text:
\begin{document}

\title[Bottom of the $L^2$ spectrum of the Laplacian
on locally symmetric spaces]{Bottom of the $L^2$ spectrum of
the Laplacian \\ on locally symmetric spaces}

\author{Jean-Philippe ANKER and Hong-Wei ZHANG}

\keywords{Locally symmetric space, $L^2$ spectrum, Poincar\'e series , critical exponent,  Green function, heat kernel}

\makeatletter
\@namedef{subjclassname@2020}{\textnormal{2020}
    \it{Mathematics Subject Classification}}
\makeatother
\subjclass[2020]{22E40, 11N45, 35K08, 47A10, 58J50}

\maketitle

\vspace{-10pt}
\begin{center}
\footnotesize\it
    In memory of Michel Marias (1953-2020),
    who introduced us to locally symmetric spaces
\end{center}

\begin{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 Poincar\'e 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.
\end{abstract}

\section{Introduction}\label{intro}
We adopt the standard notation and refer to \cite{Hel2000} 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
$\Sigma_{\text{red}}=\lbrace\alpha\in\Sigma\,|\,\frac\alpha2\notin\Sigma\rbrace$.\\

Consider the classical Poincar\'e series
\begin{align}\label{PSeries}
P_{s}(xK,yK) =\sum\nolimits_{\gamma\in\Gamma}e^{-sd(xK,\gamma yK)}
\qquad\forall\,s>0,\,\forall\,x,y\in G
\end{align}
and denote by
$\delta(\Gamma)=\inf\lbrace s>0\,|\,P_s(xK,yK) <+\infty\rbrace$
its critical exponent, which is independent of $xK$ and $yK$.
Recall that $\delta(\Gamma)\in[0,2\hspace{.2mm}\|\rho\|]$ may be also defined by
\begin{align*}
\delta(\Gamma)=\limsup_{R\to+\infty}\frac{\log N_{R}(xK,yK)}R
\qquad\forall\,x,y\in G,
\end{align*}
where $N_{R}(xK,yK)=|\lbrace\gamma\in\Gamma\,|\,d(xK,\gamma yK)\le R\rbrace|$ 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 (\cite{Els1973a}, \cite{Els1973b}, \cite{Els1974}), Patterson \cite{Pat1976}, Sullivan \cite{Sul1987} and Corlette \cite{Cor1990}, expresses $\lambda_0(Y)$ in terms of $\rho$ and $\delta(\Gamma)$ in rank one.

\begin{theorem}\label{corlette}
In the rank one case ($\ell=1$), we have
\begin{align}\label{rank 1}
\lambda_0(Y) =
\begin{cases}
\|\rho\|^2
&\qquad\textnormal{if}\;\;0\le\delta(\Gamma)\le\|\rho\|,\\
\|\rho\|^2-(\delta(\Gamma)-\|\rho\|)^2
&\qquad\textnormal{if}\;\;\|\rho\|\le\delta(\Gamma)\le2\hspace{.2mm}\|\rho\|.
\end{cases}
\end{align}
\end{theorem}

This result was extended in higher rank as follows by Leuzinger \cite{Leu2004} and Weber \cite{Web2008}.
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 \ref{corlette}.

\begin{theorem}\label{leuzinger}
In the general case ($\ell\ge1$), the following estimates hold:
\begin{itemize}[leftmargin=*]
\item[$\bullet$]
Upper bound:
\begin{align*}
\lambda_0(Y)\le
\begin{cases}
\,\|\rho\|^2
&\qquad\textnormal{if}\;\;0\le\delta(\Gamma)\le\|\rho\|,\\
\,\|\rho\|^2-(\delta(\Gamma)-\|\rho\|)^2
&\qquad\textnormal{if}\;\;\|\rho\|\le\delta(\Gamma)\le2\hspace{.2mm}\|\rho\|.
\end{cases}
\end{align*}
\item[$\bullet$]
Lower bound:
\begin{align*}
\lambda_0(Y)\ge
\begin{cases}
\,\|\rho\|^2
&\qquad\textnormal{if}\;\;0\le\delta(\Gamma)\le\rho_{\text{\rm min}},\\
\,\max\hspace{.5mm}\lbrace0,\|\rho\|^2-(\delta(\Gamma)-\rho_{\text{\rm min}})^2\rbrace
&\qquad\textnormal{if}\;\;\rho_{\text{\rm min}}\le\delta(\Gamma)\le2\hspace{.2mm}\|\rho\|.
\end{cases}
\end{align*}
\end{itemize}

\noindent In other terms,
\begin{align*}
\begin{cases}
\,\lambda_0(Y)=\|\rho\|^2
&\textnormal{if}\;\;\delta(\Gamma)\in\bigl[\hspace{.2mm}0,\rho_{\text{\rm min}}\bigr],\\[5pt]
\,\lambda_0(Y)\in
\bigl[\hspace{.2mm}\|\rho\|^2\!-(\delta(\Gamma)\hspace{-.5mm}-\hspace{-.5mm}\rho_{\text{\rm min}})^2,
\|\rho\|^2\hspace{.2mm}\bigr]
&\textnormal{if}\;\;\delta(\Gamma)\in\bigl[\hspace{.3mm}\rho_{\text{\rm min}},\|\rho\|\bigr],\\[5pt]
\,\lambda_0(Y)\in
\bigl[\hspace{.2mm}\|\rho\|^2\!-\hspace{-.5mm}(\delta(\Gamma)\hspace{-.5mm}-\hspace{-.5mm}\rho_{\text{\rm min}})^2,
\|\rho\|^2\!-\hspace{-.5mm}(\delta(\Gamma)\hspace{-.5mm}-\hspace{-.5mm}\|\rho\|)^2\hspace{.2mm}\bigr]
&\textnormal{if}\;\;\delta(\Gamma)\in\bigl[\hspace{.2mm}\|\rho\|,\|\rho\|\hspace{-.5mm}+\hspace{-.5mm}\rho_{\text{\rm min}}\bigr],\\[5pt]
\,\lambda_0(Y)\in\bigl[\hspace{.2mm}0,
\|\rho\|^2\!-\hspace{-.5mm}(\delta(\Gamma)\hspace{-.5mm}-\hspace{-.5mm}\|\rho\|)^2\hspace{.2mm}\bigr]
&\textnormal{if}\;\;\delta(\Gamma)\in\bigl[\hspace{.2mm}
\|\rho\|\hspace{-.5mm}+\hspace{-.5mm}\rho_{\text{\rm min}},2\hspace{.2mm}\|\rho\|\hspace{.2mm}\bigr].
\end{cases}
\end{align*}
\end{theorem}

\vspace{10pt}
In this paper, we first improve the lower bound of $\lambda_0(Y)$ in Theorem \ref{leuzinger} by a slight modification of the classical Poincar\'e series \eqref{PSeries}. Let $\deltaprime{(\Gamma)}$ denote
the critical exponent of the modified Poincaré series \eqref{PprimeSeries}
associated to the polyhedral distance \eqref{d'}.

\begin{theorem}\label{TheoremFirstImprovement}
The following lower bound holds for the bottom $\lambda_0(Y)$
of the $L^2$ spectrum of $-\Delta$ on $Y\!=\Gamma\backslash G/K$:
\begin{align*}
\lambda_0(Y)\ge
\begin{cases}
\,\|\rho\|^2
&\qquad\textnormal{if}\;\;0\le\deltaprime{(\Gamma)}\le\|\rho\|,\\
\,\|\rho\|^2-(\deltaprime{(\Gamma)}-\|\rho\|)^2
&\qquad\textnormal{if}\;\;\|\rho\|\le\deltaprime{(\Gamma)}\le2\hspace{.2mm}\|\rho\|.
\end{cases}
\end{align*}
\end{theorem}

We obtain next a plain analog of Theorem \ref{corlette} by considering a more involved family of Poincar\'e series. We denote by $\deltasecond{(\Gamma)}$ the critical exponent of
$\Psecond_s(xK, yK)$, see \eqref{PsecondSeries}.

\begin{theorem}\label{second improvement}
The following characterization holds for the bottom $\lambda_0(Y)$
of the $L^2$ spectrum of $-\Delta$ on $Y\!=\Gamma\backslash G/K$:
\begin{align}\label{higher rank charac}
\lambda_0(Y)=\begin{cases}
\,\|\rho\|^2
&\qquad\textnormal{if}\;\;0\le\deltasecond{(\Gamma)}\le\|\rho\|,\\
\,\|\rho\|^2- (\deltasecond{(\Gamma)}-\|\rho\|)^2
&\qquad\textnormal{if}\;\;\|\rho\|\le\deltasecond{(\Gamma)}\le 2\hspace{.2mm}\|\rho\|.
\end{cases}
\end{align}
\end{theorem}

\begin{remark}
If $\Gamma$ is a lattice, i.e., $Y\!=\Gamma\backslash G/K$ has finite volume,
then $\lambda_0(Y)=0$ and $\deltasecond(\Gamma)=2\hspace{.2mm}\|\rho\|$, hence $\deltaprime(\Gamma)=2\hspace{.2mm}\|\rho\|$.
Furthermore $\delta(\Gamma)=2\hspace{.2mm}\|\rho\|$ \cite[Theorem~7.4]{Alb1999}.
As pointed out by Corlette \cite{Cor1990} in rank one and by Leuzinger \cite{Leu2003} in higher rank,
if $G$ has Kazhdan's property (T), then the following conditions are actually equivalent:
\vspace{1mm}

\centerline{
(a) $\Gamma$ is a lattice,\quad
(b) $\lambda_0(Y)=0$,\quad
(c) $\delta(\Gamma)=2\hspace{.2mm}\|\rho\|$,\quad
(d) $\deltasecond(\Gamma)=2\hspace{.2mm}\|\rho\|$.
}
\end{remark}

\begin{remark}
As for the Green function, the heat kernel
\begin{align*}
h_t^Y(\Gamma xK,\Gamma yK)=\sum\nolimits_{\gamma\in\Gamma}h_t{(Ky^{-1}\gamma^{-1}xK)}
\end{align*}
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 \cite{AnJi1999,AnOs2003}.
By adapting straightforwardly the methods carried out in \cite{DaMa1988,Web2008}, and by applying Theorem \ref{second improvement} instead of Theorem \ref{corlette} and Theorem \ref{leuzinger}, 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 $x,y\in G$:\\

\noindent{\rm (i)}
Assume that $\deltasecond{(\Gamma)}<\|\rho\|$
and let $\deltasecond{(\Gamma)}<s<\|\rho\|$.
Then
\begin{align*}
h_t^Y( \Gamma xK,\Gamma yK)
\lesssim t^{-\frac n2}\,(1+t)^{\frac{n-D}2}\,e^{-\|\rho\|^2t}\,
e^{-\frac{d (\Gamma xK,\Gamma yK)^2}{4t}}\,\Psecond_s(xK, yK).
\end{align*}

\noindent{\rm (ii)}
Assume that $\|\rho\|\le\deltasecond{(\Gamma)}<2\hspace{.2mm}\|\rho\|$
and let $\deltasecond{(\Gamma)}-\|\rho\|<s_1<s_2<\|\rho\|$.
Then
\begin{align*}
h_t^Y(\Gamma xK,\Gamma yK)
\lesssim t^{-\frac n2}\,e^{-\left(\|\rho\|^2-s_2^2\right)t}\,\Psecond_{\|\rho\|+s_1}(xK,yK).
\end{align*}

\noindent{\rm (iii)}
Assume that $\deltasecond{(\Gamma)}<2\hspace{.2mm}\|\rho\|$.
Let $s>\deltasecond{(\Gamma)}$ and $\epsilon>0$. Then
\begin{align*}
h_t^Y( \Gamma xK,\Gamma yK)
\lesssim t^{-\frac n2}\,
e^{-\left({\|\rho\|^2-(\deltasecond(\Gamma)-\|\rho\|)^2}-2\epsilon\right)t}\,
e^{-\frac{d(\Gamma xK,\Gamma yK)^2}{4(1+\epsilon)t}}\,\Psecond_s(xK, xK)^{\frac{1}{2}}\Psecond_s(yK, yK)^{\frac{1}{2}}.
\end{align*}
\end{remark}

\vspace{10pt}
\section{First improvement}

In this section, we replace the Riemannian distance $d$ on $X$ by a \textit{polyhedral} distance
\begin{align}\label{d'}
\textstyle
\dprime(xK,yK)=\langle\frac{\rho}{\|\rho\|},(y^{-1}x)^+\rangle
\qquad\forall\,x,y\in G,
\end{align}
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 \cite{Ank1990} and \cite{Ank1991}. More general polyhedral sets were considered in \cite{Ank1992} and \cite{AAS2010}.

\begin{proposition}
$\dprime$ is a $G$-invariant distance on $X$.
\end{proposition}

\begin{proof}
Notice first that \eqref{d'} 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 $\dprime$ is straightforward from the definition \eqref{d'}. The symmetry
\begin{align*}
\dprime(xK,yK)=\dprime(yK,xK)
\end{align*}
follows from
\begin{align}\label{distance properties}
(y^{-1})^+=-w_0. y^+
\quad\text{and}\quad
-w_{0}.\rho=\rho,
\end{align}
where $w_0$ denotes the longest element in the Weyl group. Let us check the triangular inequality
\begin{align*}
\dprime(xK,yK)\le\dprime(xK,zK)+\dprime(zK,yK).
\end{align*}
By $G$-invariance, we may reduce to the case where $zK = eK$.
According to Lemma \ref{distance lemma} below,
\begin{align*}
x^{+} + (y^{-1})^{+} - (y^{-1}x)^{+}
\end{align*}
belongs to the cone generated by the positive roots.
\end{proof}

\begin{lemma}\label{distance lemma}
For every $x,y \in G$, we have the following inclusion
\begin{align}\label{co}
\co[W.(xy)^+]\subset\co[W.(x^++ y^+)]
\end{align}
between convex hulls.
\end{lemma}

\begin{proof}
The inclusion \eqref{co} amounts to the fact that
\begin{align*}
x^{+} + y^{+} - (xy)^{+}
\end{align*}
belongs to the cone generated by the positive roots or, equivalently, to the inequality
\begin{align}\label{co2}
\langle\lambda,(xy)^+\rangle\le\langle\lambda,x^+\rangle+\langle\lambda,y^+\rangle
\qquad\forall\,\lambda\in\overline{\mathfrak{a}^{+}}.
\end{align}
It is enough to prove \eqref{co2} 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 \cite[Proposition 7.15]{Kna2002}), there exists an inner product on $V$ such that
\begin{align*}
\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}
\end{align*}
As $\lambda$ is the highest weight of $\pi$, then
\begin{align*}
e^{\langle \lambda, (xy)^{+} \rangle} = \| \pi(xy) \|
\le \| \pi(x) \| \| \pi(y) \| =
e^{\langle \lambda, x^{+} \rangle} e^{\langle \lambda, y^{+} \rangle}
= e^{\langle \lambda, x^{+} + y^{+} \rangle}.
\end{align*}
\end{proof}

\begin{remark}
The distance $\dprime$ is comparable to the Riemannian distance $d$. Specifically,
\begin{align}\label{d and d'}
\textstyle
\frac{\rho_{\text{min}}}{\|\rho\|}d(xK,yK)\le\dprime(xK,yK)\le d(xK,yK)
\qquad\forall\,x,y\in G.
\end{align}
This follows indeed from
\begin{align*}
\textstyle
\frac{\rho_{\text{min}}}{\|\rho\|}\|H\|\le\langle\frac{\rho}{\|\rho\|},H\rangle\le\|H\|
\qquad\forall\,H\in\overline{\mathfrak{a}^+}.
\end{align*}
\end{remark}

\vspace{10pt}
The volume of balls
\begin{align*}
\Bprime_r(xK)=\lbrace yK\in X\,|\,\dprime(yK,xK)\le r\rbrace
\end{align*}
was determined in \cite[Lemma 6]{Ank1990}.
For the reader's convenience, we recall the statement and its proof.

\begin{lemma}\label{ball vol}
For every $x\in G$ and $r>0$, we have
\footnote{The symbol $f\asymp g$ between two non-negative expressions
means that there exist constants $0<A\le B<+\infty$ such that $Ag\le f\le Bg$.}
\begin{align*}
|\Bprime_r(xK)|\asymp
\begin{cases}
\,r^{n}
&\qquad\textnormal{if}\;\;0<r<1,\\
\,r^{\ell-1}e^{2\hspace{.2mm}\|\rho\|r}
&\qquad\textnormal{if}\;\;r\ge1.
\end{cases}
\end{align*}
\end{lemma}

\begin{remark}
Notice the different large scale behavior,
in comparison with the classical ball volume
\begin{align*}
|B_r(xK)|\asymp
\begin{cases}
\,r^{n}
&\qquad\textnormal{if}\;\;0<r<1,\\
\,r^{\frac{\ell-1}2}e^{2\hspace{.2mm}\|\rho\|r}
&\qquad\textnormal{if}\;\;r\ge1,
\end{cases}
\end{align*}
see for instance \cite{Str1981} or \cite{Kni1997}.
\end{remark}

\begin{proof}
By translation invariance, we may assume that $x=e$.
Recall the integration formula
\begin{align}\label{IntegrationFormula}
\int_{X}dx\,f(x)=\const\int_{K} dk \
\int_{\mathfrak{a}^+}dH\,\omega(H)\,f(k(\exp H)K),
\end{align}
in the Cartan decomposition, with density
\begin{align*}
\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},
\end{align*}
since $\sinh{t}\sim\tfrac{t}{1+t}e^{t}$ for all $t\ge0$.
Thus
\begin{align*}
|\Bprime_r(eK)|
\,=\,\const\int_{\lbrace H\in\mathfrak{a}^+|\langle\rho,H\rangle\le\|\rho\|r\rbrace}dH\,\omega(H)
\,\asymp\,\int_{\lbrace H\in\mathfrak{a}^+|\|H\|\le r\rbrace}dH\,
\prod_{\alpha\in\Sigma^+}\langle\alpha,H\rangle^{m_\alpha}
\asymp r^n
\end{align*}
if $r$ is small.
Let us turn to $r$ large. On the one hand, we estimate from above
\begin{align*}
| \Bprime_r(eK) |\lesssim
\int_{\lbrace H\in\mathfrak{a}^+|\langle\rho,H\rangle\le\|\rho\|r\rbrace}dH\,e^{2\langle\rho,H\rangle}
\asymp \int_0^{2\hspace{.2mm}\|\rho\|r}ds\,s^{\ell-1 }e^s
\asymp r^{\ell -1} e^{2 \| \rho \| r}.
\end{align*}
On the other hand, let $H_0\in\mathfrak{a}^+$. As
\begin{align*}
\omega(H)\asymp e^{2\langle\rho,H\rangle}
\qquad\forall\,H\in H_0+\overline{\mathfrak{a}^+},
\end{align*}
we estimate from below
\begin{align*}
|\Bprime_r(eK) |\gtrsim
\int_{\lbrace H\in H_0+\mathfrak{a}^+|\langle\rho,H\rangle\le\|\rho\|r\rbrace}dH\,e^{2 \langle\rho,H\rangle}
\gtrsim\int_{C_0}^{2\hspace{.2mm}\|\rho\|r}ds\,s^{\ell-1 }e^s
\asymp r^{\ell-1} e^{2\hspace{.2mm}\|\rho\|r},
\end{align*}
where $C_{0} > 0$ is a constant depending on $H_0$.
\end{proof}

Consider now the modified Poincar\'e series
\begin{align}\label{PprimeSeries}
\Pprime_s(xK,yK)=\sum\nolimits_{\gamma\in\Gamma}e^{-s\dprime(xK,\gamma yK)}
\qquad\forall\,s>0,\,\forall\,x,y \in G
\end{align}
associated with $\dprime$, its critical exponent
\begin{align}\label{delta' def}
\deltaprime(\Gamma)=\inf\lbrace s>0\,|\,\Pprime_s(xK,yK) <+\infty\rbrace
\end{align}
and the modified orbital counting function
\begin{align}\label{N'CountingFunction}
\NRprime(xK,yK)=|\lbrace\gamma\in\Gamma\,|\,\dprime(xK,\gamma yK)\le R\rbrace|
\qquad \forall\,R\ge0,\,\forall\,x,y\in G.
\end{align}
The following proposition shows that \eqref{PprimeSeries}, \eqref{delta' def} and \eqref{N'CountingFunction} share the properties of their classical counterparts.

\begin{proposition}\label{delta' prop}
The following assertions hold:
\begin{enumerate}[leftmargin=8mm]
\item[\rm (i)]
$\deltaprime{(\Gamma)}$ does not depend on the choice of $x$ and $y$.
\item[\rm (ii)]
$0\le\deltaprime{(\Gamma)}\le2\hspace{.2mm}\|\rho\|$.
\item[\rm (iii)]
For every $x,y \in G$,
\begin{align}\label{limsup}
\deltaprime(\Gamma)=\limsup_{R\to+\infty}\frac{\log\NRprime(xK,yK)}R.
\end{align}
\end{enumerate}

\begin{remark}
It follows from \eqref{d and d'} that
\begin{align*}
P_s(xK,yK)\le\Pprime_s(xK,yK)\le P_{\frac{\rho_{\text{min}}}{\|\rho\|}s}(xK,yK)
\end{align*}
and
\begin{align*}
N_R(xK,yK)\le\NRprime(xK,yK)\le N_{\frac{\|\rho\|}{\rho_{\text{min}}}R}(xK,yK).
\end{align*}
Hence
\begin{align*}
0\le\delta{(\Gamma)}\le\deltaprime{(\Gamma)}
\le \frac{\|\rho\|}{\rho_{\text{min}}}\,\delta{(\Gamma)}.
\end{align*}
\end{remark}
\end{proposition}

\begin{proof}

\noindent(i) follows from the triangular inequality.
More precisely, let $x_1,y_1,x_2,y_2\in G$ and $s>0$.
Then
\begin{align*}
\dprime(x_2K,\gamma y_2K)\le\dprime(x_2K,x_1K)+\dprime(x_1K,\gamma y_1K)+
\underbrace{
\dprime(\gamma y_1K,\gamma y_2K)
}_{\dprime(y_1K,\,y_2K)}
\quad\forall\,\gamma\in\Gamma,
\end{align*}
hence
\begin{align*}
\underbrace{\sum\nolimits_{\gamma\in\Gamma}e^{-s\,\dprime(x_1K,\gamma y_1K)}}_{\Pprime_s(x_1K,y_1K)}
\le e^{s\lbrace\dprime(x_1K,x_2K)+\dprime(y_1K,y_2K)\rbrace}
\underbrace{\sum\nolimits_{\gamma\in\Gamma}e^{-s\,\dprime(x_2K,\gamma y_2K)}}_{\Pprime_s(x_2K,y_2K)}.
\end{align*}

\noindent (ii) According to (i), let us show, without loss of generality, that $\Pprime_s(eK,eK)<+\infty$ for every $s>2\hspace{.2mm}\|\rho\|$.
According to Lemma \ref{ball disjoint} below, there exists $r>0$ such that the balls
$\Bprime_{r}(\gamma K)$, with $\gamma\in\Gamma$, are pairwise disjoint in $G/K$.
Let us apply the integration formula \eqref{IntegrationFormula} to the function
\begin{align*}
f_{s}(xK)=\sum\nolimits_{\gamma\in\Gamma}e^{-s\dprime(\gamma{K},eK)}\mathbf{1}_{\Bprime_r(\gamma K)}(xK).
\end{align*}
On the one hand, as
\begin{align*}
\bigl|\dprime(xK,eK)-\dprime(\gamma K,eK)\bigr|\le r
\qquad\forall\,xK\in\Bprime_r(\gamma K),
\end{align*}
we have
\begin{align*}
\int_Xd(xK)\,f_{s}(xK)
\asymp\sum\nolimits_{\gamma\in\Gamma}e^{-s\dprime(\gamma K,eK)}
\underbrace{|\Bprime_r(\gamma K)|}_{|\Bprime_r(eK)|}
\asymp\Pprime_s(eK,eK).
\end{align*}
On the other hand,
\begin{align*}
\int_Xd(xK)\,f_{s}(xK)
&\lesssim\int_Xd(xK)\,e^{-s\dprime(xK,eK)}\\
&\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}
\end{align*}
is finite if $s>2\hspace{.2mm}\|\rho\|$.
Thus $\Pprime_s(eK,eK)<+\infty$ in that case, and consequently $\deltaprime(\Gamma)\le2\hspace{.2mm}\|\rho\|$.

\noindent(iii)
Denote the right hand side of \eqref{limsup} by $L(xK,yK)$
and let us first show that $L(xK,yK)$ is finite.
By applying Lemma \ref{ball disjoint} below to $y^{-1}\Gamma y$,
we deduce that there exists $r>0$ such that
the balls $\Bprime_r(\gamma yK)$, with $\gamma\in\Gamma$, are pairwise disjoint.
Set
\begin{align*}
\GammaRprime(xK,yK)=\lbrace\gamma\in\Gamma\,|\,\dprime(xK,\gamma yK)\le R\rbrace
\qquad\forall\,R\ge0,\,\forall\,x,y\in G.
\end{align*}
Then the ball $\Bprime_{R+r}(xK)$ contains the disjoint balls $\Bprime_r(\gamma yK)$,
with $\gamma\in\GammaRprime(xK,$ $yK)$. By computing volumes, we estimate
\begin{align*}
\NRprime(xK,yK)=|\GammaRprime(xK,yK)|\le\frac{|\Bprime_{R+r}(xK)|}{|\Bprime_r(eK)|}
\asymp(1+R)^{\ell-1}e^{2\hspace{.2mm}\|\rho\|R}.
\end{align*}
Hence $L(xK,yK)\le2\hspace{.2mm}\|\rho\|$.
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
\begin{align*}
R_2=R_1+\dprime(x_1K,x_2K)+\dprime(y_1K,y_2K).
\end{align*}
Then the triangular inequality
\begin{align*}
\dprime(x_2K,\gamma y_2K)\le
\dprime(x_2K,x_1K)+\dprime(x_1K,\gamma y_1K)+\underbrace{\dprime(\gamma y_1K,\gamma y_2K)}_{\dprime(y_1K,\,y_2K)}
\end{align*}
implies successively
\begin{align*}
\Gamma_{\hspace{-.2mm}R_1}^{\hspace{.2mm}\prime}(x_1K,y_1K)
&\subset\Gamma_{\hspace{-.2mm}R_2}^{\hspace{.2mm}\prime}(x_2K,y_2K),\\
N_{\hspace{-.2mm}R_1}^{\hspace{.2mm}\prime}(x_1K,y_1K)
&\le N_{\hspace{-.2mm}R_2}^{\hspace{.2mm}\prime}(x_2K,y_2K),\\
L(x_1K,y_1K)
&\le L(x_2K,y_2K).
\end{align*}

\noindent Let us finally prove the equality between $\deltaprime(\Gamma)$ and $L=L(eK,eK)$.
For this purpose, observe that
\begin{align}
\Pprime_s
&=1+\sum\nolimits_{R\in\N^*}\sum\nolimits_{
\gamma\in\Gamma_{\!R}^{\hspace{.2mm}\prime}\smallsetminus\hspace{.2mm}\Gamma_{\!R-1}^{\hspace{.2mm}\prime}
}\!e^{-s\,\dprime(eK,\gamma K)}
\nonumber\\
&\asymp1 +\sum\nolimits_{R\in\N^*}\bigl(\NRprime-N_{\hspace{-.2mm}R-1}^{\hspace{.2mm}\prime}\bigr)\,e^{-sR}\nonumber\\
&\asymp\sum\nolimits_{R\in\N}\NRprime\,e^{-sR},
\label{series}
\end{align}
where we have written for simplicity
\begin{align*}
\Pprime_s=\Pprime_s(eK,eK),\quad
\GammaRprime=\GammaRprime(eK,eK)
\quad\text{and}\quad
\NRprime=\NRprime(eK,eK).
\end{align*}

\noindent One the one hand, let $s>L$ and set $\epsilon=\frac{s-L}2$.
By definition of $L$,
\begin{align*}
\NRprime\lesssim e^{(L+\epsilon)R}
\qquad\forall\,R\ge0.
\end{align*}
Hence
\begin{align*}
\Pprime_s\lesssim\sum\nolimits_{R\in\N}e^{-\epsilon R}<+\infty.
\end{align*}
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
\begin{align*}
N_{\hspace{-.2mm}R_j}^{\hspace{.2mm}\prime}\ge e^{s R_j}
\qquad\forall\,j\in\N^*.
\end{align*}
Hence the series \eqref{series} diverges.
\end{proof}

\begin{remark}
Here is an example where \,$\delta\ssb<\ssb\delta^{\ssf\prime}$.
Consider the product
\vspace{.5mm}

\centerline{$
\Gamma\msf\backslash\ssf G/K\ssb
=(\ssf\Gamma_{\ssb1}\backslash\ssf G_{\vsf\vsf1}/K_{\ssb1})\msb
\times\msb(\ssf\Gamma_{\ssb2}\backslash\ssf G_{\ssf2\vsf}/K_{\vsb2})
$}\vspace{.5mm}

\noindent
of two locally symmetric spaces of rank one,
with parameters \msf$\rho_1$, $\delta_1$ and \msf$\rho_2$\vsf, $\delta_2$\ssf.
Then

\begin{equation}\label{EstimateDelta}
\delta\le\sqrt{\vsf\delta_1^{\ssf2}\!+\ssb\delta_2^{\ssf2}\msf}
\end{equation}
and
\begin{equation}\label{EstimateDeltaPrime}
\delta^{\ssf\prime}
\ge\sqrt{\rho_1^{\ssf2}\!+\ssb\rho_2^{\ssf2}\ssf}\,
\max\,\bigl\{\tfrac{\delta_1}{\rho_1},\tfrac{\delta_2}{\rho_2}\bigr\}\msf.
\end{equation}
Hence \,$\delta\ssb<\ssb\delta^{\ssf\prime}$ if \,$\tfrac{\delta_1}{\rho_1}\msb\ne\msb\tfrac{\delta_2}{\rho_2}$\vsf.
Notice that there are plenty of such products,
starting with the case where \,$\delta_1\!=\ssb2\ssf\rho_1$ and \,$\delta_2\msb=\ssb0$\ssf.
Let us first prove \eqref{EstimateDelta} and begin with some notation.
Write for simplicity
\vspace{.5mm}

\centerline{
$N_{1,\vsf R}\ssb=\msb(N_1)_{\vsb R\ssf}(e\vsf K_{\vsb1},e\vsf K_{\vsb1})$\ssf,
\ssf$N_{2\vsf,\vsf R}\ssb=\msb(N_2)_{\vsb R\ssf}(e\vsf K_{\vsb2},e\vsf K_{\vsb2})$
\ssf and \,$N_R\ssb=\msb N_{\vsb R\ssf}(e\vsf K,e\vsf K)$\ssf,
}\vspace{.5mm}

\noindent
for every \msf$R\ssb\ge\ssb0$\ssf.
Moreover, for every \msf$D\msb\subset\ssb\smash{\mathbb{R}_+^2}$,
denote by \ssf$N(D)$ the number of \,$\gamma\ssb=\ssb(\gamma_1,\gamma_{\vsf2})$
in \,$\Gamma\msb=\ssb\Gamma_{\ssb1}\msb\times\ssb\Gamma_{\ssb2}$ \ssf such that
\msf$\bigl({d}_{\vsf1\vsb}(\gamma_1K_{\ssb1},e\vsf K_{\ssb1})\vsf,{d}_2(\gamma_{\vsf2}K_{\vsb2},e\vsf K_{\vsb2})\ssb\bigr)$
belongs to $D$.
In \,$\mathbb{R}_+^2$ we consider the covering of
\msf$D_R\ssb=\ssb\{\vsf(R_1\ssb,R_2)\msb\in\msb\mathbb{R}_+^2\msf|\msf R_1^2\msb+\msb R_2^2\ssb\le\msb R^2\vsf\}$
by the two segments \,$[\ssf0\ssf,\ssb R\ssf]\msb\times\msb\{0\}$\vsf, $\{0\}\msb\times\msb[\ssf0\ssf,\ssb R\ssf]$
and by the squares \,$Q_{j_1\vsb,\ssf j_2}\msb=\ssb(j_1\vsb,j_1\!+\!1]\ssb\times\msb(j_2,j_2\!+\!1]\ssf$,
with \,$j_1^2\msb+\ssb j_2^2\msb<\msb R^2$.
Then

\begin{equation}\label{Inequality1}
N_R=N(D_R)\le N_{1,\vsf R}+N_{2\vsf,\vsf R}+\ssb\sum\nolimits_{\vsf j_1^2\vsb+j_2^2<R^2}\ssb N(Q_{j_1\vsb,\ssf j_2})\msf,
\end{equation}

\noindent
with
\begin{equation}\label{Inequality2}
N(Q_{j_1\vsb,\ssf j_2})\ssb
=\ssb(N_{1,\ssf j_1\vsb\vsb+1}\!-\msb N_{1,\ssf j_1}\vsb)\ssf(N_{2\vsf,\ssf j_2\vsb+1}\!-\msb N_{2\vsf,\ssf j_2})\ssb
\le\vsb N_{1,\ssf j_1\vsb\vsb+1}\ssf N_{2\vsf,\ssf j_2\vsb+1}\vsf.
\end{equation}

\noindent
Given \,$s_1\!>\ssb\delta_1$ and \,$s_2\msb>\ssb\delta_2$\ssf,
there exist \,$C\msb\ge\msb1$ such that
\begin{equation}\label{Inequality3}
N_{1,\vsf R}\ssb\le\ssb C\ssf e^{\ssf s_1R}
\quad\text{and}\quad
N_{2\vsf,\vsf R}\ssb\le\ssb C\ssf e^{\ssf s_2\vsf R}
\end{equation}

\noindent
for every \,$R\ssb\ge\ssb0$\ssf.
By combining \eqref{Inequality1}, \eqref{Inequality2} and \eqref{Inequality3}, we get
\begin{equation}\label{Inequality4}
N_R\le C\ssf e^{\ssf s_1R}+C\ssf e^{\ssf s_2\vsf R}
+C^{\vsf2}\sum\nolimits_{\vsf j_1^2\vsb+j_2^2<R^2}\ssb e^{\ssf s_1\vsb(j_1\ssb+1)\vsf+\vsf s_2(j_2\vsb+1)}\msf.
\end{equation}

\noindent
Up to a multiplicative constant, the right hand side of \eqref{Inequality4} is bounded above by the integral
\begin{equation}\label{Inequality5}
\int_{\ssf\mathbb{R}_+^2\vsb\vsb\cap B(0\vsf,R+2)}\hspace{-1mm}d\vsf R_1\ssf d\vsf R_2\,e^{\ssf s_1\vsb\vsb R_1\vsb+\ssf s_2R_2}
=\ssb\int_{\ssf0}^{R\vsf+2}\hspace{-1mm}dr\,r\int_{\ssf0}^{\frac\pi2}\!d\theta\,e^{\ssf r\ssf(s_1\ssb\cos\theta\ssf+\ssf s_2\sin\theta)}\msf.
\end{equation}

\noindent
As the function \,$\theta\vsb\longmapsto\ssb s_1\ssb\cos\theta\ssb+\ssb s_2\sin\theta$ \ssf
reaches its maximum $\sqrt{\vsb s_1^2\msb+\ssb s_2^2\ssf}$ at \,$\theta_0\ssb=\ssb\arctan\frac{s_2}{s_1}$,
the latter integral is itself bounded above by
\begin{equation}\label{Inequality6}
\tfrac\pi2\int_{\ssf0}^{R\vsf+2}\hspace{-1mm}dr\,r\,e^{\ssf r\vsf\sqrt{\ssb s_1^2\vsb+s_2^2}}
\le\tfrac\pi2\ssf\tfrac{R+2}{\sqrt{s_1^2\ssb+s_2^2\ssf}}\,e^{\ssf(\vsb R+2)\sqrt{\ssb s_1^2\vsb+s_2^2}}
\end{equation}

\noindent
In conclusion, we obtain
\vspace{1mm}

\centerline{$
\tfrac{\log N_R}R\le
\tfrac{2\log C\vsf+\ssf\log\pi\vsf-\ssf\log2\ssf-\frac12\ssb\log\vsf(\vsb s_1^2+s_2^2)}R+\tfrac{\log\vsf(\vsb R\vsf+\vsf2)}R+\tfrac{R\vsf+2}R\ssf\sqrt{\ssb s_1^2\!+\msb s_2^2\ssf}
$}\vspace{1.5mm}

\noindent
by combining \eqref{Inequality4}, \eqref{Inequality5} and \eqref{Inequality6},
hence \,$\delta\ssb\le\msb\sqrt{\ssb s_1^2\msb+\ssb s_2^2\ssf}$ \ssf by letting \,$R\msb\to\msb+\infty$\ssf,
and finally \,$\delta\ssb\le\msb\sqrt{\ssb\delta_1^2\!+\ssb\delta_2^2\ssf}$ \ssf
by letting \,$s_1\!\searrow\ssb\delta_1$ and \,$s_2\!\searrow\ssb\delta_2$\vsf.
Let us turn to the proof of \eqref{EstimateDeltaPrime}.
Set \,$\|\rho\|\ssb=\ssb\sqrt{\rho_1^{\ssf2}\!+\ssb\rho_2^{\ssf2}\ssf}$
and assume that \,$\tfrac{\delta_1}{\rho_1}\msb\ge\msb\tfrac{\delta_2}{\rho_2}$\vsf. As

\centerline{$
d^{\ssf\prime}(\gamma\ssf K,e\vsf K)\ssb
=\ssb\tfrac{\rho_1}{\|\rho\|}\,{d}_1(\gamma_1\vsf K_{\ssb1},e\vsf K_{\ssb1})\ssb
+\ssb\tfrac{\rho_2}{\|\rho\|}\,{d}_2(\gamma_2\vsf K_{\vsb2},e\vsf K_{\vsb2})\msf,
$}\vspace{.5mm}

\noindent
the set \,$\{\ssf\gamma\msb\in\msb\Gamma\,|\,d^{\ssf\prime}(\gamma\ssf K,e\vsf K)\ssb\le\ssb R\ssf\}$ contains the product
\vspace{.5mm}

\centerline{$
\{\ssf\gamma_{\vsf1}\hspace{-.8mm}\in\msb\Gamma_{\ssb1}\msf|\,{d}_1(\gamma_1\vsb K_{\ssb1},e\vsf K_{\ssb1})\msb
\le\msb\tfrac{\|\rho\|}{\rho_1}R\ssf\}\ssb\times\ssb\{e\}\msf,
$}

\noindent
for every \,$R\ssb\ge\ssb0$\ssf.
Hence \msf$N_{\ssb R}^{\ssf\prime}\ssb\ge\ssb N_{1\vsf,\smash{\frac{\|\rho\|}{\rho_1}}\vsb R}^{\vphantom{|}}$\msf,
where \,$N_{\ssb R}^{\ssf\prime}\ssb=\msb N_{\vsb R\ssf}^{\ssf\prime}(e\vsf K,e\vsf K)$\vsf,
and conseqently \,$\delta^{\ssf\prime}\msb\ge\ssb\tfrac{\|\rho\|}{\rho_1}\ssf\delta_1$.
\end{remark}

\begin{lemma}\label{ball disjoint}
\footnote{As 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
$\Bprime_{r}(\gamma K)$, with $\gamma\in\Gamma$,
are pairwise disjoint in $G/K$.
\end{lemma}

\begin{proof}
Let $r>0$. As $\Gamma$ is discrete in $G$, its intersection with the compact subset
\begin{align*}
\Gprime_r=\lbrace y\in G\,|\,\dprime(yK,eK) \le r\rbrace=K{\bigl(}\exp
\lbrace H\in\overline{\mathfrak{a}^+}\,|\,{\langle\rho,H\rangle\le\|\rho\|r}\rbrace \bigr)K
\end{align*}
is finite. Moreover, as $\Gamma$ is torsion-free,
\begin{align*}
\gamma^+\neq0\qquad\forall\,\gamma\in\Gamma\backslash\lbrace e\rbrace.
\end{align*}
Hence there exists $r>0$ such that $\Gamma\cap\Gprime_{2r}=\lbrace e\rbrace$,
which implies that the sets $\gamma\Gprime_r$ are pairwise disjoint in $G$.
In other words, the balls $\Bprime_r(\gamma K)$ are pairwise disjoint in $G/K$.
\end{proof}

\vspace{10pt}
By using $\deltaprime{(\Gamma)}$,
we prove Theorem \ref{TheoremFirstImprovement},
which improves the lower bound in Theorem \ref{leuzinger}.

\begin{proof}[Proof of Theorem \ref{TheoremFirstImprovement}]
Let us resume the approach in \cite[Section 4]{Cor1990} and \cite[Section 3]{Leu2004}.
It consists in studying the convergence of the positive series
\begin{align}\label{GreenSeries}
g_\zeta^{\Gamma} (\Gamma xK,\Gamma yK)=\sum\nolimits_{\gamma\in\Gamma}g_\zeta(Ky^{-1}\gamma^{-1}xK),
\end{align}
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 \cite[Theorem 4.2.2]{AnJi1999} that
\begin{align}\label{GreenFunction}
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\|}
\end{align}
for $H\in\overline{\mathfrak{a}^+}$ large, let say $\|H\|\ge\frac12$,
while
\begin{align*}
g_\zeta(\exp H)\asymp\begin{cases}
\|H\|^{-(n-2)}
&\text{if }\,n>2\\
\log\frac1{\|H\|}
&\text{if }\,n=2\\
\end{cases}\end{align*}
for $H$ small, lets say $0<\|H\|\le\frac12$.
Thus \eqref{GreenSeries} converges if and only if
\begin{equation}\label{SeriesGamma}\begin{aligned}
\sum\nolimits_{\gamma\in\Gamma}\,
&\Bigl\{\prod\nolimits_{\alpha\in\Sigma_{\text{red}}^+}\!
\bigl(1\!+\hspace{-.5mm}\langle\alpha,(y^{-1}\gamma^{-1}x)^+\rangle\bigr)\Bigr\}\,\times\\
&\times\,d(xK,\gamma yK)^{-\frac{\ell-1}2-|\Sigma_{\text{red}}^+|}\,
e^{-\|\rho\|\dprime(xK,\gamma yK)-\zeta d(xK,\gamma yK)}
\end{aligned}\end{equation}
converges.
Let us compare the series \eqref{SeriesGamma}
with the Poincar\'e series \eqref{PSeries} and \eqref{PprimeSeries}.
On the one hand, as $\|(y^{-1}\gamma^{-1}x)^+\|=d(xK,\gamma yK)$,
\eqref{SeriesGamma} is bounded from above by
$\Pprime_{\|\rho\|+\zeta}(xK, yK)$.
On the other hand, as
\begin{align*}
d(xK,\gamma yK)^{-\frac{\ell-1}2-|\Sigma_{\text{red}}^+|}
\gtrsim e^{-\epsilon d(xK,\gamma yK)}
\end{align*}
for every $\epsilon>0$, \eqref{SeriesGamma} is bounded from below by
$P_{\|\rho\|+\zeta+\epsilon}(xK, yK)$.
Hence \eqref{SeriesGamma} converges
if $\|\rho\|+\zeta>\deltaprime{(\Gamma)}$,
i.e., $\zeta>\deltaprime{(\Gamma)}-\|\rho\|$,
while $\eqref{SeriesGamma}$ diverges
if $\zeta<\delta{(\Gamma)}-\|\rho\|$.
We conclude by using the fact \cite[Section 4]{Cor1990} that
$\lambda_0(Y)$ is the supremum of $\|\rho\|^2-\zeta^2$
over all $\zeta>0$ such that \eqref{GreenSeries} converges.
\end{proof}

Next statement is obtained by combining this lower bound with the upper bound in Theorem \ref{leuzinger}.

\begin{corollary}\label{CorollaryFirstImprovement}
The following estimates hold for $\lambda_0(Y)$:
\begin{align*}\begin{cases}
\,\lambda_0(Y)=\|\rho\|^2
&\qquad\textnormal{if}\;\;\deltaprime{(\Gamma)}\le\|\rho\|,\\
\,\|\rho\|^2-(\deltaprime{(\Gamma)}-\|\rho\| )^2\le\lambda_0(Y)\le\|\rho\|^2
&\qquad\textnormal{if}\;\;
\delta{(\Gamma)}\le\|\rho\|\le\deltaprime{(\Gamma)},\\
\,\|\rho\|^2-(\deltaprime{(\Gamma)}-\|\rho\|)^2
\le\lambda_0(Y)\le\|\rho\|^2-(\delta{(\Gamma)}-\|\rho\| )^2
&\qquad\textnormal{if}\;\;\|\rho\|\le\delta{(\Gamma)}.
\end{cases}
\end{align*}
\end{corollary}

\section{Second improvement}

In this section, we obtain the actual higher rank analog of Theorem \ref{corlette} by considering a further family of distances on $X$,
which reflects the large scale behavior \eqref{GreenFunction} of the Green function.
Specifically, for every $s>0$ and $x,y\in G$, let
\begin{equation}\label{ds}\begin{aligned}
\dsecond_{s}(xK,yK)
&=\min\lbrace s,\|\rho\|\rbrace\,\dprime(xK,yK)
+\max\lbrace s-\|\rho\|,0\rbrace\,d(xK,yK)\\
&=\begin{cases}
\,s\,\dprime(xK,yK)
&\text{if }\,0<s\le\|\rho\|,\\
\,\|\rho\|\,\dprime(xK,yK)+(s-\|\rho\|)\,d(xK,yK)
&\text{if }\,s\ge\|\rho\|.\\
\end{cases}\end{aligned}\end{equation}
Then \eqref{ds} defines a $G$-invariant distance on $X$ such that
\begin{align}\label{d' ds d}
s\,\dprime(xK,yK)
\le \dsecond_{s}(xK,yK)
\le s\,d(xK,yK)
\qquad\forall\,s>0,\,\forall\,x,y\in G.
\end{align}
Consider the associated Poincar\'e series
\begin{align}\label{PsecondSeries}
\Psecond_s(xK, yK) =\sum\nolimits_{\gamma\in\Gamma}e^{-\dsecond_{s}(xK,\gamma yK)}
\qquad\forall\,s>0,\,\forall\,x,y\in G
\end{align}
and its critical exponent
\begin{align*}
\deltasecond{(\Gamma)}=\inf\lbrace s>0\,|\,\Psecond_s(xK,yK)<+\infty\rbrace.
\end{align*}
It follows from \eqref{d' ds d} that
\begin{align}\label{delta delta'' delta'}
0\le\delta{(\Gamma)}
\le\deltasecond{(\Gamma)}
\le\deltaprime{(\Gamma)}\le2\hspace{.2mm}\|\rho\|.
\end{align}

\begin{proof}[Proof of Theorem \ref{second improvement}]
In the proof of Theorem \ref{TheoremFirstImprovement}
and Corollary \ref{CorollaryFirstImprovement},
we compared the series \eqref{GreenSeries}, or equivalently \eqref{SeriesGamma},
with the Poincar\'e series \eqref{PSeries} and \eqref{PprimeSeries}.
If we consider instead the Poincar\'e series \eqref{PsecondSeries},
we obtain in the same way that
\eqref{SeriesGamma} is bounded from above by $\Psecond_{\|\rho\|+\zeta}(xK, yK)$
and from below by $\Psecond_{\|\rho\|+\zeta+\epsilon}(xK, yK)$,
for every $\epsilon>0$.
Hence \eqref{SeriesGamma} converges if $\zeta>\deltasecond(\Gamma)-\|\rho\|$,
while $\eqref{SeriesGamma}$ diverges if $\zeta<\deltasecond(\Gamma)-\|\rho\|$.
We conclude as in the above-mentioned proof.
\end{proof}

\vspace{10pt}
\noindent\textbf{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 \textit{Analysis and Partial Differential Equations} during his postdoc stay at Ghent University.

\printbibliography

\vspace{20pt}
\address{
\noindent\textsc{Jean-Philippe Anker:}
\href{mailto:anker@univ-orleans.fr}
{anker@univ-orleans.fr}\\
Institut Denis Poisson,
Universit\'e d'Orl\'eans, Universit\'e de Tours \& CNRS,
Orl\'eans, France}
\vspace{10pt}

\address{
\noindent\textsc{Hong-Wei Zhang:}
\href{mailto:hongwei.zhang@ugent.be}
{hongwei.zhang@ugent.be}\\
Department of Mathematics,
Ghent University,
Ghent, Belgium\\
Institut Denis Poisson,
Universit\'e d'Orl\'eans, Universit\'e de Tours \& CNRS,
Orl\'eans, France}

\end{document}


text:
\begin{document}

\title{Rank partition functions\\ and truncated theta identities}
\author{Mircea Merca\\

	\footnotesize Department of Mathematics, University of Craiova, 200585 Craiova, Romania\\
	\footnotesize Academy of Romanian Scientists, Ilfov 3, Sector 5, Bucharest, Romania\\
	\footnotesize mircea.merca@profinfo.edu.ro

}
\date{}
\maketitle

\begin{abstract}
In $1944$, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition.
A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G. E. Andrews and F. G. Garvan.
In this paper, we introduce truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank. As corollaries we derive new infinite families of linear inequalities for the partition function $p(n)$.
The number of Garden of Eden partitions are also considered in this context in order to provide other infinite families of linear inequalities for $p(n)$.
\\
\\
{\bf Keywords:}  partitions, Dyson's rank, theta series, inequalities
\\
\\
{\bf MSC 2010:}   05A17, 11P81, 11P83
\end{abstract}

\section{Introduction}

A partition of a positive integer $n$ is any non-increasing sequence of positive
integers whose sum is $n$ \cite{Andrews76}. Let $p(n)$ denote the number of partitions of $n$ with the usual convention that
$p(0)=1$ and $p(n)=0$ when $n$ is not a non-negative integer.
Ramanujan proved that for every positive integer $n$, we have:
\begin{align*}
p(5n+4) &\equiv 0 \pmod 5\\
p(7n+5) &\equiv 0 \pmod 7\\
p(11n+6) &\equiv 0 \pmod {11}.
\end{align*}
In order to explain the last two congruences combinatorially, Dyson \cite{Dyson}
introduced the rank of a partition. The rank of a partition is defined to be its
largest part minus the number of its parts. We denote by $N(m,n)$ the number of partitions of $n$ with rank $m$.
According to Atkin and Swinnerton-Dyer \cite[eq. (2.12)]{Atkin}, the generating function for $N(m,n)$ is given by
\begin{align}\label{GF}
\sum_{n=0}^\infty N(m,n) q^n = \frac{1}{(q;q)_\infty} \sum_{n=1}^\infty (-1)^{n-1} q^{n(3n-1)/2+mn}(1-q^n).
\end{align}
Here and throughout this paper, we use the following customary $q$-series notation:
\begin{align*}
& (a;q)_n = \begin{cases}
1, & \text{for $n=0$,}\\
(1-a)(1-aq)\cdots(1-aq^{n-1}), &\text{for $n>0$;}
\end{cases}\\
& (a;q)_\infty = \lim_{n\to\infty} (a;q)_n;\\
&
\begin{bmatrix}
n\\k
\end{bmatrix}
=
\begin{cases}
\dfrac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}, &  \text{if $0\leqslant k\leqslant n$},\\
0, &\text{otherwise.}
\end{cases}
\end{align*}
We sometimes use the following compressed notations:
\begin{align*}
& (a_1,a_2,\ldots,a_r;q)_n = (a_1;q)_n(a_2,q)_n\cdots(a_r;q)_n,\\
& (a_1,a_2,\ldots,a_r;q)_\infty = (a_1;q)_n(a_2,q)_n\cdots(a_r;q)_\infty.
\end{align*}
Because the infinite product $(a;q)_\infty$
diverges when $a\neq 0$
and $|q|\geqslant 1$,
whenever $(a;q)_\infty$ appears in a formula, we shall assume $|q| <1$.

By \eqref{GF}, we immediately deduce that
\begin{equation}\label{eq:1}
\sum_{n=0}^\infty N(n) q^n = \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{n(3n+1)/2} = 1+\sum_{n=1}^\infty q^n
\begin{bmatrix} 2n-1\\n-1 \end{bmatrix},
\end{equation}
and
\begin{equation}\label{eq:3.1}
\sum_{n=0}^\infty R(n) q^n = \frac{1}{(q;q)_\infty} \sum_{n=1}^\infty (-1)^{n+1} q^{n(3n+1)/2} = \sum_{n=1}^\infty q^{n+1}
\begin{bmatrix}
2n\\n-1
\end{bmatrix},
\end{equation}
where $N(n)$ is the number of partitions of $n$ with non-negative rank and
$R(n)$ is the number of partitions of $n$ with positive rank.
We remark that the sequences $\{N(n)\}_{n>0}$ and $\{R(n)\}_{n>0}$ are known and can be seen in the
On-Line Encyclopedia of Integer Sequence \cite[A064173,A064174]{Sloane}.

Linear inequalities involving Euler’s partition function $p(n)$ have been the subject of recent studies.
In \cite{Andrews12},  Andrews and Merca considered Euler's pentagonal number theorem
\begin{align*}
\sum_{n=-\infty}^{\infty} (-1)^n q^{n(3n-1)/2} = (q;q)_\infty
\end{align*}
and proved a truncated theorem on partitions.

\begin{theorem}\label{th:1}
	For $k\geqslant 1$,
	\begin{equation*}
	\frac{1}{(q;q)_\infty} \sum_{j=0}^{k-1} (-1)^j q^{j(3j+1)/2} (1-q^{2j+1})
	= 1+(-1)^{k-1} \sum_{n=1}^\infty \frac{q^{{k\choose 2}+(k+1)n}}{(q;q)_n}
	\begin{bmatrix}
	n-1\\k-1
	\end{bmatrix}.
	\end{equation*}
\end{theorem}

As a consequence of Theorem \ref{th:1}, Andrews and Merca derived the following linear partition inequality:
For $n>0$, $k\geqslant 1$,
\begin{equation}\label{eq:1.1}
(-1)^{k-1} \sum_{j=0}^{k-1} (-1)^j
\Big( p\big(n-j(3j+1)/2\big) - p\big(n-j(3j+5)/2-1\big) \Big) \geqslant 0,
\end{equation}
with strict inequality if $n\geqslant k(3k+1)/2$.

Theorem \ref{th:1} has opened up a new study on truncated theta series and linear partition inequalities.
Other recent investigations involving truncated theta series and linear partition inequalities can be found in several papers by
Andrews and Merca \cite{Andrews17},
Chan, Ho and Mao \cite{Chan},
Guo and Zeng \cite{Guo},
He, Ji and Zang \cite{He},
Mao \cite{Mao,Mao17},
Merca \cite{Merca16},
and Merca, Wang and Yee \cite{MWY}.

In this paper, motivated by these results, we shall provide a bisected version of Theorem \ref{th:1}.
The first result contains a truncated form of the identity \eqref{eq:1}.

\begin{theorem}\label{th:2}
	For $|q|<1$ and $k\geqslant 1$, there holds
	\begin{align*}
	& \frac{1}{(q;q)_\infty} \sum_{j=0}^{k-1} (-1)^j q^{j(3j+1)/2} \\
	&\qquad = 1+\sum_{j=1}^\infty q^j \begin{bmatrix} 2j-1\\j-1 \end{bmatrix}
	+(-1)^{k-1}  \frac{q^{k(3k+1)/2}}{(q,q^3;q^3)_\infty} \sum_{j=0}^\infty \frac{q^{j(3j+3k+2)}}{(q^3;q^3)_j(q^2;q^3)_{k+j}}
	\end{align*}
	and
	\begin{align*}
	& \frac{1}{(q;q)_\infty} \sum_{j=0}^{k-1} (-1)^{j} q^{j(3j+5)/2+1}\\
	&\qquad = \sum_{j=1}^\infty q^j \begin{bmatrix} 2j-1\\j-1 \end{bmatrix}
	+(-1)^{k-1}  \frac{q^{k(3k+5)/2+1}}{(q^2,q^3;q^3)_\infty} \sum_{j=0}^\infty \frac{q^{j(3j+3k+4)}}{(q^3;q^3)_j(q;q^3)_{k+j+1}}.
	\end{align*}
\end{theorem}

An immediate consequence owing to the positivity of the sums on the right hand side of the second identity
is given by the following infinite family of linear partition inequalities.

\begin{corollary}\label{cor:1.4}
	For $n>0$, $k\geqslant 1$,
	$$ (-1)^{k-1} \left( \sum_{j=0}^{k-1} (-1)^{j} p(n-j(3j+5)/2-1)  - N(n) \right) \geqslant 0.$$
	with strict inequality if $n\geqslant k(3k+5)/2+1$.	For example,
	\begin{align*}
	& p(n-1) \geqslant N(n),\\
	& p(n-1)-p(n-5) \leqslant N(n),\\
	& p(n-1)-p(n-5)+p(n-12) \geqslant N(n),\text{ and}\\
	& p(n-1)-p(n-5)+p(n-12)-p(n-22) \leqslant N(n).
	\end{align*}
\end{corollary}

Regarding the inequality \eqref{eq:1.1}, we recall the following
partition theoretic interpretation given by Andrews and Merca \cite[Theorem 1]{Andrews12}:
\begin{equation*}
(-1)^{k-1} \sum_{j=0}^{k-1} (-1)^j
\Big( p\big(n-j(3j+1)/2\big) - p\big(n-j(3j+5)/2-1\big) \Big)= M_k(n),
\end{equation*}
where $M_k(n)$ is the number of partitions of $n$ in which
$k$ is the least integer that is not a part and there are
more parts $>k$ than there are $<k$. In \cite{Yee} has
given a combinatorial proof of this result.
We can easily deduce that Corollary \ref{cor:1.4} is equivalent to the following result.

\begin{corollary}\label{cor:1.3}
	For $n>0$, $k\geqslant 1$,
	$$ (-1)^{k-1} \left( \sum_{j=0}^{k-1} (-1)^j p(n-j(3j+1)/2)  - N(n) \right) \geqslant M_k(n),$$
	with strict inequality if  $n\geqslant k(3k+5)/2+1$.
\end{corollary}

The following theorem contains a truncated version of the identity  \eqref{eq:3.1}.
\begin{theorem}\label{th:3.1}
	For $|q|<1$ and $k> 1$, there holds
	\begin{align*}
	& \frac{1}{(q;q)_\infty} \sum_{j=1}^{k-1} (-1)^{j+1} q^{j(3j+1)/2} \\
	&\qquad = \sum_{j=1}^\infty q^{j+1} \begin{bmatrix} 2j\\j-1 \end{bmatrix}
	+(-1)^{k}  \frac{q^{k(3k+1)/2}}{(q,q^3;q^3)_\infty} \sum_{j=0}^\infty \frac{q^{j(3j+3k+2)}}{(q^3;q^3)_j(q^2;q^3)_{k+j}}
	\end{align*}
	and
	\begin{align*}
	& \frac{1}{(q;q)_\infty}\left(1-\sum_{j=0}^{k-1} (-1)^{j} q^{j(3j+5)/2+1}\right)  \\
	&\qquad =1+ \sum_{j=1}^\infty q^{j+1} \begin{bmatrix} 2j\\j-1 \end{bmatrix}
	+(-1)^{k}  \frac{q^{k(3k+5)/2+1}}{(q^2,q^3;q^3)_\infty} \sum_{j=0}^\infty \frac{q^{j(3j+3k+4)}}{(q^3;q^3)_j(q;q^3)_{k+j+1}}.
	\end{align*}
\end{theorem}

Theorem \ref{th:3.1} is not essentially a new result, it is an equivalent version of Theorem \ref{th:2}.
As a consequence of Theorem \ref{th:3.1} we remark the following equivalent form of Corollary \ref{cor:1.3}.

\begin{corollary}\label{cor:3.2}
	For $n\geqslant 0$, $k>1$,
	$$ (-1)^{k} \left( \sum_{j=1}^{k-1} (-1)^{j+1} p\big(n-j(3j+1)/2\big)  - R(n) \right) \geqslant M_k(n),$$
	with strict inequality if $n\geqslant k(3k+5)/2+1$.
\end{corollary}

Theorems \ref{th:2} and \ref{th:3.1} are good reasons to look for new infinite families of
linear inequalities for the partition function $p(n)$.
The rest of this paper is organized as follows. We will first prove Theorem \ref{th:2} in Section \ref{S2}. In Section \ref{S3}, we consider the partitions with non-negative crank and provide a truncated form of an identity of Auluck \cite{Auluck}.
Section \ref{S4} is devoted the partitions with rank $-2$ or less. Connections between partitions with rank $-2$ or less and partitions with positive crank are given in this context.

\section{Proof of Theorem \ref{th:2}}
\label{S2}

To prove the theorem, we consider the second identity by Heine's transformation  of ${_{2}}\phi_{1}$ series \cite[(III.2)]{gasper}, namely
\begin{equation}\label{eq:HT}
{_{2}}\phi_{1}\bigg(\begin{matrix}a, b\\ c\end{matrix}\,;q,z\bigg)
= \frac{(c/b,bz;q)_\infty}{(c,z;q)_\infty}
{_{2}}\phi_{1}\bigg(\begin{matrix}abz/c, b\\ bz\end{matrix}\,;q,c/b\bigg).
\end{equation}
Rewriting \eqref{eq:1} as
\begin{equation*}
\frac{1}{(q;q)_\infty} \sum_{n=0}^{k-1} (-1)^n q^{n(3n+1)/2}
= 1+\sum_{n=1}^\infty q^n \begin{bmatrix} 2n-1\\n-1 \end{bmatrix}
- \frac{1}{(q;q)_\infty} \sum_{n=k}^{\infty} (-1)^n q^{n(3n+1)/2},
\end{equation*}
we get
\allowdisplaybreaks{
\begin{align*}
& \frac{1}{(q;q)_\infty} \sum_{n=k}^\infty (-1)^n q^{n(3n+1)/2}\\
& \quad = (-1)^k \frac{ q^{k(3k+1)/2}}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{n(6k+1)/2+3n^2/2}\\

& \quad = (-1)^k \frac{ q^{k(3k+1)/2}}{(q;q)_\infty} \lim_{z\to 0} \sum_{n=0}^\infty \frac{(q^{3k+2}/z;q^3)_n}{(z;q^3)_n} z^n\\
& \quad = (-1)^k \frac{ q^{k(3k+1)/2}}{(q;q)_\infty} \lim_{z\to 0} {_{2}}\phi_{1}\bigg(\begin{matrix}q^{3},  q^{3k+2}/z\\ z\end{matrix}\,;q^{3},z\bigg)\\
& \quad = (-1)^k \frac{ q^{k(3k+1)/2}}{(q;q)_\infty} \\
& \quad\qquad \times\lim_{z\to 0} \frac{(z^2/q^{3k+2},q^{3k+2};q^3)_\infty}{(z;q^3)^2_\infty}
{_{2}}\phi_{1}\bigg(\begin{matrix}q^{3k+5}/z,  q^{3k+2}/z\\ q^{3k+2}\end{matrix}\,;q^{3},z^2/q^{3k+2}\bigg)\\
& \quad = (-1)^k q^{k(3k+1)/2}\frac{(q^{3k+2};q^3)_\infty}{(q;q)_\infty} \lim_{z\to 0} \sum_{n=0}^\infty \frac{(q^{3k+5}/z,q^{3k+2}/z;q^3)_n}{(q^3,q^{3k+2};q^3)_n} \left(\frac{z^2}{q^{3k+2}} \right)^n \\
& \quad = (-1)^k q^{k(3k+1)/2}\frac{(q^{3k+2};q^3)_\infty}{(q;q)_\infty} \lim_{z\to 0} \sum_{n=0}^\infty \frac{(-1)^n q^{3n(n+1)/2} (q^{3k+2}/z;q^3)_n z^n}{(q^3,q^{3k+2};q^3)_n} \\
& \quad = (-1)^k q^{k(3k+1)/2}\frac{(q^{3k+2};q^3)_\infty}{(q;q)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+2)}}{(q^3,q^{3k+2};q^3)_n} \\
& \quad = (-1)^k q^{k(3k+1)/2}\frac{(q^{2};q^3)_\infty}{(q;q)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+2)}}{(q^3;q^3)_n(q^2;q^3)_{n+k}}.
\end{align*}}
The first identity is proved.

Considering Euler's pentagonal number theorem, the identity \eqref{eq:1} becomes
\begin{equation*}
\frac{1}{(q;q)_\infty} \sum_{n=0}^{k-1} (-1)^n q^{n(3n+5)/2+1} = \sum_{n=1}^\infty q^n
\begin{bmatrix}
2n-1\\n-1
\end{bmatrix}
- \frac{1}{(q;q)_\infty} \sum_{n=k}^\infty (-1)^n q^{n(3n+5)/2+1}.
\end{equation*}
The proof of the second identity is quite similar to the proof of the first one.
We can write
\begin{align*}
& \frac{1}{(q;q)_\infty} \sum_{n=k}^\infty (-1)^n q^{n(3n+5)/2+1}\\
& \quad = (-1)^k \frac{ q^{k(3k+5)/2+1}}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{n(6k+5)/2+3n^2/2}\\

& \quad = (-1)^k \frac{ q^{k(3k+5)/2+1}}{(q;q)_\infty} \lim_{z\to 0} \sum_{n=0}^\infty \frac{(q^{3k+4}/z;q^3)_n}{(z;q^3)_n} z^n\\
& \quad = (-1)^k \frac{ q^{k(3k+5)/2+1}}{(q;q)_\infty} \lim_{z\to 0} {_{2}}\phi_{1}\bigg(\begin{matrix}q^{3},  q^{3k+4}/z\\ z\end{matrix}\,;q^{3},z\bigg)\\
& \quad = (-1)^k \frac{ q^{k(3k+5)/2+1}}{(q;q)_\infty} \\
& \quad\qquad \times\lim_{z\to 0} \frac{(z^2/q^{3k+4},q^{3k+4};q^3)_\infty}{(z;q^3)^2_\infty}
{_{2}}\phi_{1}\bigg(\begin{matrix}q^{3k+7}/z,  q^{3k+4}/z\\ q^{3k+4}\end{matrix}\,;q^{3},z^2/q^{3k+4}\bigg)\\
& \quad = (-1)^k q^{k(3k+5)/2+1}\frac{(q^{3k+4};q^3)_\infty}{(q;q)_\infty} \lim_{z\to 0} \sum_{n=0}^\infty \frac{(q^{3k+7}/z,q^{3k+4}/z;q^3)_n}{(q^3,q^{3k+4};q^3)_n} \left(\frac{z^2}{q^{3k+4}} \right)^n \\
& \quad = (-1)^k q^{k(3k+5)/2+1}\frac{(q^{3k+4};q^3)_\infty}{(q;q)_\infty} \lim_{z\to 0} \sum_{n=0}^\infty \frac{(-1)^n q^{3n(n+1)/2} (q^{3k+4}/z;q^3)_n z^n}{(q^3,q^{3k+4};q^3)_n} \\
& \quad = (-1)^k q^{k(3k+5)/2+1}\frac{(q^{3k+4};q^3)_\infty}{(q;q)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+4)}}{(q^3,q^{3k+4};q^3)_n} \\
& \quad = (-1)^k q^{k(3k+5)/2+1}\frac{(q;q^3)_\infty}{(q;q)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+4)}}{(q^3;q^3)_n(q;q^3)_{n+k+1}}.
\end{align*}
This concludes the proof.

\section{Truncated identity of Auluck}
\label{S3}

In 1988, Andrews and Garvan \cite{Andrews88} defined the crank of an integer partition as follows.
The crank of a partition is the largest part of the partition if there are no ones as parts and otherwise is the number of parts larger than the number of ones minus the number of ones.
If $M(m,n)$ denotes the number of partitions of $n$ with crank $m$, then \cite{Andrews88}:
\begin{align}\label{eq:3.1a}
\sum_{n=0}^\infty \sum_{m=-\infty}^\infty M(m,n) z^m q^n = \frac{(q;q)_\infty}{(zq;q)_\infty (q/z;q)_\infty}.
\end{align}
In this section we denote by $C(n)$ the number of partition of $n$ with non-negative crank.
Recently, Uncu \cite{Uncu} proved that the number of partitions into even number of distinct parts whose odd-indexed parts’ sum is $n$ is
equal to the number of partitions of $n$ with non-negative crank. In this context he provided the following result.

\begin{theorem}\label{Th:Uncu}
	The generating function for partitions with non-negative crank is
	\begin{align*}
	\sum_{n=0}^\infty C(n) q^n = \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty  (-1)^n q^{n(n+1)/2}.
	\end{align*}
\end{theorem}

We remark that this result was proved independently by Ballantine and Merca \cite{BM} in a paper that investigate connections between least $r$-gaps in partitions and partitions with non-negative rank and non-negative crank. In this paper they proved that the number of partitions of $n$ with nonnegative crank is even except when $n$ is twice a generalized pentagonal number.
Very recently,  Andrews and Newman \cite{Andrews19} considered \eqref{eq:3.1a} and provided a different proof for Theorem \ref{Th:Uncu}.
In 2011, Andrews \cite{Andrews11} remarked that the following theta identity
\begin{align}\label{eq:3.2}
\frac{1}{(q;q)_\infty} \sum_{n=0}^\infty  (-1)^n q^{n(n+1)/2}  = \sum_{n=0}^\infty \frac{q^{n(n+1)}}{(q;q)^2_n}
\end{align}
is effectively equivalent to an identity of Auluck \cite[eq. (10)]{Auluck} published in $1951$.
We have the following truncated form of the identity \eqref{eq:3.2}.

\begin{theorem}\label{Th:3.2}
	For $k\geqslant 1$,
	\begin{equation*}
	\frac{1}{(q;q)_\infty} \sum_{n=0}^{k-1} (-1)^n q^{n(n+1)/2}
	= \sum_{n=0}^\infty \frac{q^{n(n+1)}}{(q;q)^2_n} + (-1)^{k-1} q^{k(k+1)/2} \sum_{n=0}^\infty \frac{q^{n(n+k+1)}}{(q;q)_n(q;q)_{n+k}}.
	\end{equation*}
\end{theorem}

\begin{proof}
	The proof of this theorem is quite similar to the proof of Theorem \ref{th:2}.
	The identity \eqref{eq:3.2} can be written as:
	$$
	\frac{1}{(q;q)_\infty} \sum_{n=0}^{k-1}  (-1)^n q^{n(n+1)/2} = \sum_{n=0}^\infty \frac{q^{n(n+1)}}{(q;q)^2_n}-\frac{1}{(q;q)_\infty} \sum_{n=k}^\infty  (-1)^n q^{n(n+1)/2}.
	$$
	We have
	\allowdisplaybreaks{
		\begin{align*}
		& \frac{1}{(q;q)_\infty} \sum_{n=k}^\infty  (-1)^n q^{n(n+1)/2}\\
		&\quad = (-1)^k \frac{q^{k(k+1)/2}}{(q;q)_\infty} \sum_{n=0}^\infty  (-1)^n q^{n(2k+1)/2+n^2/2}\\
		&\quad = (-1)^k \frac{q^{k(k+1)/2}}{(q;q)_\infty} \lim_{z\to 0} \sum_{n=0}^\infty \frac{(q^{k+1}/z;q)_n}{(z;q)_n}\\
		&\quad = (-1)^k \frac{q^{k(k+1)/2}}{(q;q)_\infty} \lim_{z\to 0} \frac{(z^2/q^{k+1},q^{k+1};q)_\infty}{(z;q)^2_\infty} \sum_{n=0}^\infty \frac{(q^{k+2}/z,q^{k+1}/z;q)_n}{(q;q)_n(q^{k+1};q)_n}\left(\frac{z^2}{q^{k+1}} \right)^n
		\tag{By Heine's transformation \eqref{eq:HT}}\\
		&\quad = (-1)^k q^{k(k+1)/2} \frac{(q^{k+1};q)_\infty}{(q;q)_\infty} \sum_{n=0}^\infty \frac{q^{n(n+k+1)}}{(q,q^{k+1};q)_n}\\
		&\quad = (-1)^k q^{k(k+1)/2} \sum_{n=0}^\infty \frac{q^{n(n+k+1)}}{(q;q)_n(q;q)_{n+k}}.
		\end{align*}
	}
	This concludes the proof.
\end{proof}

In analogy with Corollary \ref{cor:1.3}, we derive a new infinite family of linear inequalities for $p(n)$.

\begin{corollary}
	For $n\geqslant 0$, $k\geqslant 1$,
	$$ (-1)^{k-1} \left( \sum_{j=0}^{k-1} (-1)^{j} p\big(n-j(j+1)/2\big)  - C(n) \right) \geqslant 0,$$
	with strict inequality if $n\geqslant k(k+1)/2$. For example,
	\begin{align*}
	& p(n) \geqslant C(n),\\
	& p(n)-p(n-1) \leqslant C(n),\\
	& p(n)-p(n-1)+p(n-3) \geqslant C(n), \text{ and}\\
	& p(n)-p(n-1)+p(n-3)-p(n-6) \leqslant C(n).
	\end{align*}
\end{corollary}

\section{Garden of Eden partitions}
\label{S4}

In $2007$, B. Hopkins and J. A. Sellers \cite{Hopkins} provided  a formula that counts the number of partitions of $n$ that have rank $-2$ or less.
Following the terminology of cellular automata and combinatorial game theory, they call these Garden of Eden partitions.
These partitions arise naturally in analyzing the game \textit{Bulgarian solitaire} which was popularized by Gardner \cite{Gardner} in $1983$.
By \eqref{GF}, Hopkins and Sellers obtained
\begin{equation}\label{eq:4.1}
\sum_{n=0}^\infty ge(n) q^n = \frac{1}{(q;q)_\infty} \sum_{n=1}^\infty (-1)^{n-1} q^{3n(n+1)/2},
\end{equation}
where $ge(n)$ counts the Garden of Eden partitions of $n$.  We remark the following theta identity.

\begin{theorem}\label{TH:4.1}
	For $|q|<1$,
	$$ \frac{1}{(q;q)_\infty} \sum_{n=1}^\infty (-1)^{n-1} q^{3n(n+1)/2}
	= \frac{1}{(q,q^2;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{3(n+1)^2}}{(q^3;q^3)_n(q^3;q^3)_{n+1}}.$$
\end{theorem}

\begin{proof}
	We can write
	\begin{align*}
	& \frac{1}{(q;q)_\infty} \sum_{n=1}^\infty (-1)^{n-1} q^{3n(n+1)/2} \\
	& \quad= \frac{q^3}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^{n} q^{3n^2/2 + 9n/2} \\
	& \quad= \frac{q^3}{(q;q)_\infty} \lim_{z\to 0} \sum_{n=0}^\infty \frac{(q^6/z;q^3)_n}{(z;q^3)_n}z^n\\
	& \quad= \frac{q^3}{(q;q)_\infty} \lim_{z\to 0} \frac{(z^2/q^6,q^6;q^3)_\infty}{(z;q^3)^2_\infty}
	\sum_{n=0}^\infty \frac{(q^n/z,q^6/z;q^3)_n}{(q^3,q^6;q^3)_n}\left(\frac{z^2}{q^6} \right)^n
	\tag{By Heine's transformation \eqref{eq:HT}}\\
	& \quad= \frac{q^3(q^6;q^3)_\infty}{(q;q)_\infty} \lim_{z\to 0} \sum_{n=0}^\infty \frac{(-1)^n q^{3n(n+1)/2}(q^6/z;q^3)_n}{(q^3,q^6;q^3)_n}z^n\\
	& \quad= \frac{q^3(q^6;q^3)_\infty}{(q;q)_\infty} \sum_{n=0}^\infty \frac{ q^{3n(n+2)}}{(q^3,q^6;q^3)_n}\\
	& \quad= \frac{(q^3;q^3)_\infty}{(q;q)_\infty} \sum_{n=0}^\infty \frac{ q^{3(n+1)^2}}{(q^3;q^3)_n(q^3;q^3)_{n+1}}.
	\end{align*}
\end{proof}

Relating to Theorem \ref{TH:4.1}, we remark that
\begin{align}\label{Eq:9}
\sum_{n=0}^\infty D(n) q^n =\frac{1}{(q;q)_\infty} \sum_{n=1}^\infty (-1)^{n-1} q^{n(n+1)/2} = \sum_{n=0}^\infty \frac{q^{(n+1)^2}}{(q;q)_n(q;q)_{n+1}}
\end{align}
is the generating function for the partitions with positive crank.
It is an easy exercise to deduce Theorem \ref{TH:4.1} from \eqref{Eq:9} and vice versa.
Connections between Garden of Eden partitions and partitions with positive crank can be easily derived considering Theorem \ref{TH:4.1}.

\begin{corollary}
	For $n\geqslant 0$,
	\begin{align*}
	ge(n) = \sum_{k=0}^{\lfloor n/3 \rfloor} D(j) p_3(n-3j),
	\end{align*}
	where $p_3(n)$ counts partitions of $n$ in which no parts are multiples of $3$.
\end{corollary}

We have the following truncated form of Theorem \ref{TH:4.1}.

\begin{theorem}\label{TH:4.2}
	For $|q|<1$, $k\geqslant 1$,
	\begin{align*}
	& \frac{1}{(q;q)_\infty} \sum_{n=1}^k (-1)^{n-1} q^{3n(n+1)/2} = \frac{1}{(q,q^2;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{3(n+1)^2}}{(q^3;q^3)_n(q^3;q^3)_{n+1}}\\
	& \qquad\qquad\qquad\qquad + (-1)^{k-1} \frac{q^{3(k+1)(k+2)/2}}{(q,q^2;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{3n(n+k+2)}}{(q^3;q^3)_n(q^{3};q^3)_{n+k+1}}.
	\end{align*}
\end{theorem}

\begin{proof}
	The identity \eqref{eq:4.1} can be written as:
	$$\frac{1}{(q;q)_\infty} \sum_{n=1}^k (-1)^{n-1} q^{3n(n+1)/2} =
	\sum_{n=0}^\infty ge(n) q^n - \frac{q^3}{(q;q)_\infty} \sum_{n=k}^\infty (-1)^{n} q^{3n(n+3)/2}.$$
	We have
	\allowdisplaybreaks{
		\begin{align*}
		& \frac{q^3}{(q;q)_\infty} \sum_{n=k}^\infty (-1)^{n} q^{3n(n+3)/2}\\
		& \quad = (-1)^k \frac{q^{3(k+1)(k+2)/2}}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{3n(2k+3)/2+3n^2/2}\\
		& \quad = (-1)^k \frac{q^{3(k+1)(k+2)/2}}{(q;q)_\infty} \lim_{z\to 0} \sum_{n=0}^\infty \frac{(q^{3(k+2)/2}/z;q^{3})_n}{(z;q^3)_n}z^n\\
		& \quad = (-1)^k \frac{q^{3(k+1)(k+2)/2}}{(q;q)_\infty} \\
		& \quad\quad \times \lim_{z\to 0} \frac{(z^2/q^{3(k+2)},q^{3(k+2)};q^3)_\infty}{(z;q^3)^2_\infty}
		\sum_{n=0}^\infty \frac{(q^{3(k+3)}/z,q^{3(k+2)}/z;q^3)_n}{(q^3,q^{3(k+2)};q^3)_n} \left( \frac{z^2}{q^{3(k+2)}}\right)^n
		\tag{By Heine's transformation \eqref{eq:HT}}\\
		& \quad = (-1)^k q^{3(k+1)(k+2)/2} \frac{(q^{3(k+2)};q^3)_\infty}{(q;q)_\infty}
		\lim_{z\to 0} \sum_{n=0}^\infty (-1)^n \frac{q^{3n(n+1)/2}(q^{3(k+2)}/z;q^3)_n}{(q^3,q^{3(k+2)};q^3)_n} z^n\\
		& \quad = (-1)^k q^{3(k+1)(k+2)/2} \frac{(q^{3(k+2)};q^3)_\infty}{(q;q)_\infty} \sum_{n=0}^\infty \frac{q^{3n(n+k+2)}}{(q^3,q^{3(k+2)};q)_n}\\
		& \quad = (-1)^k q^{3(k+1)(k+2)/2} \frac{(q^3;q^3)_\infty}{(q;q)_\infty} \sum_{n=0}^\infty \frac{q^{3n(n+k+2)}}{(q^3;q^3)_n(q^{3};q^3)_{n+k+1}}.
		\end{align*}
	}
	The proof follows easily considering Theorem \ref{TH:4.1}.
\end{proof}

On the one hand, as a consequence of Theorem \ref{TH:4.2}, we remark a new infinite family of linear inequalities for the partition function $p(n)$.

\begin{corollary}\label{Th:4.1}
	For $n\geqslant 0$, $k\geqslant 1$,
	$$ (-1)^{k-1} \left( \sum_{j=1}^{k} (-1)^{j-1} p\big(n-3j(j+1)/2\big)  - ge(n) \right) \geqslant 0,$$
	with strict inequality if $n\geqslant 3(k+1)(k+2)/2$. For example,
	\begin{align*}
	& p(n-3) \geqslant ge(n),\\
	& p(n-3)-p(n-9) \leqslant ge(n),\\
	& p(n-3)-p(n-9)+p(n-18) \geqslant ge(n), \text{ and}\\
	& p(n-3)-p(n-9)+p(n-18)-p(n-30) \leqslant ge(n).
	\end{align*}
\end{corollary}

On the other hand, by Theorem \ref{TH:4.2}, we deduce the following truncated version of \eqref{Eq:9}.

\begin{corollary}\label{C:4.5}
	For $|q|<1$, $k\geqslant 1$,
	\begin{align*}
	& \frac{1}{(q;q)_\infty} \sum_{n=1}^k (-1)^{n-1} q^{n(n+1)/2} \\
	& = \sum_{n=0}^\infty \frac{q^{(n+1)^2}}{(q;q)_n(q;q)_{n+1}} + (-1)^{k-1} q^{(k+1)(k+2)/2} \sum_{n=0}^\infty \frac{q^{n(n+k+2)}}{(q;q)_n(q;q)_{n+k+1}}.
	\end{align*}
\end{corollary}

This result allows us to deduce the following infinite family of linear inequalities for the partition function $p(n)$.

\begin{corollary}\label{C:4.6}
	For $n\geqslant 0$, $k\geqslant 1$,
	$$ (-1)^{k-1} \left( \sum_{j=1}^{k} (-1)^{j-1} p\big(n-j(j+1)/2\big)  - D(n) \right) \geqslant 0,$$
	with strict inequality if $n\geqslant (k+1)(k+2)/2$. For example,
	\begin{align*}
	& p(n-1) \geqslant D(n),\\
	& p(n-1)-p(n-3) \leqslant D(n),\\
	& p(n-1)-p(n-3)+p(n-6) \geqslant D(n), \text{ and}\\
	& p(n-1)-p(n-3)+p(n-6)-p(n-10) \leqslant D(n).
	\end{align*}
\end{corollary}

\section{Concluding remarks}

New infinite families of linear inequalities for the partition function $p(n)$ have been introduced in this paper considering
two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank.
Inspired by these results,  in Section \ref{S4} we considered the partitions with rank $\leqslant -2$ (Garden of Eden partitions)
and obtained another infinite families of linear inequalities for $p(n)$.

Theorems \ref{th:1} and \ref{th:2} allow us to derive the following theta identity.

\begin{corollary}\label{C:5.1}
	For $|q|<1$ and $k\geqslant 1$, there holds
	\begin{align*}
	&\sum_{n=1}^\infty \frac{q^{{k\choose 2}+(k+1)n}}{(q;q)_n}
	\begin{bmatrix}
	n-1\\k-1
	\end{bmatrix}\\
	& = \frac{q^{k(3k+1)/2}}{(q,q^3;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+2)}}{(q^3;q^3)_n(q^2;q^3)_{n+k}}
	- \frac{q^{k(3k+5)/2+1}}{(q^2,q^3;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+4)}}{(q^3;q^3)_n(q;q^3)_{n+k+1}}.
	\end{align*}
\end{corollary}

A similar theta identity can be derived if we consider another truncated form of Euler's pentagonal number theorem given by
D. Shanks \cite{Shanks} in $1951$:
\begin{align}\label{eq:S}
1+\sum_{n=1}^{k} (-1)^n \left(q^{n(3n+1)/2} + q^{n(3n-1)/2} \right)
= \sum_{n=0}^k (-1)^n \frac{q^{{n+1\choose 2}+kn}(q;q)_k}{(q;q)_n}.
\end{align}

\begin{corollary}\label{C:5.2}
	For $|q|<1$ and $k\geqslant 0$, there holds
	\begin{align*}
	& \frac{(-1)^k}{(q;q)_\infty}\sum_{n=0}^k (-1)^n \frac{q^{{n+1\choose 2}+kn}(q;q)_k}{(q;q)_n}-(-1)^k\\
	& =  \frac{q^{k(3k+7)/2+2}}{(q,q^3;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+5)}}{(q^3;q^3)_n(q^2;q^3)_{n+k+1}}
	+ \frac{q^{k(3k+5)/2+1}}{(q^2,q^3;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+4)}}{(q^3;q^3)_n(q;q^3)_{n+k+1}}.
	\end{align*}
\end{corollary}

The Shanks identity \eqref{eq:S} and Corollary \ref{C:5.2} allow us to obtain the following infinite family of linear inequalities: For $n>0$, $k\geqslant 1$,
\begin{equation*}
(-1)^{k}\left( p(n)+ \sum_{j=1}^{k} (-1)^j
\Big( p\big(n-j(3j+1)/2\big) - p\big(n-j(3j-1)/2\big) \Big) \right) \geqslant 0,
\end{equation*}
with strict inequality if $n> k(3k+5)/2$. We remark that this inequality is weaker than the inequality \eqref{eq:1.1}. However, a partition theoretic interpretation for it would be very interesting.

Relevant to Theorem \ref{th:2} and Corollaries \ref{C:5.1} and \ref{C:5.2}, it would be very appealing to have combinatorial interpretations for
$$
\frac{q^{k(3k+1)/2}}{(q,q^3;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+2)}}{(q^3;q^3)_n(q^2;q^3)_{n+k}},
$$
$$
\frac{q^{k(3k+5)/2+1}}{(q^2,q^3;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+4)}}{(q^3;q^3)_n(q;q^3)_{n+k+1}},
$$
and
$$
\frac{q^{k(3k+7)/2+2}}{(q,q^3;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{n(3n+3k+5)}}{(q^3;q^3)_n(q^2;q^3)_{n+k+1}}.
$$
Finally, with regard to Theorems \ref{Th:3.2} and  \ref{Th:4.1}, partition theoretic interpretation for
$$
q^{k(k+1)/2} \sum_{n=0}^\infty \frac{q^{n(n+k+1)}}{(q;q)_n(q;q)_{n+k}}
$$
and
$$\frac{q^{3(k+1)(k+2)/2}}{(q,q^2;q^3)_\infty} \sum_{n=0}^\infty \frac{q^{3n(n+k+2)}}{(q^3;q^3)_n(q^{3};q^3)_{n+k+1}}$$
would be very interesting.

\section*{Acknowledgements}
The author expresses his gratitude to Professor George E. Andrews for some helpful suggestions.

config: arxiv


TITLE: degree of a sum of two algebraic numbers
QUESTION [4 upvotes]: Let $a=\sqrt{2}$ and $b=\sqrt{3}$, then $a$ and $b$ are algebraic numbers of degree 2, while the degree of $a+b$ is not 2, actually it is 4, by the standard argument: if $x=a+b$, then we can rationalize it completely and get $x^4-10x^2+1=0$, so $a+b$ is of degree 4, since the polynomial is irreducible.
Now take $a=\sqrt[2019]{2}$ and $b=\sqrt[2019]{3}$. How to prove that the degree of $a+b$ is:

greater than 2019? (if it can be done with less advanced methods then the second part)
equal $2019^2$? (if this value is correct)

REPLY [1 votes]: Let $\alpha=2^{1/2019}$, $\beta=3^{1/2019}$, $\zeta=\exp(2\pi i/2019)$, $K=\mathbb{Q}(\alpha)$, and $L=\mathbb{Q}(\zeta,\beta)$.
Here, we prove the following statement: $\alpha\notin L$ and the degree of $\alpha$ over $L$ is $2019$.
Then it implies the validity of the second claim of mathprincess since
$$\frac{(\zeta^m - 1)\alpha}{(1-\zeta^n)\beta} = 1,$$
cannot hold if $\alpha\notin L$. This also leads to the validity of the first claim and a full solution to this question.
It is known that $[L:\mathbb{Q}]=2019\phi(2019)$ (see Jacobson, Velez cited in this answer: Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}[x]$). Thus, the basis of $\mathbb{Q}(\zeta)$ over $\mathbb{Q}$ are also a basis of $L$ over $\mathbb{Q}(\beta)$. Then
$$
\textrm{disc}_{\mathbb{Q}(\beta)}L=\textrm{disc}_{\mathbb{Q}}{\mathbb{Q}(\zeta)} \bigg\vert (2019)^{\phi(2019)}.
$$
It is well-known that a rational prime $p$ does not divide the discriminant $\textrm{disc}_{\mathbb{Q}}L$ if and only if $p$ is unramified in $L$. Then we have by (Exrcise 23b of Marcus 'Number Fields'),
$$\begin{align}
\textrm{disc}_{\mathbb{Q}}L &\bigg\vert (\textrm{disc}_{\mathbb{Q}}{\mathbb{Q}(\beta)})^{\phi(2019)} N_{\mathbb{Q}}^{\mathbb{Q}(\beta)}\textrm{disc}_{\mathbb{Q(\beta)}}L\\
&\bigg\vert (3^{2018}\cdot 2019^{2019})^{\phi(2019)}((2019)^{\phi(2019)})^{2019}.\end{align}
$$
This gives $2\nmid \textrm{disc}_{\mathbb{Q}}L$.
Thus, the rational prime $2$ is unramified in $L$. Now, take any prime $\mathcal{P}\subseteq L$ lying above $2$. Then $\mathcal{P}$ divides any coefficient of $x^{2019}-2$ except for the leading coefficient, and $\mathcal{P}^2$ does not divide the constant coefficient $2$ since $2$ is unramified in $L$. Hence, by Eisenstein, $x^{2019}-2$ over $L$ is irreducible. This shows that the degree of $\alpha$ over $L$ must be $2019$. Moreover, the same argument also yields that the degree of $\alpha$ over $\mathbb{Q}(\beta)$ is $2019$.
As in mathprincess's answer, we may proceed with $\mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha,\beta)$. Then
$$\begin{align}
[\mathbb{Q}(\alpha+\beta):\mathbb{Q}]&=[\mathbb{Q}(\alpha+\beta):\mathbb{Q}(\beta)][\mathbb{Q}(\beta):\mathbb{Q}]\\
&=[\mathbb{Q}(\alpha,\beta):\mathbb{Q}(\beta)][\mathbb{Q}(\beta):\mathbb{Q}]
=2019^2.\end{align}
$$


TITLE: What is the proof of - $E[2]$ torsion group is isomophic to $Z_2 \oplus Z_2$?
QUESTION [2 upvotes]: This is from the book "Elliptic Curves" by Lawrence Washington.
$E$ is an elliptic curve over $K$
$E[n] = \{P \in E(\bar K) \mid nP = \infty\}$
where $\bar K$ is the algebraic closure of $K$
The points of $E[2]$ are ${\infty, (e_1, 0), (e_2, 0), (e_3, 0)}$
I understood how they get up to the above.
But after this the book says
As an abstract group, this is isomorphic to $Z_2 \oplus Z_2$
They haven't expanded further on this - so it looks like the above is something which should be obvious but I don't understand how the above is true.
For it to be isomorphic to $Z_2 \oplus Z_2$, I think the following needs to be shown
$(e_1, 0) + (e_2, 0) = (e_3, 0)$ and so on
How do we prove the above?

REPLY [0 votes]: From @reuns' answer, we get
$e_1 + e_2 + e_3 = 0$
Let $P = (\infty,\infty), Q = (e_1,0), R = (e_2,0),S = (e_3,0)$
Let $N = \{x, y\} = Q + R$
The formula for point addition for elliptic curves is
$m = \frac {y_2 - y_1}{x_2 - x_1}$
$m = \frac {0 - 0} {e_2 - e_1} = 0 $
$x = m^2 - e_1 - e_2 \bmod p$
$x = -e_1 - e_2$
$e_1 + e_2 + e_3 = 0$, so $e_3 = -e_1 - e_2$
So $x = e_3$
$y = m(x_1 - x) - y_1 \bmod p$
$y = m(e_1 - e_3) - 0 \bmod p$
$y = 0$
So $N = Q + R = \{e_3, 0\} = S$
Likewise we can prove that
$R + S = Q$
$S + Q = R$


TITLE: $P(X < Y)$ using conditional probability
QUESTION [0 upvotes]: Let $X$ and $Y$ be i.i.d. random variables with Exponential$(\lambda)$ distribution. Using conditioning, determine a) $P(X < Y)$ and b) the distribution function of $X + Y$.

Attempt:
I thought I'd calculate the probability with $P(X < Y) = P(X < y |Y = y)$. To do that you first determine the conditional density, then calculate $P(X < y |Y = y)$. So,
Since $X$ and $Y$ are independent, then $f_{X, Y}(x, y) = f_X(x)f_Y(y)$. Then the conditional distribution is
$$ f_{X|Y}(x|y) = \frac{f_X(x)f_Y(y)}{f_X(x)}, ~~x>0,~~y>0 $$
That is
$$ f_{X|Y}(x|y) = f_X(x),~~x>0,~~~y>0 $$
By this point I'm already thinking I'm doing it totally wrong because when is a problem that easy, right? Anyway, continuing this line of reasoning you get
$$ P(X < x|Y = y) = \int_{0}^{y}\lambda e^{-\lambda x}dx = F_X(y) = 1 - e^{-\lambda y}$$

Official Answer:
  $P(X < Y) = 1/2 $.

Since my answer for a) is wrong, I didn't even try b).

REPLY [2 votes]: We have
\begin{align}
\mathbb P(X<Y) &= \int_0^\infty \mathbb P(X<Y\mid Y=y) f_Y(y)\ \mathsf dy\\
&=\int_0^\infty \int_0^y f_X(x)\ \mathsf dx\ f_Y(y)\ \mathsf dy\\
&=\int_0^\infty \int_0^y \lambda e^{-\lambda x}\ \mathsf dx \lambda e^{-\lambda y}\ \mathsf dy\\
&=\int_0^\infty (1-e^{-\lambda y})\lambda e^{-\lambda y}\ \mathsf dy\\
&=\frac12.
\end{align}
For the second part, convolution is the simplest method; for $t\geqslant 0$ the density of $Z:=X+Y$ is
\begin{align}
f_Z(t) &= (f_X\star f_Y)(t)\\
&= \int_{\mathbb R}f_X(s)f_Y(t-s)\ \mathsf ds\\
&= \int_0^t \lambda e^{-\lambda s}\lambda e^{-\lambda(t-s)}\ \mathsf ds\\
&= \lambda^2\lambda e^{-\lambda t}\int_0^t \ \mathsf ds\\
&= \lambda(\lambda t)e^{-\lambda t}.
\end{align}
I suppose there is a way to obtain this by conditioning but the computations would likely be more complicated.


text:
\section{Fractional and Multifractional Ornstein-Uhlenbeck Process }
\label{sec:mfracOU}
FBM and MBM are used to model long memory phenomena. For describing systems which are short-range dependent, Ornstein-Uhlenbeck process can be a suitable candidate. Recall that Ornstein-Uhlenbeck process is the solution ofthe ordinary Langevin equation
\begin{align}
  D_t x(t) + \omega x(t) & = \eta(t) ,
\label{eq:mfracOU_0010}
\end{align}
where $\eta(t)$ is standard white noise
which can be regarded as the time derivative of Brownian motion in the sense of generalized function.
Assuming $x(a) = 0$, the solution of (\ref{eq:mfracOU_0010}) is given by
\begin{align}
  x(t) & = \int_a^t e^{\omega(t-u)} \eta(u)du .
\label{eq:mfracOU_0020}
\end{align}
There are several ways to fractionalize Ornstein-Uhlenbeck process.
One way is to replace the white noise by a fractional Gaussian noise in
(\ref{eq:mfracOU_0010})
or (\ref{eq:mfracOU_0020}),
\cite{Cheridito03,LimMuniandy03}
or one can apply the Lamperti transformation to fractional Brownian motion.
\cite{LimMuniandy03,Magdziarz08}

In this paper we shall consider a different type of FOU processes.
FOU process of Weyl type and Riemann-Liouville type can be defined as\cite{LimEab06}
\begin{align}
  Y_\alpha^W(t) & = \frac{1}{\Gamma(\alpha)}\int_{-\infty}^t (t - u)^{\alpha-1} e^{\omega(t-u)} \eta(u)du , \label{eq:mfracOU_0030}\\
  Y_\alpha^{RL}(t) & = \frac{1}{\Gamma(\alpha)}\int_0^t (t - u)^{\alpha-1} e^{\omega(t-u)} \eta(u)du .
\label{eq:mfracOU_0040}
\end{align}
The condition $\alpha > 1/2$ is imposed to ensure finite variance for both the FOU processes.
(\ref{eq:mfracOU_0030}) and (\ref{eq:mfracOU_0040}) can be regarded as the generalizations of (\ref{eq:mfracOU_0020}),
with $a = -\infty$ and $a = 0$.
These fractional processes are solutions to the following nonlinear fractional Langevin equation:
\begin{align}
  \bigl({_aD_t + \omega}\bigr)^\alpha Y(t) & = \eta(t) .
\label{eq:mfracOU_0050}
\end{align}

The Weyl fractional Ornstein-Uhlenbeck process $Y_\alpha^W(t)$ is stationary
centred Gaussian process with variance and covariance
\begin{subequations}
\label{eq:mfracOU_x010}
\begin{align}
  E\Bigl(\bigl[Y_\alpha^W(t)\bigr]^2\Bigr) & = \frac{\Gamma(2\alpha-1)(2\omega)^{1-2\alpha}}{\Gamma(\alpha)^2},
\label{eq:mfracOU_x010a}\\
  E\Bigl(Y_\alpha^W(t)Y_\alpha^W(s)\Bigr) & = \frac{1}{\sqrt{\pi}\Gamma(\alpha)}
                                           \left(\frac{|t-s|}{2\omega}\right)^{\alpha-1/2}
                                           K_{\alpha-1/2}\bigl(\omega|t - s|\bigr), &
                                           t & \neq s,
\label{eq:mfracOU_x010b}
\end{align}
\end{subequations}
where $K_\nu(z)$ is the modified Bessel function of second kind.\cite{LimTeo09b}
On the other hand, the Riemann-Liouville fractional Ornstein-Uhlenbeck process
$Y_\alpha^{RL}(t)$ is a non-stationary centred Gaussian process with variance and covariance
\begin{subequations}
\label{eq:mfracOU_x020}
  \begin{align}
     E\Bigl(\bigl[Y_\alpha^{RL}(t)\bigr]^2\Bigr) & = \frac{(2\omega)^{1-2\alpha}\gamma(2\alpha-1,2\omega{t})}{\Gamma(\alpha)^2},
\label{eq:mfracOU_x020a}\\
     E\Bigl(Y_\alpha^{RL}(t)Y_\alpha^{RL}(s)\Bigr) & = \frac{e^{-\omega(t+s)}s^\alpha t^{\alpha-1}}{\Gamma(\alpha+1)\Gamma(\alpha)}
                                                   \Phi_1\Bigl(1,1-\alpha,1+\alpha),\frac{s}{t},2\omega(s)\Bigr),
                                               \ t  > s,
\label{eq:mfracOU_x020b}
  \end{align}
\end{subequations}
where $\gamma(a,x)$ is the incomplete Gamma function, and
$\Phi_1(a,b,c,x,y)$ is the confluent hypergeometric function in two variables.
For discussion of properties and applications of the FOU process of Weyl and RL type, and their extension to FOU process with two indices
can be found elsewhere.\cite{LimEab06,LimLiTeo08,LimTeo09}

Just like the case of MBM, one can extend the two types of FOU processes to their corresponding multifractional OU (MOU)
processes by replacing $\alpha$ by $\alpha(t)$.
The covariance of the MOU process of Weyl type for $s < t$ is given by
\begin{gather}
  E\Bigl(Y_{\alpha(t)}^W(t)Y_{\alpha(s)}^W(s)\Bigr)  = \frac{e^{-\omega(t+s)}}{\Gamma\bigl(\alpha(t)\bigr)\Gamma\bigl(\alpha(s)\bigr)}
                                                    \int_{-\infty}^s (t - u)^{\alpha(t)-1}(s - u)^{\alpha(s)-1} e^{2\omega{u}}du \nonumber \\
                                                 = \frac{e^{-\omega(t-s)}}{\Gamma\bigl(\alpha(t)\bigr)\Gamma\bigl(\alpha(s)\bigr)}
                                                    \int_0^\infty u^{\alpha(s)-1}(u+ t - s)^{\alpha(t)-1} e^{-2\omega{u}}du \nonumber \\
                                          \qquad       = \frac{e^{-\omega(t-s)}(t-s)^{\alpha(s)+\alpha(t)-1}}{\Gamma\bigl(\alpha(t)\bigr)}
                                                    \Psi\bigl(\alpha(s),\alpha(s) + \alpha(t), 2\omega(t-s)\bigr),
\label{eq:mfracOU_0060}
\end{gather}
where
$\Psi(\alpha,y;z)$ is the confluent hypergeometric function.
In contrast to the Weyl fractional Ornstein-Uhlenbeck process, the multifractional process is in general not stationary.

For MOU process of RL type, its covariance for $s < t$ is
\begin{gather}
  E\Bigl(Y_{\alpha(t)}^{RL}(t)Y_{\alpha(s)}^{RL}(s)\Bigr)  = \frac{e^{-\omega(t+s)}}{\Gamma\bigl(\alpha(t)\bigr)\Gamma\bigl(\alpha(s)\bigr)}
                                                         \int_0^s (t - u)^{\alpha(t)-1}(s - u)^{\alpha(s)-1} e^{2\omega{u}}du \nonumber \\
                                                     = \frac{e^{-\omega(t+s)}s^{\alpha(s)}t^{\alpha(t)-1}}{\Gamma\bigl(\alpha(t)\bigr)\Gamma\bigl(\alpha(s)\bigr)}
                                                    \int_0^1 (1 - u)^{\alpha(s)-1}\Bigl(1 - \frac{s}{t}u\Bigr)^{\alpha(t)-1} e^{2\omega{us}}du \nonumber \\
                                           \mkern-27mu
          = \frac{e^{-\omega(t+s)}s^{\alpha(s)}t^{\alpha(t)-1}}{\Gamma\bigl(\alpha(s)+1\bigr)\Gamma\bigl(\alpha(s)\bigr)}
                                                       \Phi_1\bigl(1,1-\alpha(t),1+\alpha(s),s/t,2\omega(s)\bigr).
\label{eq:mfracOU_0070}
\end{gather}

The local properties of these two types of MOU processes are similar to that of the corresponding MBM.
With probability one, both the functions $Y_\alpha^W(t)$ and $Y_\alpha^{RL}(t)$  have
H\"{o}lder exponent $\alpha(t_\circ) - 1/2$  at the point $t_\circ$; and the Hausdorff dimension of the two processes is $5/2 - \alpha(t_\circ)$.\cite{LimTeo07}
In addition, MOU processes of Weyl and RL-type are locally asymptotically self-similar,
their tangent process at a point $t_\circ$ is the FBM indexed by parameter $\alpha(t_\circ) - 1/2$.
In contrast to MBM, MOU processes are SRD, that is they are short memory processes.

The remark concerning the multifractality of the multifractional Brownian motion applies to the multifractional Ornstein-Uhlenbeck process.
That is, the multifractional process is truly multifractal if the H\"{o}lder exponent is a random function, otherwise it is a multiscaling process.
However, there are many phenomena that are multiscaling instead of multifractal.


text:
\begin{document}

\begin{frontmatter}

\title{New Edgeworth-type expansions with finite sample guarantees\thanksref[1]{a2}}

\runtitle{Edgeworth-type expansions with finite sample bounds}

 \author{\fnms{Mayya} \snm{Zhilova}\thanksref[2]{a1}
 \corref{}\ead[label=e1]{mzhilova@math.gatech.edu}}
\thankstext{a2}{\emph{Submitted on June 5, 2020.}}
\thankstext{a1}{Support by the National Science Foundation grant DMS-1712990 is gratefully acknowledged}
\address{School of Mathematics\\Georgia Institute of Technology\\
Atlanta, GA 30332-0160 USA \\\printead{e1}}

\runauthor{M. Zhilova}

\affiliation{School of Mathematics\\Georgia Institute of Technology}
\begin{abstract}
We establish higher-order expansions for a difference between probability distributions of sums of i.i.d. random vectors in a Euclidean space. The derived bounds are uniform over two classes of sets: the set of all Euclidean balls and the set of all half-spaces. These results allow to account for an impact of higher-order moments or cumulants of the considered distributions; the obtained error terms depend on a sample size and a dimension explicitly. The new inequalities outperform accuracy of the normal approximation in existing Berry--Esseen inequalities under very general conditions. For symmetrically distributed random summands, the obtained results are optimal in terms of the ratio between the dimension and the sample size. The new technique which we developed for establishing nonasymptotic higher-order expansions can be interesting by itself.
Using the new higher-order inequalities, we study accuracy of the nonparametric bootstrap approximation and propose a bootstrap score test under possible model misspecification. The results of the paper also include explicit error bounds for general elliptical confidence regions for an expected value of the random summands, and optimality of the Gaussian anti-concentration inequality over the set of all Euclidean balls.
\end{abstract}

\begin{keyword}[class=MSC]
\kwd[Primary ]{62E17}
\kwd{62F40}
\kwd[; secondary ]{62F25}
\end{keyword}

\begin{keyword}
\kwd{Edgeworth series, dependence on dimension, higher-order accuracy, multivariate Berry--Esseen inequality, finite sample inference, anti-concentration inequality, bootstrap, elliptical confidence sets, linear contrasts, bootstrap score test, model misspecification}
\end{keyword}
\end{frontmatter}

\section{Introduction}
\input{ES_intro_5}
\input{ES_intro_contrib_2}
\input{ES_intro_notation}
\input{ES_mainres_multivar_6}
\input{ES_technique}
\input{ES_symm_1}
\input{ES_bootst_6}
\input{ES_ellip}
\input{ES_score_1}
\input{ES_theor_earlier_1}
\section{Proofs for Sections \ref{sect:multivar} and  \ref{sect:symm}}
\label{sect:proof_2}
\input{ES_proof_theorem_1_1}
\input{ES_proof_gener_1}
\input{ES_proof_theorem_2}
\input{ES_symm_proofs}
\input{ES_bootst_proofs_2}

\bibliographystyle{apalike}
\bibliography{references_1,ref_appr.bib,ref_appr_ES.bib}

\end{document}