X$, which is obtained as follows

$\displaystyle
f_{D}(\gamma)=\frac{1}{\bar{\gamma}}f_{X}\left(\frac{\gamma}{\bar{\gamma}}\right).$
(35)

Next plugging (35) and (19) into $\mathcal{M}$, and following the similar line
as in (25), the proof is finished. ∎

######  Remark 2.

When $\rho\rightarrow\infty$, i.e., at high SNR regime, the asymptotic
behavior of ER given in (16) is further simplified as

$\displaystyle\mathcal{R}^{\infty}\approx-\frac{1}{A}\log_{2}\left(\frac{R_{2}^{2+\delta}-R_{1}^{2+\delta}}{\bar{\gamma}\left(2\delta+4\right)(L-1)(A-1)(R_{2}^{2}-R_{1}^{2})}\right).$
(36)

###### Proof.

When $\rho$ is at high SNR regime, both $\frac{R_{2}^{\delta}}{4\bar{\gamma}}$
and $\frac{R_{1}^{\delta}}{4\bar{\gamma}}$ tend to $0$. Thus, the expression
in (36) is obtained by expanding the Meijer’s $G$-function at $0$ and using
the fact given in [19]. ∎

##  IV Numerical Analysis

In this section, we provide numerical evaluations of the theoretical ER and
compare it with the results obtained using Monte Carlo simulations with
$10^{6}$ runs.

We start with the analysis of effective rate for different values of $\rho$
and $L$. For this, we assume $A=5.4$, $\delta=3.4$, $R_{1}=2$ m and $R_{2}=5$
m. The results are depicted in Fig. 2. We observe that the analytically
determined effective rate is close to the results obtained via extensive
simulations.

Figure 2: Effective rate $\mathcal{R}$ vs $\rho$ for selected $L$.

Furthermore, we observe that the effective rate increases with increasing
number of RIS elements. This is expected, since the spatial diversity
increases with increasing $L$, which can be used to improve the performance.
In addition, we observe that the results of the asymptotic approximation
deviate substantially from the simulation and theory at low SNR regime.
Specifically, the approximation underestimates the effective rate, especially
with a low number of RIS elements. However, with practical value of $L$, this
effect vanishes, such that the low complexity expression in (36) becomes
accurate even at low SNR.

(a) $d_{RD}=2$m (b) $d_{RD}=5$m

Figure 3: Effective rate $\mathcal{R}$ vs. $\rho$ (a) $d_{RD}=2$; and (b)
$d_{RD}=5$.

In Fig. 3, the results are depicted for three scenarios assuming $L=40$: (i)
without impacts from location and phase error uncertainty; (ii) with phase
error only; and (iii) with both uncertainties. We observe a substantial
performance degradation in terms of effective rate in presence of the phase
and location uncertainties. Also, the location uncertainty affects the
effective rate performance much less than the phase uncertainty.
Interestingly, the result obtained in [10] without taking into account these
practical impairments leads to a dramatic overestimation of the expected
system performance, which may even render the whole network design based on
such an idealistic effective rate prediction incorrect.

Fig. 4 investigates the impact of the coverage area on the effective rate.
Here, $\rho=5$ dB has been assumed. One can observe that the effective rate
decreases rapidly with increasing coverage area, since the average distance
between the user and RIS increases, thus increasing the path loss. In
practice, Fig. 4 being a theoretical proof, can be utilized to deduce e.g. the
required number of RIS elements in order to satisfy a given network demand
within a certain coverage area. Consequently, no complicated numerical
optimization is required for this design problem anymore.

Figure 4: Effective rate $\mathcal{R}$ vs. coverage area
$\pi(R_{2}^{2}-R_{1}^{2})$ for selected $L$.

Considering four different kinds of mobile radio environments [20, Chapter
3.9], Fig. 5 plots $\mathcal{R}$ against the decay rate $\theta$ for selected
values of path loss exponent $\delta$. For this, we fix $L=20$, $R_{2}=8$ m,
and $A=\frac{\theta TB}{\ln 2}$ with $TB=1$ while all other parameters remain
unchanged. Obviously, the effective rate demonstrates a performance
degradation as $\theta$ increases. In other words, a smaller delay constraint
results in a higher effective rate performance. Correspondingly, a better
quality of service can be supported for delay-tolerant services. Starting from
$\theta\approx 10^{0}$ 1/bit, the effective rate starts to drop to very low
values, i.e., the delay-sensitive communications services, such as video
conferencing or gaming, cannot be supported. In addition, we observe that for
a large value of $\delta$, i.e., in case of poor propagation conditions, the
performance drops significantly compared to low path loss exponents.

Figure 5: Effective rate $\mathcal{R}$ vs. $\theta$ for selected $\delta$.

##  V Conclusion

In this paper, we derived two novel closed-form expressions of effective rate
for RIS-assisted wireless networks under the impacts of location uncertainty
and phase errors. Our derived closed-form results are validated via Monte-
Carlo simulations and their accuracy is displayed to be high. We further
observed a performance degradation in case of increasing decay rate and
compared the system performance in different environments. Using the derived
expression, it is possible to facilitate the network designers with respect to
the coverage area, placement of RIS, etc. under realistic conditions of phase
errors and location uncertainty.

## Acknowledgment

This work has been supported by the Luxembourg National Research Fund (FNR)
projects, entitled Exploiting Interference for Physical Layer Security in 5G
Networks (CIPHY), and Reconfigurable Intelligent Surfaces for Smart Cities
(RISOTTI). This work was done while Long Kong was a research associate at the
University of Luxembourg. He is now a postdoctoral fellow at the University of
Ottawa.

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inal]shorthand [strwidth=3]labelname trwidth=2,strside=right]year

# Bottom of the $L^{2}$ spectrum of the Laplacian
on locally symmetric spaces

Jean-Philippe ANKER and Hong-Wei ZHANG  Jean-Philippe Anker:anker@univ-
orleans.fr
Institut Denis Poisson, Université d’Orléans, Université de Tours & CNRS,
Orléans, France  Hong-Wei<EMAIL_ADDRESS>
Department of Mathematics, Ghent University, Ghent, Belgium
Institut Denis Poisson, Université d’Orléans, Université de Tours & CNRS,
Orléans, France

###### Abstract.

We estimate the bottom of the $L^{2}$ spectrum of the Laplacian on locally
symmetric spaces in terms of the critical exponents of appropriate Poincaré
series. Our main result is the higher rank analog of a characterization due to
Elstrodt, Patterson, Sullivan and Corlette in rank one. It improves upon
previous results obtained by Leuzinger and Weber in higher rank.

###### Key words and phrases:

Locally symmetric space, $L^{2}$ spectrum, Poincaré series , critical
exponent, Green function, heat kernel

###### 2020 Mathematics Subject Classification:

22E40, 11N45, 35K08, 47A10, 58J50

In memory of Michel Marias (1953-2020), who introduced us to locally symmetric
spaces

##  1\. Introduction

We adopt the standard notation and refer to [Hel00] for more details. Let $G$
be a semi-simple Lie group, connected, noncompact, with finite center, and $K$
be a maximal compact subgroup of $G$. The homogeneous space $X=G/K$ is a
Riemannian symmetric space of noncompact type. Let
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be the Cartan decomposition of
the Lie algebra of $G$. The Killing form of $\mathfrak{g}$ induces a
$K$-invariant inner product $\langle\,.\,,\,.\,\rangle$ on $\mathfrak{p}$,
hence a $G$-invariant Riemannian metric on $G/K$. Fix a maximal abelian
subspace $\mathfrak{a}$ in $\mathfrak{p}$. We identify $\mathfrak{a}$ with its
dual $\mathfrak{a}^{*}$ by means of the inner product inherited from
$\mathfrak{p}$. Let $\Gamma$ be a discrete torsion-free subgroup of $G$ that
acts freely and properly discontinuously on $X$. Then $Y\\!=\Gamma\backslash
X$ is a locally symmetric space, whose Riemannian structure is inherited from
$X$. We denote by $d(\,.\,,\,.\,)$ the joint Riemannian distance on $X$ and
$Y$, by $n$ their joint dimension, and by $\ell$ their joint rank, which is
the dimension of $\mathfrak{a}$.

Let $\Sigma\subset\mathfrak{a}$ be the root system of
$(\mathfrak{g},\mathfrak{a})$ and let $W$ be the associated Weyl group. Choose
a positive Weyl chamber $\mathfrak{a}^{+}\subset\mathfrak{a}$ and let
$\Sigma^{+}{\subset\Sigma}$ be the corresponding subsystem of positive roots.
Denote by $\rho=\frac{1}{2}\sum_{\alpha\in\Sigma^{+}}m_{\alpha}\alpha$ the
half sum of positive roots counted with their multiplicities. Occasionally we
shall need the reduced root system
$\Sigma_{\text{red}}=\\{\alpha\in\Sigma\,|\,\frac{\alpha}{2}\notin\Sigma\\}$.

Consider the classical Poincaré series

$\displaystyle P_{s}(xK,yK)=\sum\nolimits_{\gamma\in\Gamma}e^{-sd(xK,\gamma
yK)}\qquad\forall\,s>0,\,\forall\,x,y\in G$ (1.1)

and denote by $\delta(\Gamma)=\inf\\{s>0\,|\,P_{s}(xK,yK)<+\infty\\}$ its
critical exponent, which is independent of $xK$ and $yK$. Recall that
$\delta(\Gamma)\in[0,2\hskip 0.56905pt\|\rho\|]$ may be also defined by

$\displaystyle\delta(\Gamma)=\limsup_{R\to+\infty}\frac{\log
N_{R}(xK,yK)}{R}\qquad\forall\,x,y\in G,$

where $N_{R}(xK,yK)=|\\{\gamma\in\Gamma\,|\,d(xK,\gamma yK)\leq R\\}|$ denotes
the orbital counting function. Finally, let $\Delta_{Y}$ be the Laplace-
Beltrami operator on $Y$ and let $\lambda_{0}(Y)$ be the bottom of the $L^{2}$
spectrum of $-\Delta_{Y}$. The following celebrated result, due to Elstrodt
([Els73], [Els73a], [Els74]), Patterson [Pat76], Sullivan [Sul87] and Corlette
[Cor90], expresses $\lambda_{0}(Y)$ in terms of $\rho$ and $\delta(\Gamma)$ in
rank one.

######  Theorem 1.1.

In the rank one case ($\ell=1$), we have

$\displaystyle\lambda_{0}(Y)=\begin{cases}\|\rho\|^{2}&\qquad\textnormal{if}\;\;0\leq\delta(\Gamma)\leq\|\rho\|,\\\
\|\rho\|^{2}-(\delta(\Gamma)-\|\rho\|)^{2}&\qquad\textnormal{if}\;\;\|\rho\|\leq\delta(\Gamma)\leq
2\hskip 0.56905pt\|\rho\|.\end{cases}$ (1.2)

This result was extended in higher rank as follows by Leuzinger [Leu04] and
Weber [Web08]. Let
$\rho_{\text{min}}=\min_{H\in\overline{\mathfrak{a}^{+}},\|H\|=1}\langle\rho,H\rangle\in(0,\|\rho\|]$.
Notice that $\rho_{\text{min}}={\|\rho\|}$ in rank one and thus the following
theorem reduces to Theorem 1.1.

######  Theorem 1.2.

In the general case ($\ell\geq 1$), the following estimates hold:

  * $\bullet$

Upper bound:

$\displaystyle\lambda_{0}(Y)\leq\begin{cases}\,\|\rho\|^{2}&\qquad\textnormal{if}\;\;0\leq\delta(\Gamma)\leq\|\rho\|,\\\
\,\|\rho\|^{2}-(\delta(\Gamma)-\|\rho\|)^{2}&\qquad\textnormal{if}\;\;\|\rho\|\leq\delta(\Gamma)\leq
2\hskip 0.56905pt\|\rho\|.\end{cases}$

  * $\bullet$

Lower bound:

$\displaystyle\lambda_{0}(Y)\geq\begin{cases}\,\|\rho\|^{2}&\qquad\textnormal{if}\;\;0\leq\delta(\Gamma)\leq\rho_{\text{\rm
min}},\\\ \,\max\hskip
1.42262pt\\{0,\|\rho\|^{2}-(\delta(\Gamma)-\rho_{\text{\rm
min}})^{2}\\}&\qquad\textnormal{if}\;\;\rho_{\text{\rm
min}}\leq\delta(\Gamma)\leq 2\hskip 0.56905pt\|\rho\|.\end{cases}$

In other terms,

$\displaystyle\begin{cases}\,\lambda_{0}(Y)=\|\rho\|^{2}&\textnormal{if}\;\;\delta(\Gamma)\in\bigl{[}\hskip
0.56905pt0,\rho_{\text{\rm min}}\bigr{]},\\\\[5.0pt]
\,\lambda_{0}(Y)\in\bigl{[}\hskip
0.56905pt\|\rho\|^{2}\\!-(\delta(\Gamma)-\rho_{\text{\rm
min}})^{2},\|\rho\|^{2}\hskip
0.56905pt\bigr{]}&\textnormal{if}\;\;\delta(\Gamma)\in\bigl{[}\hskip
0.85358pt\rho_{\text{\rm min}},\|\rho\|\bigr{]},\\\\[5.0pt]
\,\lambda_{0}(Y)\in\bigl{[}\hskip
0.56905pt\|\rho\|^{2}\\!-(\delta(\Gamma)-\rho_{\text{\rm
min}})^{2},\|\rho\|^{2}\\!-(\delta(\Gamma)-\|\rho\|)^{2}\hskip
0.56905pt\bigr{]}&\textnormal{if}\;\;\delta(\Gamma)\in\bigl{[}\hskip
0.56905pt\|\rho\|,\|\rho\|+\rho_{\text{\rm min}}\bigr{]},\\\\[5.0pt]
\,\lambda_{0}(Y)\in\bigl{[}\hskip
0.56905pt0,\|\rho\|^{2}\\!-(\delta(\Gamma)-\|\rho\|)^{2}\hskip
0.56905pt\bigr{]}&\textnormal{if}\;\;\delta(\Gamma)\in\bigl{[}\hskip
0.56905pt\|\rho\|+\rho_{\text{\rm min}},2\hskip 0.56905pt\|\rho\|\hskip
0.56905pt\bigr{]}.\end{cases}$

In this paper, we first improve the lower bound of $\lambda_{0}(Y)$ in Theorem
1.2 by a slight modification of the classical Poincaré series (1.1). Let
$\delta^{\hskip 0.56905pt\prime}{(\Gamma)}$ denote the critical exponent of
the modified Poincaré series (2.7) associated to the polyhedral distance
(2.1).

######  Theorem 1.3.

The following lower bound holds for the bottom $\lambda_{0}(Y)$ of the $L^{2}$
spectrum of $-\Delta$ on $Y\\!=\Gamma\backslash G/K$:

$\displaystyle\lambda_{0}(Y)\geq\begin{cases}\,\|\rho\|^{2}&\qquad\textnormal{if}\;\;0\leq\delta^{\hskip
0.56905pt\prime}{(\Gamma)}\leq\|\rho\|,\\\ \,\|\rho\|^{2}-(\delta^{\hskip
0.56905pt\prime}{(\Gamma)}-\|\rho\|)^{2}&\qquad\textnormal{if}\;\;\|\rho\|\leq\delta^{\hskip
0.56905pt\prime}{(\Gamma)}\leq 2\hskip 0.56905pt\|\rho\|.\end{cases}$

We obtain next a plain analog of Theorem 1.1 by considering a more involved
family of Poincaré series. We denote by $\delta^{\hskip
0.56905pt\prime\prime}{(\Gamma)}$ the critical exponent of $P^{\hskip
1.13809pt\prime\prime}_{s}(xK,yK)$, see (3.3).

######  Theorem 1.4.

The following characterization holds for the bottom $\lambda_{0}(Y)$ of the
$L^{2}$ spectrum of $-\Delta$ on $Y\\!=\Gamma\backslash G/K$:

$\displaystyle\lambda_{0}(Y)=\begin{cases}\,\|\rho\|^{2}&\qquad\textnormal{if}\;\;0\leq\delta^{\hskip
0.56905pt\prime\prime}{(\Gamma)}\leq\|\rho\|,\\\
\,\|\rho\|^{2}-(\delta^{\hskip
0.56905pt\prime\prime}{(\Gamma)}-\|\rho\|)^{2}&\qquad\textnormal{if}\;\;\|\rho\|\leq\delta^{\hskip
0.56905pt\prime\prime}{(\Gamma)}\leq 2\hskip 0.56905pt\|\rho\|.\end{cases}$
(1.3)

######  Remark 1.5.

If $\Gamma$ is a lattice, i.e., $Y\\!=\Gamma\backslash G/K$ has finite volume,
then $\lambda_{0}(Y)=0$ and $\delta^{\hskip
0.56905pt\prime\prime}(\Gamma)=2\hskip 0.56905pt\|\rho\|$, hence
$\delta^{\hskip 0.56905pt\prime}(\Gamma)=2\hskip 0.56905pt\|\rho\|$.
Furthermore $\delta(\Gamma)=2\hskip 0.56905pt\|\rho\|$ [Alb99, Theorem 7.4].
As pointed out by Corlette [Cor90] in rank one and by Leuzinger [Leu03] in
higher rank, if $G$ has Kazhdan’s property (T), then the following conditions
are actually equivalent:

(a) $\Gamma$ is a lattice, (b) $\lambda_{0}(Y)=0$, (c) $\delta(\Gamma)=2\hskip
0.56905pt\|\rho\|$, (d) $\delta^{\hskip 0.56905pt\prime\prime}(\Gamma)=2\hskip
0.56905pt\|\rho\|$.

######  Remark 1.6.

As for the Green function, the heat kernel

$\displaystyle h_{t}^{Y}(\Gamma xK,\Gamma
yK)=\sum\nolimits_{\gamma\in\Gamma}h_{t}{(Ky^{-1}\gamma^{-1}xK)}$

on a locally symmetric space $Y\\!=\Gamma\backslash G/K$ can be expressed and
estimated by using the heat kernel $h_{t}$ on the symmetric space $X=G/K$,
whose behavior is well understood [AnJi99, AnOs03]. By adapting
straightforwardly the methods carried out in [DaMa88, Web08], and by applying
Theorem 1.4 instead of Theorem 1.1 and Theorem 1.2, we refine the Gaussian
bounds of $h_{t}^{Y}$ and get rid in particular of $\rho_{\text{\rm min}}$.
The following estimates hold for all $t>0$ and all $x,y\in G$:
(i) Assume that $\delta^{\hskip 0.56905pt\prime\prime}{(\Gamma)}<\|\rho\|$ and
let $\delta^{\hskip 0.56905pt\prime\prime}{(\Gamma)}<s<\|\rho\|$. Then

$\displaystyle h_{t}^{Y}(\Gamma xK,\Gamma yK)\lesssim
t^{-\frac{n}{2}}\,(1+t)^{\frac{n-D}{2}}\,e^{-\|\rho\|^{2}t}\,e^{-\frac{d(\Gamma
xK,\Gamma yK)^{2}}{4t}}\,P^{\hskip 1.13809pt\prime\prime}_{s}(xK,yK).$

(ii) Assume that $\|\rho\|\leq\delta^{\hskip
0.56905pt\prime\prime}{(\Gamma)}<2\hskip 0.56905pt\|\rho\|$ and let
$\delta^{\hskip
0.56905pt\prime\prime}{(\Gamma)}-\|\rho\|<s_{1}<s_{2}<\|\rho\|$. Then

$\displaystyle h_{t}^{Y}(\Gamma xK,\Gamma yK)\lesssim
t^{-\frac{n}{2}}\,e^{-\left(\|\rho\|^{2}-s_{2}^{2}\right)t}\,P^{\hskip
1.13809pt\prime\prime}_{\|\rho\|+s_{1}}(xK,yK).$

(iii) Assume that $\delta^{\hskip 0.56905pt\prime\prime}{(\Gamma)}<2\hskip
0.56905pt\|\rho\|$. Let $s>\delta^{\hskip 0.56905pt\prime\prime}{(\Gamma)}$
and $\varepsilon>0$. Then

$\displaystyle h_{t}^{Y}(\Gamma xK,\Gamma yK)\lesssim
t^{-\frac{n}{2}}\,e^{-\left({\|\rho\|^{2}-(\delta^{\hskip
0.56905pt\prime\prime}(\Gamma)-\|\rho\|)^{2}}-2\varepsilon\right)t}\,e^{-\frac{d(\Gamma
xK,\Gamma yK)^{2}}{4(1+\varepsilon)t}}\,P^{\hskip
1.13809pt\prime\prime}_{s}(xK,xK)^{\frac{1}{2}}P^{\hskip
1.13809pt\prime\prime}_{s}(yK,yK)^{\frac{1}{2}}.$

##  2\. First improvement

In this section, we replace the Riemannian distance $d$ on $X$ by a polyhedral
distance

$\displaystyle\textstyle d^{\hskip
0.85358pt\prime}(xK,yK)=\langle\frac{\rho}{\|\rho\|},(y^{-1}x)^{+}\rangle\qquad\forall\,x,y\in
G,$ (2.1)

where $(y^{-1}x)^{+}$ denotes the $\overline{\mathfrak{a}^{+}}$-component of
$y^{-1}x$ in the Cartan decomposition $G=K(\exp\overline{\mathfrak{a}^{+}})K$.
The corresponding balls, which reflect the volume growth of $X$ at infinity,
played an important role in [Ank90] and [Ank91]. More general polyhedral sets
were considered in [Ank92] and [AAS10].

######  Proposition 2.1.

$d^{\hskip 0.85358pt\prime}$ is a $G$-invariant distance on $X$.

###### Proof.

Notice first that (2.1) descends from $G\times{G}$ to $X\times{X}$, as the map
$z\mapsto{z^{+}}$ is $K$-bi-invariant on ${G}$. The $G$-invariance of
$d^{\hskip 0.85358pt\prime}$ is straightforward from the definition (2.1). The
symmetry

$\displaystyle d^{\hskip 0.85358pt\prime}(xK,yK)=d^{\hskip
0.85358pt\prime}(yK,xK)$

follows from

$\displaystyle(y^{-1})^{+}=-w_{0}.y^{+}\quad\text{and}\quad-w_{0}.\rho=\rho,$
(2.2)

where $w_{0}$ denotes the longest element in the Weyl group. Let us check the
triangular inequality

$\displaystyle d^{\hskip 0.85358pt\prime}(xK,yK)\leq d^{\hskip
0.85358pt\prime}(xK,zK)+d^{\hskip 0.85358pt\prime}(zK,yK).$

By $G$-invariance, we may reduce to the case where $zK=eK$. According to Lemma
2.2 below,

$\displaystyle x^{+}+(y^{-1})^{+}-(y^{-1}x)^{+}$

belongs to the cone generated by the positive roots. ∎

######  Lemma 2.2.

For every $x,y\in G$, we have the following inclusion

$\displaystyle\operatorname{co}[W.(xy)^{+}]\subset\operatorname{co}[W.(x^{+}+y^{+})]$
(2.3)

between convex hulls.

###### Proof.

The inclusion (2.3) amounts to the fact that

$\displaystyle x^{+}+y^{+}-(xy)^{+}$

belongs to the cone generated by the positive roots or, equivalently, to the
inequality

$\displaystyle\langle\lambda,(xy)^{+}\rangle\leq\langle\lambda,x^{+}\rangle+\langle\lambda,y^{+}\rangle\qquad\forall\,\lambda\in\overline{\mathfrak{a}^{+}}.$
(2.4)

It is enough to prove (2.4) for all highest weights $\lambda$ of irreducible
finite-dimensional complex representations $\pi:G\longrightarrow\text{GL}(V)$
with $K$-fixed vectors. According to Weyl’s unitary trick (see for instance
[Kna02, Proposition 7.15]), there exists an inner product on $V$ such that

$\displaystyle\textstyle\begin{cases}\pi(k)\text{ is unitary}&{\forall}\,k\in
K,\\\ \pi(a)\text{ is self-
adjoint}&{\forall}\,a\in\exp\mathfrak{a}.\end{cases}$

As $\lambda$ is the highest weight of $\pi$, then

$\displaystyle
e^{\langle\lambda,(xy)^{+}\rangle}=\|\pi(xy)\|\leq\|\pi(x)\|\|\pi(y)\|=e^{\langle\lambda,x^{+}\rangle}e^{\langle\lambda,y^{+}\rangle}=e^{\langle\lambda,x^{+}+y^{+}\rangle}.$

∎

######  Remark 2.3.

The distance $d^{\hskip 0.85358pt\prime}$ is comparable to the Riemannian
distance $d$. Specifically,

$\displaystyle\textstyle\frac{\rho_{\text{min}}}{\|\rho\|}d(xK,yK)\leq
d^{\hskip 0.85358pt\prime}(xK,yK)\leq d(xK,yK)\qquad\forall\,x,y\in G.$ (2.5)

This follows indeed from

$\displaystyle\textstyle\frac{\rho_{\text{min}}}{\|\rho\|}\|H\|\leq\langle\frac{\rho}{\|\rho\|},H\rangle\leq\|H\|\qquad\forall\,H\in\overline{\mathfrak{a}^{+}}.$

The volume of balls

$\displaystyle B^{\hskip 0.56905pt\prime}_{r}(xK)=\\{yK\in X\,|\,d^{\hskip
0.85358pt\prime}(yK,xK)\leq r\\}$

was determined in [Ank90, Lemma 6]. For the reader’s convenience, we recall
the statement and its proof.

######  Lemma 2.4.

For every $x\in G$ and $r>0$, we have 111The symbol $f\asymp g$ between two
non-negative expressions means that there exist constants $0<A\leq B<+\infty$
such that $Ag\leq f\leq Bg$.

$\displaystyle|B^{\hskip
0.56905pt\prime}_{r}(xK)|\asymp\begin{cases}\,r^{n}&\qquad\textnormal{if}\;\;0<r<1,\\\
\,r^{\ell-1}e^{2\hskip 0.56905pt\|\rho\|r}&\qquad\textnormal{if}\;\;r\geq
1.\end{cases}$

######  Remark 2.5.

Notice the different large scale behavior, in comparison with the classical
ball volume

$\displaystyle|B_{r}(xK)|\asymp\begin{cases}\,r^{n}&\qquad\textnormal{if}\;\;0<r<1,\\\
\,r^{\frac{\ell-1}{2}}e^{2\hskip
0.56905pt\|\rho\|r}&\qquad\textnormal{if}\;\;r\geq 1,\end{cases}$

see for instance [Str81] or [Kni97].

###### Proof.

By translation invariance, we may assume that $x=e$. Recall the integration
formula

$\displaystyle\int_{X}dx\,f(x)=\operatorname{const.}\int_{K}dk\
\int_{\mathfrak{a}^{+}}dH\,\omega(H)\,f(k(\exp H)K),$ (2.6)

in the Cartan decomposition, with density

$\displaystyle\omega(H)=\prod_{\alpha\in\Sigma^{+}}(\sinh\langle\alpha,H\rangle)^{m_{\alpha}}\asymp\prod_{\alpha\in\Sigma^{+}}\\!\left(\tfrac{\langle\alpha,H\rangle}{1+\langle\alpha,H\rangle}\right)^{m_{\alpha}}e^{2\langle\rho,H\rangle},$

since $\sinh{t}\sim\tfrac{t}{1+t}e^{t}$ for all $t\geq 0$. Thus

$\displaystyle|B^{\hskip
0.56905pt\prime}_{r}(eK)|\,=\,\operatorname{const.}\int_{\\{H\in\mathfrak{a}^{+}|\langle\rho,H\rangle\leq\|\rho\|r\\}}dH\,\omega(H)\,\asymp\,\int_{\\{H\in\mathfrak{a}^{+}|\|H\|\leq
r\\}}dH\,\prod_{\alpha\in\Sigma^{+}}\langle\alpha,H\rangle^{m_{\alpha}}\asymp
r^{n}$

if $r$ is small. Let us turn to $r$ large. On the one hand, we estimate from
above

$\displaystyle|B^{\hskip
0.56905pt\prime}_{r}(eK)|\lesssim\int_{\\{H\in\mathfrak{a}^{+}|\langle\rho,H\rangle\leq\|\rho\|r\\}}dH\,e^{2\langle\rho,H\rangle}\asymp\int_{0}^{2\hskip
0.56905pt\|\rho\|r}ds\,s^{\ell-1}e^{s}\asymp r^{\ell-1}e^{2\|\rho\|r}.$

On the other hand, let $H_{0}\in\mathfrak{a}^{+}$. As

$\displaystyle\omega(H)\asymp e^{2\langle\rho,H\rangle}\qquad\forall\,H\in
H_{0}+\overline{\mathfrak{a}^{+}},$

we estimate from below

$\displaystyle|B^{\hskip 0.56905pt\prime}_{r}(eK)|\gtrsim\int_{\\{H\in
H_{0}+\mathfrak{a}^{+}|\langle\rho,H\rangle\leq\|\rho\|r\\}}dH\,e^{2\langle\rho,H\rangle}\gtrsim\int_{C_{0}}^{2\hskip
0.56905pt\|\rho\|r}ds\,s^{\ell-1}e^{s}\asymp r^{\ell-1}e^{2\hskip
0.56905pt\|\rho\|r},$

where $C_{0}>0$ is a constant depending on $H_{0}$. ∎

Consider now the modified Poincaré series

$\displaystyle P^{\hskip
1.13809pt\prime}_{s}(xK,yK)=\sum\nolimits_{\gamma\in\Gamma}e^{-sd^{\hskip
0.85358pt\prime}(xK,\gamma yK)}\qquad\forall\,s>0,\,\forall\,x,y\in G$ (2.7)

associated with $d^{\hskip 0.85358pt\prime}$, its critical exponent

$\displaystyle\delta^{\hskip 0.56905pt\prime}(\Gamma)=\inf\\{s>0\,|\,P^{\hskip
1.13809pt\prime}_{s}(xK,yK)<+\infty\\}$ (2.8)

and the modified orbital counting function

$\displaystyle N_{R}^{\hskip
0.56905pt\prime}(xK,yK)=|\\{\gamma\in\Gamma\,|\,d^{\hskip
0.85358pt\prime}(xK,\gamma yK)\leq R\\}|\qquad\forall\,R\geq
0,\,\forall\,x,y\in G.$ (2.9)

The following proposition shows that (2.7), (2.8) and (2.9) share the
properties of their classical counterparts.

######  Proposition 2.6.

The following assertions hold:

  1. (i)

$\delta^{\hskip 0.56905pt\prime}{(\Gamma)}$ does not depend on the choice of
$x$ and $y$.

  2. (ii)

$0\leq\delta^{\hskip 0.56905pt\prime}{(\Gamma)}\leq 2\hskip
0.56905pt\|\rho\|$.

  3. (iii)

For every $x,y\in G$,

$\displaystyle\delta^{\hskip
0.56905pt\prime}(\Gamma)=\limsup_{R\to+\infty}\frac{\log N_{R}^{\hskip
0.56905pt\prime}(xK,yK)}{R}.$ (2.10)

######  Remark 2.7.

It follows from (2.5) that

$\displaystyle P_{s}(xK,yK)\leq P^{\hskip 1.13809pt\prime}_{s}(xK,yK)\leq
P_{\frac{\rho_{\text{min}}}{\|\rho\|}s}(xK,yK)$

and

$\displaystyle N_{R}(xK,yK)\leq N_{R}^{\hskip 0.56905pt\prime}(xK,yK)\leq
N_{\frac{\|\rho\|}{\rho_{\text{min}}}R}(xK,yK).$

Hence

$\displaystyle 0\leq\delta{(\Gamma)}\leq\delta^{\hskip
0.56905pt\prime}{(\Gamma)}\leq\frac{\|\rho\|}{\rho_{\text{min}}}\,\delta{(\Gamma)}.$

###### Proof.

(i) follows from the triangular inequality. More precisely, let
$x_{1},y_{1},x_{2},y_{2}\in G$ and $s>0$. Then

$\displaystyle d^{\hskip 0.85358pt\prime}(x_{2}K,\gamma y_{2}K)\leq d^{\hskip
0.85358pt\prime}(x_{2}K,x_{1}K)+d^{\hskip 0.85358pt\prime}(x_{1}K,\gamma
y_{1}K)+\underbrace{d^{\hskip 0.85358pt\prime}(\gamma y_{1}K,\gamma
y_{2}K)}_{d^{\hskip
0.85358pt\prime}(y_{1}K,\,y_{2}K)}\quad\forall\,\gamma\in\Gamma,$

hence

$\displaystyle\underbrace{\sum\nolimits_{\gamma\in\Gamma}e^{-s\,d^{\hskip
0.85358pt\prime}(x_{1}K,\gamma y_{1}K)}}_{P^{\hskip
1.13809pt\prime}_{s}(x_{1}K,y_{1}K)}\leq e^{s\\{d^{\hskip
0.85358pt\prime}(x_{1}K,x_{2}K)+d^{\hskip
0.85358pt\prime}(y_{1}K,y_{2}K)\\}}\underbrace{\sum\nolimits_{\gamma\in\Gamma}e^{-s\,d^{\hskip
0.85358pt\prime}(x_{2}K,\gamma y_{2}K)}}_{P^{\hskip
1.13809pt\prime}_{s}(x_{2}K,y_{2}K)}.$

(ii) According to (i), let us show, without loss of generality, that
$P^{\hskip 1.13809pt\prime}_{s}(eK,eK)<+\infty$ for every $s>2\hskip
0.56905pt\|\rho\|$. According to Lemma 2.9 below, there exists $r>0$ such that
the balls $B^{\hskip 0.56905pt\prime}_{r}(\gamma K)$, with $\gamma\in\Gamma$,
are pairwise disjoint in $G/K$. Let us apply the integration formula (2.6) to
the function

$\displaystyle f_{s}(xK)=\sum\nolimits_{\gamma\in\Gamma}e^{-sd^{\hskip
0.85358pt\prime}(\gamma{K},eK)}\mathbf{1}_{B^{\hskip
0.56905pt\prime}_{r}(\gamma K)}(xK).$

On the one hand, as

$\displaystyle\bigl{|}d^{\hskip 0.85358pt\prime}(xK,eK)-d^{\hskip
0.85358pt\prime}(\gamma K,eK)\bigr{|}\leq r\qquad\forall\,xK\in B^{\hskip
0.56905pt\prime}_{r}(\gamma K),$

we have

$\displaystyle\int_{X}d(xK)\,f_{s}(xK)\asymp\sum\nolimits_{\gamma\in\Gamma}e^{-sd^{\hskip
0.85358pt\prime}(\gamma K,eK)}\underbrace{|B^{\hskip
0.56905pt\prime}_{r}(\gamma K)|}_{|B^{\hskip 0.56905pt\prime}_{r}(eK)|}\asymp
P^{\hskip 1.13809pt\prime}_{s}(eK,eK).$

On the other hand,

$\displaystyle\int_{X}d(xK)\,f_{s}(xK)$
$\displaystyle\lesssim\int_{X}d(xK)\,e^{-sd^{\hskip 0.85358pt\prime}(xK,eK)}$
$\displaystyle\asymp\int_{\mathfrak{a}^{+}}dH\,\omega(H)\,e^{-s\langle\frac{\rho}{\|\rho\|},H\rangle}\lesssim\int_{\mathfrak{a}^{+}}dH\,e^{-(\frac{s}{\|\rho\|}-2)\langle\rho,H\rangle}$

is finite if $s>2\hskip 0.56905pt\|\rho\|$. Thus $P^{\hskip
1.13809pt\prime}_{s}(eK,eK)<+\infty$ in that case, and consequently
$\delta^{\hskip 0.56905pt\prime}(\Gamma)\leq 2\hskip 0.56905pt\|\rho\|$.

(iii) Denote the right hand side of (2.10) by $L(xK,yK)$ and let us first show
that $L(xK,yK)$ is finite. By applying Lemma 2.9 below to $y^{-1}\Gamma y$, we
deduce that there exists $r>0$ such that the balls $B^{\hskip
0.56905pt\prime}_{r}(\gamma yK)$, with $\gamma\in\Gamma$, are pairwise
disjoint. Set

$\displaystyle\Gamma_{\\!R}^{\hskip
0.56905pt\prime}(xK,yK)=\\{\gamma\in\Gamma\,|\,d^{\hskip
0.85358pt\prime}(xK,\gamma yK)\leq R\\}\qquad\forall\,R\geq
0,\,\forall\,x,y\in G.$

Then the ball $B^{\hskip 0.56905pt\prime}_{R+r}(xK)$ contains the disjoint
balls $B^{\hskip 0.56905pt\prime}_{r}(\gamma yK)$, with
$\gamma\in\Gamma_{\\!R}^{\hskip 0.56905pt\prime}(xK,$ $yK)$. By computing
volumes, we estimate

$\displaystyle N_{R}^{\hskip 0.56905pt\prime}(xK,yK)=|\Gamma_{\\!R}^{\hskip
0.56905pt\prime}(xK,yK)|\leq\frac{|B^{\hskip
0.56905pt\prime}_{R+r}(xK)|}{|B^{\hskip
0.56905pt\prime}_{r}(eK)|}\asymp(1+R)^{\ell-1}e^{2\hskip 0.56905pt\|\rho\|R}.$

Hence $L(xK,yK)\leq 2\hskip 0.56905pt\|\rho\|$. Let us next show that
$L(xK,yK)$ is actually independent of $x,y\in G$. Given
$x_{1},y_{1},x_{2},y_{2}\in G$ and $R_{1}>0$, let

$\displaystyle R_{2}=R_{1}+d^{\hskip 0.85358pt\prime}(x_{1}K,x_{2}K)+d^{\hskip
0.85358pt\prime}(y_{1}K,y_{2}K).$

Then the triangular inequality

$\displaystyle d^{\hskip 0.85358pt\prime}(x_{2}K,\gamma y_{2}K)\leq d^{\hskip
0.85358pt\prime}(x_{2}K,x_{1}K)+d^{\hskip 0.85358pt\prime}(x_{1}K,\gamma
y_{1}K)+\underbrace{d^{\hskip 0.85358pt\prime}(\gamma y_{1}K,\gamma
y_{2}K)}_{d^{\hskip 0.85358pt\prime}(y_{1}K,\,y_{2}K)}$

implies successively

$\displaystyle\Gamma_{R_{1}}^{\hskip 0.56905pt\prime}(x_{1}K,y_{1}K)$
$\displaystyle\subset\Gamma_{R_{2}}^{\hskip 0.56905pt\prime}(x_{2}K,y_{2}K),$
$\displaystyle N_{R_{1}}^{\hskip 0.56905pt\prime}(x_{1}K,y_{1}K)$
$\displaystyle\leq N_{R_{2}}^{\hskip 0.56905pt\prime}(x_{2}K,y_{2}K),$
$\displaystyle L(x_{1}K,y_{1}K)$ $\displaystyle\leq L(x_{2}K,y_{2}K).$

Let us finally prove the equality between $\delta^{\hskip
0.56905pt\prime}(\Gamma)$ and $L=L(eK,eK)$. For this purpose, observe that

$\displaystyle P^{\hskip 1.13809pt\prime}_{s}$
$\displaystyle=1+\sum\nolimits_{R\in\mathbb{N}^{*}}\sum\nolimits_{\gamma\in\Gamma_{\\!R}^{\hskip
0.56905pt\prime}\smallsetminus\hskip 0.56905pt\Gamma_{\\!R-1}^{\hskip
0.56905pt\prime}}\\!e^{-s\,d^{\hskip 0.85358pt\prime}(eK,\gamma K)}$
$\displaystyle\asymp 1+\sum\nolimits_{R\in\mathbb{N}^{*}}\bigl{(}N_{R}^{\hskip
0.56905pt\prime}-N_{R-1}^{\hskip 0.56905pt\prime}\bigr{)}\,e^{-sR}$
$\displaystyle\asymp\sum\nolimits_{R\in\mathbb{N}}N_{R}^{\hskip
0.56905pt\prime}\,e^{-sR},$ (2.11)

where we have written for simplicity

$\displaystyle P^{\hskip 1.13809pt\prime}_{s}=P^{\hskip
1.13809pt\prime}_{s}(eK,eK),\quad\Gamma_{\\!R}^{\hskip
0.56905pt\prime}=\Gamma_{\\!R}^{\hskip
0.56905pt\prime}(eK,eK)\quad\text{and}\quad N_{R}^{\hskip
0.56905pt\prime}=N_{R}^{\hskip 0.56905pt\prime}(eK,eK).$

One the one hand, let $s>L$ and set $\varepsilon=\frac{s-L}{2}$. By definition
of $L$,

$\displaystyle N_{R}^{\hskip 0.56905pt\prime}\lesssim
e^{(L+\varepsilon)R}\qquad\forall\,R\geq 0.$

Hence

$\displaystyle P^{\hskip
1.13809pt\prime}_{s}\lesssim\sum\nolimits_{R\in\mathbb{N}}e^{-\varepsilon
R}<+\infty.$

One the other hand, let $s<L$. By definition of $L$, there exists a sequence
of integers $1<R_{1}<R_{2}<\cdots\to+\infty$ such that

$\displaystyle N_{R_{j}}^{\hskip 0.56905pt\prime}\geq
e^{sR_{j}}\qquad\forall\,j\in\mathbb{N}^{*}.$

Hence the series (2.11) diverges. ∎

######  Remark 2.8.

Here is an example where $\delta<\delta^{\hskip 0.85358pt\prime}$. Consider
the product

$\Gamma\hskip 1.42262pt\backslash\hskip 0.85358ptG/K=(\hskip
0.85358pt\Gamma_{1}\backslash\hskip 0.85358ptG_{\hskip 0.28453pt\hskip
0.28453pt1}/K_{1})\times(\hskip 0.85358pt\Gamma_{2}\backslash\hskip
0.85358ptG_{\hskip 0.85358pt2\hskip 0.28453pt}/K_{2})$

of two locally symmetric spaces of rank one, with parameters $\rho_{1}$,
$\delta_{1}$ and $\rho_{2}$ , $\delta_{2}$ . Then

$\delta\leq\sqrt{\hskip 0.28453pt\delta_{1}^{\hskip
0.85358pt2}\\!+\delta_{2}^{\hskip 0.85358pt2}\hskip 1.42262pt}$ (2.12)

and

$\delta^{\hskip 0.85358pt\prime}\geq\sqrt{\rho_{1}^{\hskip
0.85358pt2}\\!+\rho_{2}^{\hskip 0.85358pt2}\hskip
0.85358pt}\,\max\,\bigl{\\{}\tfrac{\delta_{1}}{\rho_{1}},\tfrac{\delta_{2}}{\rho_{2}}\bigr{\\}}\hskip
1.42262pt.$ (2.13)

Hence $\delta<\delta^{\hskip 0.85358pt\prime}$ if
$\tfrac{\delta_{1}}{\rho_{1}}\neq\tfrac{\delta_{2}}{\rho_{2}}$ . Notice that
there are plenty of such products, starting with the case where
$\delta_{1}\\!=2\hskip 0.85358pt\rho_{1}$ and $\delta_{2}=0$ . Let us first
prove (2.12) and begin with some notation. Write for simplicity

$N_{1,\hskip 0.28453ptR}=(N_{1})_{R\hskip 0.85358pt}(e\hskip
0.28453ptK_{1},e\hskip 0.28453ptK_{1})$ , $N_{2\hskip 0.28453pt,\hskip
0.28453ptR}=(N_{2})_{R\hskip 0.85358pt}(e\hskip 0.28453ptK_{2},e\hskip
0.28453ptK_{2})$ and $N_{R}=N_{R\hskip 0.85358pt}(e\hskip 0.28453ptK,e\hskip
0.28453ptK)$ ,

for every $R\geq 0$ . Moreover, for every
$D\subset\smash{\mathbb{R}_{+}^{2}}$, denote by $N(D)$ the number of
$\gamma=(\gamma_{1},\gamma_{\hskip 0.28453pt2})$ in
$\Gamma=\Gamma_{1}\times\Gamma_{2}$ such that $\bigl{(}{d}_{\hskip
0.28453pt1}(\gamma_{1}K_{1},e\hskip 0.28453ptK_{1})\hskip
0.28453pt,{d}_{2}(\gamma_{\hskip 0.28453pt2}K_{2},e\hskip
0.28453ptK_{2})\bigr{)}$ belongs to $D$. In $\mathbb{R}_{+}^{2}$ we consider
the covering of $D_{R}=\\{\hskip
0.28453pt(R_{1},R_{2})\in\mathbb{R}_{+}^{2}\hskip 1.42262pt|\hskip
1.42262ptR_{1}^{2}+R_{2}^{2}\leq R^{2}\hskip 0.28453pt\\}$ by the two segments
$[\hskip 0.85358pt0\hskip 0.85358pt,R\hskip 0.85358pt]\times\\{0\\}$ ,
$\\{0\\}\times[\hskip 0.85358pt0\hskip 0.85358pt,R\hskip 0.85358pt]$ and by
the squares $Q_{j_{1},\hskip
0.85358ptj_{2}}=(j_{1},j_{1}\\!+\\!1]\times(j_{2},j_{2}\\!+\\!1]\hskip
0.85358pt$, with $j_{1}^{2}+j_{2}^{2}<R^{2}$. Then

$N_{R}=N(D_{R})\leq N_{1,\hskip 0.28453ptR}+N_{2\hskip 0.28453pt,\hskip
0.28453ptR}+\sum\nolimits_{\hskip
0.28453ptj_{1}^{2}+j_{2}^{2}<R^{2}}N(Q_{j_{1},\hskip 0.85358ptj_{2}})\hskip
1.42262pt,$ (2.14)

with

$N(Q_{j_{1},\hskip 0.85358ptj_{2}})=(N_{1,\hskip
0.85358ptj_{1}+1}\\!-N_{1,\hskip 0.85358ptj_{1}})\hskip 0.85358pt(N_{2\hskip
0.28453pt,\hskip 0.85358ptj_{2}+1}\\!-N_{2\hskip 0.28453pt,\hskip
0.85358ptj_{2}})\leq N_{1,\hskip 0.85358ptj_{1}+1}\hskip 0.85358ptN_{2\hskip
0.28453pt,\hskip 0.85358ptj_{2}+1}\hskip 0.28453pt.$ (2.15)

Given $s_{1}\\!>\delta_{1}$ and $s_{2}>\delta_{2}$ , there exist $C\geq 1$
such that

$N_{1,\hskip 0.28453ptR}\leq C\hskip 0.85358pte^{\hskip
0.85358pts_{1}R}\quad\text{and}\quad N_{2\hskip 0.28453pt,\hskip
0.28453ptR}\leq C\hskip 0.85358pte^{\hskip 0.85358pts_{2}\hskip 0.28453ptR}$
(2.16)

for every $R\geq 0$ . By combining (2.14), (2.15) and (2.16), we get

$N_{R}\leq C\hskip 0.85358pte^{\hskip 0.85358pts_{1}R}+C\hskip
0.85358pte^{\hskip 0.85358pts_{2}\hskip 0.28453ptR}+C^{\hskip
0.28453pt2}\sum\nolimits_{\hskip 0.28453ptj_{1}^{2}+j_{2}^{2}<R^{2}}e^{\hskip
0.85358pts_{1}(j_{1}+1)\hskip 0.28453pt+\hskip 0.28453pts_{2}(j_{2}+1)}\hskip
1.42262pt.$ (2.17)

Up to a multiplicative constant, the right hand side of (2.17) is bounded
above by the integral

$\int_{\hskip 0.85358pt\mathbb{R}_{+}^{2}\cap B(0\hskip 0.28453pt,R+2)}d\hskip
0.28453ptR_{1}\hskip 0.85358ptd\hskip 0.28453ptR_{2}\,e^{\hskip
0.85358pts_{1}R_{1}+\hskip 0.85358pts_{2}R_{2}}=\int_{\hskip
0.85358pt0}^{R\hskip 0.28453pt+2}dr\,r\int_{\hskip
0.85358pt0}^{\frac{\pi}{2}}\\!d\theta\,e^{\hskip 0.85358ptr\hskip
0.85358pt(s_{1}\cos\theta\hskip 0.85358pt+\hskip
0.85358pts_{2}\sin\theta)}\hskip 1.42262pt.$ (2.18)

As the function $\theta\longmapsto s_{1}\cos\theta+s_{2}\sin\theta$ reaches
its maximum $\sqrt{s_{1}^{2}+s_{2}^{2}\hskip 0.85358pt}$ at
$\theta_{0}=\arctan\frac{s_{2}}{s_{1}}$, the latter integral is itself bounded
above by

$\tfrac{\pi}{2}\int_{\hskip 0.85358pt0}^{R\hskip 0.28453pt+2}dr\,r\,e^{\hskip
0.85358ptr\hskip 0.28453pt\sqrt{s_{1}^{2}+s_{2}^{2}}}\leq\tfrac{\pi}{2}\hskip
0.85358pt\tfrac{R+2}{\sqrt{s_{1}^{2}+s_{2}^{2}\hskip 0.85358pt}}\,e^{\hskip
0.85358pt(R+2)\sqrt{s_{1}^{2}+s_{2}^{2}}}$ (2.19)

In conclusion, we obtain

$\tfrac{\log N_{R}}{R}\leq\tfrac{2\log C\hskip 0.28453pt+\hskip
0.85358pt\log\pi\hskip 0.28453pt-\hskip 0.85358pt\log 2\hskip
0.85358pt-\frac{1}{2}\log\hskip
0.28453pt(s_{1}^{2}+s_{2}^{2})}{R}+\tfrac{\log\hskip 0.28453pt(R\hskip
0.28453pt+\hskip 0.28453pt2)}{R}+\tfrac{R\hskip 0.28453pt+2}{R}\hskip
0.85358pt\sqrt{s_{1}^{2}\\!+s_{2}^{2}\hskip 0.85358pt}$

by combining (2.17), (2.18) and (2.19), hence
$\delta\leq\sqrt{s_{1}^{2}+s_{2}^{2}\hskip 0.85358pt}$ by letting
$R\to+\infty$ , and finally
$\delta\leq\sqrt{\delta_{1}^{2}\\!+\delta_{2}^{2}\hskip 0.85358pt}$ by letting
$s_{1}\\!\searrow\delta_{1}$ and $s_{2}\\!\searrow\delta_{2}$ . Let us turn to
the proof of (2.13). Set $\|\rho\|=\sqrt{\rho_{1}^{\hskip
0.85358pt2}\\!+\rho_{2}^{\hskip 0.85358pt2}\hskip 0.85358pt}$ and assume that
$\tfrac{\delta_{1}}{\rho_{1}}\geq\tfrac{\delta_{2}}{\rho_{2}}$ . As

$d^{\hskip 0.85358pt\prime}(\gamma\hskip 0.85358ptK,e\hskip
0.28453ptK)=\tfrac{\rho_{1}}{\|\rho\|}\,{d}_{1}(\gamma_{1}\hskip
0.28453ptK_{1},e\hskip
0.28453ptK_{1})+\tfrac{\rho_{2}}{\|\rho\|}\,{d}_{2}(\gamma_{2}\hskip
0.28453ptK_{2},e\hskip 0.28453ptK_{2})\hskip 1.42262pt,$

the set $\\{\hskip 0.85358pt\gamma\in\Gamma\,|\,d^{\hskip
0.85358pt\prime}(\gamma\hskip 0.85358ptK,e\hskip 0.28453ptK)\leq R\hskip
0.85358pt\\}$ contains the product

$\\{\hskip 0.85358pt\gamma_{\hskip 0.28453pt1}\in\Gamma_{1}\hskip
1.42262pt|\,{d}_{1}(\gamma_{1}K_{1},e\hskip
0.28453ptK_{1})\leq\tfrac{\|\rho\|}{\rho_{1}}R\hskip
0.85358pt\\}\times\\{e\\}\hskip 1.42262pt,$

for every $R\geq 0$ . Hence $N_{R}^{\hskip 0.85358pt\prime}\geq N_{1\hskip
0.28453pt,\smash{\frac{\|\rho\|}{\rho_{1}}}R}^{\vphantom{|}}$ , where
$N_{R}^{\hskip 0.85358pt\prime}=N_{R\hskip 0.85358pt}^{\hskip
0.85358pt\prime}(e\hskip 0.28453ptK,e\hskip 0.28453ptK)$ , and conseqently
$\delta^{\hskip 0.85358pt\prime}\geq\tfrac{\|\rho\|}{\rho_{1}}\hskip
0.85358pt\delta_{1}$.

######  Lemma 2.9.

222As observed by the referee, Lemma 3 still holds without the torsion-free
assumption, provided that $\gamma$ runs through
$\Gamma\backslash(\Gamma\cap{K})$.

There exists $r>0$ such that the balls $B^{\hskip 0.56905pt\prime}_{r}(\gamma
K)$, with $\gamma\in\Gamma$, are pairwise disjoint in $G/K$.

###### Proof.

Let $r>0$. As $\Gamma$ is discrete in $G$, its intersection with the compact
subset

$\displaystyle G^{\hskip 0.56905pt\prime}_{r}=\\{y\in G\,|\,d^{\hskip
0.85358pt\prime}(yK,eK)\leq
r\\}=K{\bigl{(}}\exp\\{H\in\overline{\mathfrak{a}^{+}}\,|\,{\langle\rho,H\rangle\leq\|\rho\|r}\\}\bigr{)}K$

is finite. Moreover, as $\Gamma$ is torsion-free,

$\displaystyle\gamma^{+}\neq
0\qquad\forall\,\gamma\in\Gamma\backslash\\{e\\}.$

Hence there exists $r>0$ such that $\Gamma\cap G^{\hskip
0.56905pt\prime}_{2r}=\\{e\\}$, which implies that the sets $\gamma G^{\hskip
0.56905pt\prime}_{r}$ are pairwise disjoint in $G$. In other words, the balls
$B^{\hskip 0.56905pt\prime}_{r}(\gamma K)$ are pairwise disjoint in $G/K$. ∎

By using $\delta^{\hskip 0.56905pt\prime}{(\Gamma)}$, we prove Theorem 1.3,
which improves the lower bound in Theorem 1.2.

###### Proof of Theorem 1.3.

Let us resume the approach in [Cor90, Section 4] and [Leu04, Section 3]. It
consists in studying the convergence of the positive series

$\displaystyle g_{\zeta}^{\Gamma}(\Gamma xK,\Gamma
yK)=\sum\nolimits_{\gamma\in\Gamma}g_{\zeta}(Ky^{-1}\gamma^{-1}xK),$ (2.20)

which expresses the kernel $g_{\zeta}^{\Gamma}$ of
$(-\Delta-\|\rho\|^{2}+\zeta^{2})^{-1}$ on the locally symmetric space
$Y\\!=\Gamma\backslash G/K$ in terms of the corresponding Green function
$g_{\zeta}$ on the symmetric space $X=G/K$. Here $\zeta>0$ and $\Gamma
xK\neq\Gamma yK$. Recall [AnJi99, Theorem 4.2.2] that

$\displaystyle g_{\zeta}(\exp
H)\asymp\Bigl{\\{}\prod\nolimits_{\alpha\in\Sigma_{\text{red}}^{+}}\\!\bigl{(}1+\langle\alpha,H\rangle\bigr{)}\Bigr{\\}}\,\|H\|^{-\frac{\ell-1}{2}-|\Sigma_{\text{red}}^{+}|}\,e^{-\langle\rho,H\rangle-\zeta\|H\|}$
(2.21)

for $H\in\overline{\mathfrak{a}^{+}}$ large, let say $\|H\|\geq\frac{1}{2}$,
while

$\displaystyle g_{\zeta}(\exp H)\asymp\begin{cases}\|H\|^{-(n-2)}&\text{if
}\,n>2\\\ \log\frac{1}{\|H\|}&\text{if }\,n=2\\\ \end{cases}$

for $H$ small, lets say $0<\|H\|\leq\frac{1}{2}$. Thus (2.20) converges if and
only if

$\displaystyle\sum\nolimits_{\gamma\in\Gamma}$
$\displaystyle\Bigl{\\{}\prod\nolimits_{\alpha\in\Sigma_{\text{red}}^{+}}\\!\bigl{(}1\\!+\langle\alpha,(y^{-1}\gamma^{-1}x)^{+}\rangle\bigr{)}\Bigr{\\}}\,\times$
(2.22) $\displaystyle\times\,d(xK,\gamma
yK)^{-\frac{\ell-1}{2}-|\Sigma_{\text{red}}^{+}|}\,e^{-\|\rho\|d^{\hskip
0.85358pt\prime}(xK,\gamma yK)-\zeta d(xK,\gamma yK)}$

converges. Let us compare the series (2.22) with the Poincaré series (1.1) and
(2.7). On the one hand, as $\|(y^{-1}\gamma^{-1}x)^{+}\|=d(xK,\gamma yK)$,
(2.22) is bounded from above by $P^{\hskip
1.13809pt\prime}_{\|\rho\|+\zeta}(xK,yK)$. On the other hand, as

$\displaystyle d(xK,\gamma
yK)^{-\frac{\ell-1}{2}-|\Sigma_{\text{red}}^{+}|}\gtrsim e^{-\varepsilon
d(xK,\gamma yK)}$

for every $\varepsilon>0$, (2.22) is bounded from below by
$P_{\|\rho\|+\zeta+\varepsilon}(xK,yK)$. Hence (2.22) converges if
$\|\rho\|+\zeta>\delta^{\hskip 0.56905pt\prime}{(\Gamma)}$, i.e.,
$\zeta>\delta^{\hskip 0.56905pt\prime}{(\Gamma)}-\|\rho\|$, while
$\eqref{SeriesGamma}$ diverges if $\zeta<\delta{(\Gamma)}-\|\rho\|$. We
conclude by using the fact [Cor90, Section 4] that $\lambda_{0}(Y)$ is the
supremum of $\|\rho\|^{2}-\zeta^{2}$ over all $\zeta>0$ such that (2.20)
converges. ∎

Next statement is obtained by combining this lower bound with the upper bound
in Theorem 1.2.

######  Corollary 2.10.

The following estimates hold for $\lambda_{0}(Y)$:

$\displaystyle\begin{cases}\,\lambda_{0}(Y)=\|\rho\|^{2}&\qquad\textnormal{if}\;\;\delta^{\hskip
0.56905pt\prime}{(\Gamma)}\leq\|\rho\|,\\\ \,\|\rho\|^{2}-(\delta^{\hskip
0.56905pt\prime}{(\Gamma)}-\|\rho\|)^{2}\leq\lambda_{0}(Y)\leq\|\rho\|^{2}&\qquad\textnormal{if}\;\;\delta{(\Gamma)}\leq\|\rho\|\leq\delta^{\hskip
0.56905pt\prime}{(\Gamma)},\\\ \,\|\rho\|^{2}-(\delta^{\hskip
0.56905pt\prime}{(\Gamma)}-\|\rho\|)^{2}\leq\lambda_{0}(Y)\leq\|\rho\|^{2}-(\delta{(\Gamma)}-\|\rho\|)^{2}&\qquad\textnormal{if}\;\;\|\rho\|\leq\delta{(\Gamma)}.\end{cases}$

##  3\. Second improvement

In this section, we obtain the actual higher rank analog of Theorem 1.1 by
considering a further family of distances on $X$, which reflects the large
scale behavior (2.21) of the Green function. Specifically, for every $s>0$ and
$x,y\in G$, let

$\displaystyle d^{\hskip 1.13809pt\prime\prime}_{s}(xK,yK)$
$\displaystyle=\min\\{s,\|\rho\|\\}\,d^{\hskip
0.85358pt\prime}(xK,yK)+\max\\{s-\|\rho\|,0\\}\,d(xK,yK)$ (3.1)
$\displaystyle=\begin{cases}\,s\,d^{\hskip 0.85358pt\prime}(xK,yK)&\text{if
}\,0<s\leq\|\rho\|,\\\ \,\|\rho\|\,d^{\hskip
0.85358pt\prime}(xK,yK)+(s-\|\rho\|)\,d(xK,yK)&\text{if }\,s\geq\|\rho\|.\\\
\end{cases}$

Then (3.1) defines a $G$-invariant distance on $X$ such that

$\displaystyle s\,d^{\hskip 0.85358pt\prime}(xK,yK)\leq d^{\hskip
1.13809pt\prime\prime}_{s}(xK,yK)\leq
s\,d(xK,yK)\qquad\forall\,s>0,\,\forall\,x,y\in G.$ (3.2)

Consider the associated Poincaré series

$\displaystyle P^{\hskip
1.13809pt\prime\prime}_{s}(xK,yK)=\sum\nolimits_{\gamma\in\Gamma}e^{-d^{\hskip
1.13809pt\prime\prime}_{s}(xK,\gamma yK)}\qquad\forall\,s>0,\,\forall\,x,y\in
G$ (3.3)

and its critical exponent

$\displaystyle\delta^{\hskip
0.56905pt\prime\prime}{(\Gamma)}=\inf\\{s>0\,|\,P^{\hskip
1.13809pt\prime\prime}_{s}(xK,yK)<+\infty\\}.$

It follows from (3.2) that

$\displaystyle 0\leq\delta{(\Gamma)}\leq\delta^{\hskip
0.56905pt\prime\prime}{(\Gamma)}\leq\delta^{\hskip
0.56905pt\prime}{(\Gamma)}\leq 2\hskip 0.56905pt\|\rho\|.$ (3.4)

###### Proof of Theorem 1.4.

In the proof of Theorem 1.3 and Corollary 2.10, we compared the series (2.20),
or equivalently (2.22), with the Poincaré series (1.1) and (2.7). If we
consider instead the Poincaré series (3.3), we obtain in the same way that
(2.22) is bounded from above by $P^{\hskip
1.13809pt\prime\prime}_{\|\rho\|+\zeta}(xK,yK)$ and from below by $P^{\hskip
1.13809pt\prime\prime}_{\|\rho\|+\zeta+\varepsilon}(xK,yK)$, for every
$\varepsilon>0$. Hence (2.22) converges if $\zeta>\delta^{\hskip
0.56905pt\prime\prime}(\Gamma)-\|\rho\|$, while $\eqref{SeriesGamma}$ diverges
if $\zeta<\delta^{\hskip 0.56905pt\prime\prime}(\Gamma)-\|\rho\|$. We conclude
as in the above-mentioned proof. ∎

Acknowledgements. The authors are grateful to the referee for checking
carefully the manuscript and making several helpful suggestions of
improvement. The second author acknowledges financial support from the
University of Orléans during his Ph.D. and from the Methusalem Programme
Analysis and Partial Differential Equations during his postdoc stay at Ghent
University.

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