Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

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\begin{document}
\title{On a quantitative version of the Oppenheim conjecture.}

\author{Alex Eskin}
\address{Department of Mathematics, University of Chicago, Chicago IL~60637,
USA}
\email{eskin@math.uchicago.edu}
\thanks{Research of the first author partially supported by an NSF
    postdoctoral fellowship and by BSF grant 94-00060/1}

\author{Gregory Margulis}
\address{Department of Mathematics, Yale University, New Haven CT,
  USA}
\email{margulis@math.yale.edu}
\thanks{Research of the second author partially supported
                  by NSF grants DMS-9204270 and DMS-9424613}

\author{Shahar Mozes}
\address{Institute of Mathematics, Hebrew University, Jerusalem~91904,
  ISRAEL}
\email{mozes@math.huji.ac.il}
\thanks{Research of the third author partially supported by
the Israel Science foundation and by BSF grant 94-00060/1}
%                {\em Preliminary draft -- not for distribution}

\subjclass{Primary 11J25, 22E40}

\date{December 6, 1995}

\renewcommand{\currentvolume}{1}
\renewcommand{\currentissue}{3}

\setcounter{page}{124}


\begin{abstract}
The Oppenheim conjecture, proved by Margulis in 1986, states that the
set of values at integral points of an indefinite quadratic form in
three or more variables is dense, provided the form is not
proportional to a rational form. In this paper we study the
distribution of values of such a form. We show that if the signature
of the form is not $(2,1)$ or $(2,2)$ then the values are uniformly
distributed on the real line, provided the form is not proportional to
a rational form. In the cases where the  signature is
$(2,1)$ or $(2,2)$  we show that no such universal formula exists, and
give asymptotic upper bounds which are in general best possible. 
\end{abstract}

\maketitle
\bigskip


Let $Q$ be an indefinite nondegenerate  quadratic form in $n$
variables. Let $\cL_Q = Q(\zed^n)$
denote the set of values of $Q$ at integral points. The Oppenheim
conjecture, proved by Margulis
(cf. \cite{Margulis:number}) states that if 
$n \ge 3$, and $Q$ is not proportional to a form with rational
coefficients, then $\cL_Q$ is dense. The Oppenheim conjecture enjoyed
attention and many studies since it was conjectured in 1929 mostly using
analytic number theory methods. In this paper\footnote{The full
  version is available electronically at
{\tt http://www.math.uchicago.edu/\symbol{126}eskin}} 
we study some finer questions related to the distribution the values
of $Q$ at integral points. 


\bold{\S\thesection.}
Let $\nu$ be a continuous positive function on the sphere $\{ v \in
\reals^n \ | \ \|v \| = 1 \}$, and let $\Omega = \{ v \in \reals^n \ | \
\| v \| < \nu(v/\|v\|) \}$. 
We denote by $T\Omega$ the dilate of $\Omega$ by $T$. 
Define the following set:
%\begin{displaymath}
%\cL_Q(T\Omega) = \{ Q(x) \ | \ x \in \zed^n \cap T\Omega \}. 
%\end{displaymath}
$$
V_{(a,b)}^Q(\reals)=\{x \in \reals^n \ | \ a 0$ is
a constant depending on $Q$ and $\Omega$. Thus one might expect that 
for any interval $[a,b]$, as $T \to \infty$, 
\begin{equation}
\label{eq:guess}
| V_{(a,b)}(\zed) \cap T\Omega  | \sim c_{Q,\Omega} (b-a) T^{n-2}
\end{equation}
where $c_{Q,\Omega}$ is a constant depending on $Q$ and $\Omega$. 
This may be interpreted as ``uniform distribution'' of the sets
$Q(\zed^n \cap T\Omega)$ in the real line.
Our main
result is that (\ref{eq:guess}) holds if  $Q$ is not proportional to
a rational form, and has signature $(p,q)$ with $p \ge 3$, $q \ge 1$.
We also determine the constant $c_{Q,\Omega}$.



If $Q$ is an indefinite quadratic form in $n$ variables, $\Omega$ is as
above and $(a,b)$
is an interval, we show 
that there exists a constant $\lambda = \lambda_{Q,\Omega}$ so that
as $T \to \infty$, 
\begin{equation}
\label{eq:volume}
\Vol( V_{(a,b)}(\reals) \cap T \Omega ) \sim
\lambda_{Q,\Omega}(b-a)T^{n-2} 
\end{equation}

Our main result is the following:
\begin{theorem}
\label{theorem:main}
Let $Q$ be an indefinite quadratic form of signature $(p,q)$, with
$p \ge 3$ and $q \ge 1$. Suppose $Q$ is not proportional to a rational
form. Then for any interval $(a,b)$, as $T \to \infty$, 
\begin{displaymath}
| V_{(a,b)}(\zed) \cap T\Omega  |
 \sim \lambda_{Q,\Omega} (b-a) T^{n-2}
\end{displaymath}
where $n=p+q$, and $\lambda_{Q,\Omega}$ is as in {\rm (\ref{eq:volume})}.
\end{theorem}
Only the upper bound in this formula is new: the asymptotically exact 
lower bound
was proved in \cite{Dani:Margulis:distribution}. 
Also a lower bound with a  smaller constant was obtained independently
by M. Ratner, and by S. G. Dani jointly with S. Mozes
(both unpublished).
  

If the signature of $Q$ is $(2,1)$ or $(2,2)$ then no universal
formula like (\ref{eq:guess}) holds. In fact, we have the following
theorem:
\begin{theorem}
\label{theorem:counterexample}
Let $\Omega_0$ be the unit ball, and let $q = 1$ or $2$. Then
for every $\epsilon > 0$ and every interval $(a,b)$ there exists
a quadratic form $Q$ of signature $(2,q)$ not proportional to a
rational form, and a constant $c > 0$
such that for an
infinite sequence $T_j \to \infty$, 
\begin{displaymath}
| V_{(a,b)}(\zed)\cap T\Omega_0 |
 > c T_j^q ( \log T_j)^{1-\epsilon}.
\end{displaymath}
\end{theorem}
The case $q=1$, $b\le 0$ of Theorem~\ref{theorem:counterexample} was
noticed by P. Sarnak and worked out in detail in
\cite{Brennan:thesis}. The quadratic forms constructed are of the form
$x_1^2 + x_2^2 - \alpha x_3^2$, or $x_1^2 + x_2^2 - \alpha(x_3^2 +
x_4^2)$, where $\alpha$ is extremely well approximated by squares of
rational numbers.


However in the $(2,1)$ and $(2,2)$ cases, we can still establish an
upper bound of the form
$c T^q \log T$. This upper bound is effective, and is uniform 
over compact sets in the set of quadratic forms. We also give
an effective uniform upper bound for the case $p \ge 3$.
\begin{theorem}
\label{theorem:upper:bound}
Let $\cO(p,q)$ denote the space of quadratic forms of signature
$(p,q)$ and discriminant $\pm 1$, let $n=p+q$, $(a,b)$ be an interval, 
and let $\cD$ be a compact subset of $\cO(p,q)$. 
Let $\nu$ be a continuous positive function on the unit sphere  
and let $\Omega = \{ v \in \reals^n \ | \
\| v \| < \nu(v/\|v\|) \}$. 
Then, if $p \ge 3$ there exists
a constant $c$ depending only on $\cD$, $(a,b)$ and $\Omega$ such that for
any $Q \in \cD$ and all $T > 1$, 
\begin{displaymath}
| V_{(a,b)}(\zed) \cap T\Omega |
< c T^{n-2}
\end{displaymath}
If $p =2$ and $q=1$ or $q=2$, then there exists
a constant $c > 0$ depending only on $\cD$, $(a,b)$ and $\Omega$ such that for
any $Q \in \cD$ and all $T > 2$, 
\begin{displaymath}
| V_{(a,b)}\cap T\Omega \cap \zed^n |
< c T^{n-2} \log T
\end{displaymath}
\end{theorem}
Also, for the $(2,1)$ and $(2,2)$ cases, we have the following ``almost
everywhere'' result:
\begin{theorem}
\label{theorem:almost:everywhere}
For almost all quadratic forms $Q$ of signature $(p,q) =(2,1)$ or $(2,2)$
\begin{displaymath}
| V_{(a,b)}(\zed)\cap T\Omega  |
\sim \lambda_{Q,\Omega} (b-a) T^{n-2}
\end{displaymath}
where $n=p+q$, and $\lambda_{Q,\Omega}$ is as in {\rm (\ref{eq:volume})}.
\end{theorem}
Theorem~\ref{theorem:almost:everywhere} may be proved using a
recent general result of Nevo and Stein \cite{Nevo:Stein:balls}; we
present a self-contained argument which suffices for the application in
the full paper. In [Sar], P. Sarnak proved that an analogous asymptotic formula
holds for almost all forms within a specific two-parameter family;
this family arises in problems related to Quantum Chaos. 
\par\noindent

Finally, following \cite{Dani:Margulis:distribution} 
we give a ``uniform'' version of 
Theorem~\ref{theorem:main}:
\begin{theorem}
\label{theorem:uniform}
Let $\cD$ be a compact subset of $\cO(p,q)$, with $p \ge 3$. 
Let $n=p+q$, and let
$\Omega$ be as in Theorem~\ref{theorem:upper:bound}. 
Then for every interval $[a,b]$
and every $\theta >0$, there exists a finite subset $\cP$ of $\cD$
such that each $Q \in \cP$ is a scalar multiple of a rational form
and for any compact subset $\cF$ of $\cD -\cP$ there exists $T_0$ such
that for all $Q$ in $\cF$ and $T \ge T_0$,
\begin{displaymath}
( 1 - \theta) \lambda_{Q,\Omega} (b-a)T^{n-2} \le
| V_{(a,b)}(\zed)\cap T\Omega |
\le ( 1 + \theta )  \lambda_{Q,\Omega} (b-a)T^{n-2}
\end{displaymath}
where $\lambda_{Q,\Omega}$ is as in {\rm (\ref{eq:volume})}.
\end{theorem}
As in Theorem~\ref{theorem:main} only the upper bound is new; the
asymptotically exact 
lower bound, which holds even for $SO(2,1)$ and $SO(2,2)$, was proved in
\cite{Dani:Margulis:distribution}.

\begin{remark}
\label{remark:primitive}
If we consider $| V_{(a,b)}(\reals)\cap T\Omega \cap \cP(\zed^n) |$ instead of 
$| V_{(a,b)}(\zed)\cap T\Omega |$ (where $\cP(\zed^n)$ denotes the
set of primitive lattice points, 
then Theorem~\ref{theorem:main} and Theorem~\ref{theorem:uniform} hold
 provided one
replaces $\lambda_{Q,\Omega}$ by $\lambda'_{Q,\Omega} =
\lambda_{Q,\Omega}/\zeta(n)$, where $\zeta$ is the Riemann zeta function.
\end{remark}

\begin{remark}
\label{remark:non:effective}
Theorem~\ref{theorem:main} and Theorem~\ref{theorem:uniform}, as well
as lower bounds in \cite{Dani:Margulis:distribution} and even the
proof of the Oppenheim conjecture in \cite{Margulis:number}, 
are not effective.  It seems to be a very difficult problem to
give effective versions of these results.
\end{remark}




\bold{Passage to the space of lattices.}
We now relate the counting problem of
Theorem~\ref{theorem:main} to a certain integral expression involving
the orthogonal group of the quadratic form and the space of lattices
$SL(n,\reals)/SL(n,\zed)$. Roughly this is done as follows. Let $f$ be
a bounded function on $\reals^n-\{0\}$ vanishing outside a compact subset. For
$g \in SL(n,\reals)$ let
\begin{equation}
\label{eq:toy}
\tilde{f}(g) = \sum_{v \in \zed^n} f(g v)
\end{equation}
The proof is based on the identity of the form
\begin{equation}
\label{eq:toy:identity}
\int_K\tilde{f}(a_t k ) \,dk  = \sum_{v \in \zed^n} 
\int_K f( a_t k v)  \, dk
\end{equation}
obtained by integrating (\ref{eq:toy}). 
In (\ref{eq:toy:identity})
 $\{ a_t \}$ is a certain diagonal subgroup of the
orthogonal group of $Q$, and $K$ is a maximal compact subgroup of the
orthogonal group of $Q$. Then for an appropriate function $f$, 
the right hand side is then related to the number of lattice points 
$v \in [e^t/2,e^t]\partial\Omega$ with $a < Q(v) < b$.  
We then establish the 
asymptotics of the left-hand side using the ergodic theory of
unipotent flows and some other techniques.

\bold{Lattices.}
Let $\Delta$ be a lattice in ${\bb R}^n$.  We say that a subspace $L$
of ${\bb R}^n$ is $\Delta$-{\it rational} if $L \cap \Delta$ is a
lattice in $L$.  For any $\Delta$-rational subspace $L$, we denote by
$d_\Delta (L)$ or simply by $d (L)$ the volume of $L / (L \cap
\Delta)$.  Let us note that $d (L)$ is equal to the norm of $e_1
\wedge \cdots \wedge e_\ell$ in the exterior power $\bigwedge^\ell ({\bb
  R}^n)$ where $\ell = \dim L$ and $(e_1, \cdots, e_\ell)$ is a basis
over $\zed$ of $L \cap \Delta$.  If $L = \{ 0 \}$ we write $d (L) = 1$.
A lattice is $\Delta$ {\em unimodular} if $d_\Delta({\bb R}^n) = 1$. The space
of unimodular lattices is isomorphic to $SL(n,\reals)/SL(n,\zed)$. 

Let us introduce the following notation:
\begin{align*}
  \alpha_i (\Delta) & = \sup \Big \{ \frac{1}{d (L)} \Big | \text{ $L$
    is a $\Delta$-rational subspace of dimension $i$ } \Big \}, \quad
  0 \le i \le n, \notag \\ \alpha (\Delta) & = \max\limits_{0 \le i \le n}
  \alpha_i (\Delta).
\end{align*}

The following lemma is known as the ``Lipshitz Principle'':
\begin{lemma}[{\cite[Lemma 2]{Schmidt:Asymptotic}}]
\label{lemma:Lipshitz}
Let $f$ be a bounded function on ${\bb R}^n$ vanishing outside a
compact subset.  Then there exists a positive constant $c = c(f)$
such that
\begin{equation}
\label{eq:Lipschitz}
  \tilde{f} (\Delta) < c \alpha (\Delta)
\end{equation}
for any lattice $\Delta$ in ${\bb R}^n$. Here $\tilde{f}$ is the
function on the space of lattices defined in (\ref{eq:toy}). 
\end{lemma}


\bold{Quadratic Forms.}
Let $n \ge 3$, and let $p \ge 2$. We denote $n-p$ by $q$, and assume $
q> 0$. 
Let $\{e_1,e_2,\dots e_n\}$ be the standard basis of $\reals^n$. Let
$Q_0$ be the quadratic form defined by
\begin{displaymath}
%\label{eq:def:Q0}
Q_0 \left(\sum_{i=1}^n v_i e_i \right) = 2 v_1 v_n + \sum_{i=2}^p v_i^2 -
\sum_{i = p+1}^{n-1} v_i^2 \quad\text{ for all $v_1, \dots ,v_n \in \reals$}. 
\end{displaymath}
It is straightforward to verify that $Q_0$ has signature $(p,q)$. 
Let $G = SL(n,\reals)$, the group of $n \cross n$ matrices of
determinant $1$. For each quadratic form $Q$ and $g \in G$, let 
$Q^g$ denote the quadratic form defined by $Q^g(v) = Q(g v)$ for all
$v \in \reals^n$. By the well known classification of quadratic forms
over $\reals$, for each $Q \in \cO(p,q)$ there exists $g \in G$ such
that $Q = Q_0^g$. For any quadratic form $Q$ let $SO(Q)$ denote the
special orthogonal group corresponding to $Q$; namely $\{ g \in G \ |
\ Q^g = Q \}$. Let $H = SO(Q_0)$. Then the map $H \bs G \to \cO(p,q)$
given by $H g \to Q_0^g$ is a homeomorphism.

For $t \in \reals$, let $a_t$ be the linear map so that
$a_t e_1 = e^{-t} e_1$, $a_t e_n = e^{t} e_n$, and
$a_t e_i = e_i$, $2 \le i \le n-1$. Then the one-parameter group 
$\{ a_t \}$ is contained in  $H$.
Let $\hat{K}$ be the subgroup of $G$ consisting of orthogonal matrices, and
let $K = H \cap \hat{K}$. It is easy to check that $K$ is a maximal
compact subgroup of $H$, and consists of all $h \in H$ leaving invariant the
subspace spanned by $\{ e_1 + e_n, e_2, \dots, e_p \}$. We denote by
$m$ the normalized Haar measure on $K$.

\bold{The main theorems.}
To prove Theorem~\ref{theorem:main} one may use the following theorem:
\begin{theorem} 
\label{theorem:convergence}
Suppose $p \ge 3$, $q \ge 1$.  Let $\phi$ be a
continuous function on $G/\Gamma \approx$
the space of lattices in ${\bb R}^n$ with determinant 1.  Assume that
for some $s$, $0 < s < 2$ and some $C > 0$,
\begin{displaymath}
  | \phi (\Delta) | < C \alpha (\Delta)^{s}, \; \quad\text{ for all }
  \Delta \in G/\Gamma 
\end{displaymath}
Let $x_0 \in G/\Gamma$ be a unimodular lattice such
that $H x_0$ is not closed. Let $\nu$ be any continuous function on
$K$. Then
\begin{displaymath}
\lim\limits_{t \to + \infty} \; \int\limits_K \phi (a_t k x_0) \nu(k)\,dm (k) =
\int\limits_K \nu \,dm \; \int\limits_{G / \Gamma} \; \phi (y) \,d\mu (y).
\end{displaymath}
\end{theorem}
To prove Theorem~\ref{theorem:uniform} we use the following
generalization:
\begin{theorem}
\label{theorem:convergence:uniform}
Suppose $p \ge 3$, $q \ge 1$. Let $\phi$, $\nu$ be as in
Theorem~\ref{theorem:convergence}, and let $\cC$ be any compact set in
$G/\Gamma$. Then for any $\epsilon >0$ there
exist finitely many points $x_1,\dots,x_\ell \in G/\Gamma$ such that
\begin{itemize}
\item[{\rm (i)}] The orbits $Hx_1, \dots,H x_\ell$ are closed and have
  finite $H$-invariant measure. 
\item[{\rm (ii)}] For any compact subset $F$ of $\cC \setminus
  \bigcup_{1 \le i \le \ell} H x_i$, there exists $t_0 > 0$, so that
  for all $x \in F$ and $t >t_0$, 
\begin{displaymath}
%\label{eq:convergence:uniform}
\left| \int\limits_{K} \phi(a_t k x) \nu(k) \, dm(k) -
  \int\limits_{G/\Gamma} \phi \, d\mu \int\limits_K \nu \,dm \right| \le \epsilon
\end{displaymath}
\end{itemize}
\end{theorem}


If the function $\phi$ is bounded, then
Theorem~\ref{theorem:convergence} and
Theorem~\ref{theorem:convergence:uniform} follow easily from 
\cite[Theorem 3]{Dani:Margulis:distribution}). 
This theorem is a refined version of Ratner's uniform distribution
theorem \cite{Ratner:distribution}; the proof uses 
Ratner's measure classification theorem (see
\cite{Ratner:solvable, Ratner:semisimple,Ratner:measure}), Dani's
theorem on the behavior of unipotent orbits at infinity
\cite{Dani:orbits:I,Dani:orbits:II}, and ``linearization''
techniques. 


Both \cite[Theorem 3]{Dani:Margulis:distribution} and 
Ratner's uniform distribution theorem
hold for bounded continuous
functions, but not for arbitrary continuous functions from
$L^1(G/\Gamma)$.  
However, for a non-negative bounded 
continuous function $f$ on $\reals^n$, the function
$\tilde{f}$ defined in (\ref{eq:toy}) is non-negative, continuous,
and $L^1$ but unbounded (it is in $L^s(G/\Gamma)$ for $1 \le s < n$,
where $G = SL(n,\reals)$, and $\Gamma = SL(n,\zed)$). 
As it was done in \cite{Dani:Margulis:distribution} it is possible to 
obtain asymptotically exact lower bounds by considering bounded
continuous functions $\phi \le \tilde{f}$. However, 
to carry
out the integrals in (\ref{eq:toy:identity}) and prove 
the upper bounds in the theorems stated above we need
to examine carefully the situation at the ``cusp'' of $G/\Gamma$, i.e
outside of compact sets. 

By (\ref{eq:Lipschitz})
the function $\tilde{f}(g)$ on the space of unimodular lattices $G/\Gamma$
is majorized by the function $\alpha(g)$. 
The function $\alpha$ is more convenient
since it is invariant under the
left action of the maximal compact subgroup $\hat{K}$ of $G$, and its
growth rate at infinity is known explicitly.  
Theorems~\ref{theorem:main} and \ref{theorem:uniform} are proved
by combining \cite[Theorem 3]{Dani:Margulis:distribution} with the
following integrability estimate:
\begin{theorem}
\label{theorem:onepar}
If $p \ge 3$, $q \ge 1$ and $0 < s < 2$, or if $p = 2$, $q \ge 1$
and $0 < s < 1$, then for any lattice $\Delta$ in ${\bb R}^n$
\begin{displaymath}
  \sup\limits_{t > 0} \int\limits_K \alpha (a_t k \Delta)^{s} \,dm (k) <
  \infty.
\end{displaymath}
The upper bound is uniform as $\Delta$ varies over compact sets
in the space of lattices.
\end{theorem}
This result can be interpreted as follows. 
For a lattice $\Delta$ in $G/\Gamma$ and for $h \in H$, let
$f(h) = \alpha(h \Delta)$. Since $\alpha$ is left-$\hat{K}$ invariant,
$f$ is a function on the symmetric space $X = K \bs H$. 
Theorem~\ref{theorem:onepar} is the statement that if 
if $p \ge 3$, then the averages of $f^s$, $0  1} \frac{1}{t} \; \int\limits_K \alpha (a_t k \Delta)
\,dm (k) < \infty, 
\end{displaymath}
The upper bound is uniform as $\Delta$ varies over compact sets
in the space of lattices.
\end{theorem}

\stepcounter{section}
\bold{\S\thesection.}
We now outline the proof of Theorems~\ref{theorem:onepar} and
\ref{theorem:SO(2,1):largecircle}. 
\relax From its definition, the function $\alpha(g)$ is
the maximum over $1 
\le i \le n$ of left-$\hat{K}$ invariant functions $\alpha_i(g)$. The
main idea of the proof is to show that the $\alpha_i$ satisfy a system
of integral inequalities which imply the desired bound.

If $p \ge 3$ and  $0  0$ 
there exist $t > 0$, and $\omega > 1$ so that
the the functions $\alpha_i^s$ satisfy the following system of integral
inequalities in the space of lattices:
\begin{equation}
\label{eq:alpha:vector:inequalities}
A_t \alpha_i^s \le c_i \alpha_i^s + \omega^2 \max_{0 \le j \le n-i,i} \sqrt{\alpha_{i+j}^s \alpha_{i-j}^s}
\end{equation}
where $A_t$ is the averaging operator $(A_t f)(\Delta) = \int_K f(a_t k \Delta
)$, and $c_i \le c$. 
If $(p,q) = (2,1)$ or $(2,2)$
and $s = 1$, then (\ref{eq:alpha:vector:inequalities}) also holds (for
suitably modified functions $\alpha_i$), but some of the constants
$c_i$ cannot be made smaller than $1$. 

Let 
$f_i(h) = \alpha_i(h \Delta)$, so that each $f_i$
is a function on the symmetric space $X$. 
When one restricts to an orbit of $H$,
(\ref{eq:alpha:vector:inequalities}) becomes:
\begin{equation}
\label{eq:vector:inequalities}
A_t f_i^s \le c_i f_i^s + \omega^2 \max_{0 \le j \le n-i,i} \sqrt{f_{i+j}^s f_{i-j}^s}
\end{equation}
If $\rank X =1$, then $(A_t f)(h)$ can be interpreted as the
average of $f$ over the sphere of radius $t$ in $X$, centered at $h$. 
We show that if the $f_i$ satisfy
(\ref{eq:vector:inequalities}) then for any $\epsilon > 0$, 
the function $f = f_{\epsilon,s} = \sum_{0 \le i \le n}
\epsilon^{i(n-i)}f_i^s$ satisfies the scalar inequality:
\begin{equation}
\label{eq:scalar:inequality}
A_t f \le c f + b
\end{equation}
where $t$, $c$ and $b$ are constants. 
We show that if $c$ is
sufficiently small, then (\ref{eq:scalar:inequality}) for a fixed $t$
together with the uniform continuity of $\log f$ imply that $(A_r
f)(1)$ is bounded as a function of $r$, which is the conclusion of
Theorem~\ref{theorem:onepar}. If $c=1$, which will occur in the
$SO(2,1)$ and $SO(2,2)$ cases, then (\ref{eq:scalar:inequality})
implies that $(A_r f)(1)$ is growing at most linearly with the radius,
which is the conclusion of Theorem~\ref{theorem:SO(2,1):largecircle}. 

Throughout the proof we consider the functions $\alpha(g)^s$ for $0
< s < 2$ even though for the application to quadratic forms we
only need $s=1+\delta$. This yields a better integrability result, and is
also necessary for the proofs of Theorem~\ref{theorem:convergence} and
Theorem~\ref{theorem:convergence:uniform}. 

\bold{Acknowledgments:}
The authors would like to thank Peter Sarnak for useful conversations.

 

%\bibliographystyle{math}
%\bibliography{eskin}

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\end{thebibliography}


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