Pointwise equidistribution with an error rate and with respect to unbounded functions
Dmitry Kleinbock
Ronggang Shi
Barak Weiss
Consider $G=\SL_{ d (\mathbb R)$ and $ \Gamma=\SL_{ d (\mathbb Z)$. It was recently shown by the second-named author \cite{s that for some diagonal subgroups $\{g_t\\subset G$ and unipotent subgroups $U\subset G$, $g_t$-trajectories of almost all points on all $U$-orbits on $G/\Gamma$ are equidistributed with respect to continuous compactly supported functions $\varphi$ on $G/\Gamma$. In this paper we strengthen this result in two directions: by exhibiting an error rate of equidistribution when $\varphi$ is smooth and compactly supported, and by proving equidistribution with respect to certain unbounded functions, namely Siegel transforms of Riemann integrable functions on $\R^d$. For the first part we use a method based on effective double equidistribution of $g_t$-translates of $U$-orbits, which generalizes the main result of \cite{km12. The second part is based on Schmidt's results on counting of lattice points. Number-theoretic consequences involving spiraling of lattice approximations, extending recent work of Athreya, Ghosh and Tseng \cite{agt1, are derived using the equidistribution result.
1505.06717.tex
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\begin{document}

\title{Pointwise equidistribution with an error rate and with respect to unbounded functions}


% Information for first author
\author{Dmitry Kleinbock}
\address{Department of Mathematics, Brandeis University, Waltham MA 02454, USA}
\email{kleinboc@brandeis.edu}


\author{Ronggang Shi}
% Address of record for the research reported here
\address{School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, and
School of Mathematical Sciences, Xiamen University, Xiamen 361005, PR China}
% Current address
%\curraddr{Department of Mathematics, Ohio State
 %University, Columbus, Ohio 43210}
 \email{ronggang@xmu.edu.cn}
% \thanks will become a 1st page footnote.
%\thanks{The author is supported by NSFC (11201388), NSFC (11271278),
% ERC starter grant DLGAPS 279893.}

\author{Barak Weiss}
\address{School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel}
\email{barakw@post.tau.ac.il}

% General info
\subjclass[2000]{Primary   28A33; Secondary 37C85, 22E40.}

\date{}

%\dedicatory{This paper is dedicated to my advisor.}

\keywords{homogeneous dynamics,  equidistribution, ergodic theorem, 
%DK
Diophantine approximation}


\begin{abstract} 
Consider $G=\SL_{ d }(\mathbb R)$ and $ \Gamma=\SL_{ d }(\mathbb Z)$.
It was recently shown by the second-named author \cite{s} that for
some diagonal subgroups $\{g_t\}\subset G$ and unipotent subgroups
$U\subset G$, $g_t$-trajectories of almost all points on all
$U$-orbits on $G/\Gamma$ are equidistributed  with respect to
continuous compactly supported functions $\varphi$ on $G/\Gamma$. In
this paper we strengthen this result in two directions: by exhibiting
an error rate of equidistribution when $\varphi$ is smooth and
compactly supported, and by proving equidistribution with respect to
certain unbounded functions, namely Siegel transforms of Riemann
integrable functions on $\R^d$.  
For the first part we use a method based on effective double
equidistribution of $g_t$-translates of $U$-orbits, which generalizes
the main result of \cite{km12}. The second part is based on Schmidt's
results on counting of lattice points. Number-theoretic
consequences  involving spiraling of lattice approximations, extending
recent work of Athreya, Ghosh and Tseng \cite{agt1}, are derived using the equidistribution
result. 
\end{abstract}



\maketitle



\section{Introduction}\label{sec;intro}

\markright{} Fix an integer $d \ge 2$,  let
$G=\SL_{ d }(\mathbb R)$ and $ \Gamma=\SL_{ d }(\mathbb Z)$,  and
denote by $ \Xn$ the \hs\  $G/\Gamma$, which can be identified with
the 
space of unimodular lattices in $\R^d$. 
The group $G$ acts on $\Xn$ by
left translations preserving the probability measure $\mu$ induced by Haar measure. This action
has been intensively studied due to its intrinsic interest as a
dynamical system and for its number-theoretic applications. See
\cite[Chapter 5]{survey} for a survey of this topic. 

Let $D=\{g_t\}$ be a 
one-parameter subgroup of $G$, and let $\varphi$
be a real-valued function on $\Xn$. We say that $\Lambda
\in \Xn$ is {\em $(D^+, \varphi)$-generic} if 
\eq{generic}{
\lim_{T\to \infty}\frac{1}{T}\int_0^T \varphi(g_t\Lambda)  \dd t= \int_{\Xn}
\varphi \dd\mu\,, 
}
and, for a collection $\mathcal{S}$ of functions, that $\Lambda$ is
{\em $(D^+, \mathcal{S})$-generic}  if it is $(D^+, \varphi)$-generic
for every $\varphi \in \mathcal{S}$. Let $C_c(\Xn)$ denote the space of
continuous compactly supported functions on $\Xn$. It is a well-known consequence of
Moore's ergodicity theorem and Birkhoff's ergodic theorem  that
for any unbounded subgroup $D$ of $G$, $\mu$-almost every $\Lambda\in\Xn$ is $\big(D^+,
C_c(\Xn)\big)$-generic. Also, using effective mixing estimates for the
$D$-action on $X$ one can conclude that whenever $\varphi$ belongs to
the space $C_c^\infty(\Xn)$ of smooth compactly supported functions on
$\Xn$, the convergence in \equ{generic} takes place with a certain
rate for $\mu$-a.e.\ $\Lambda\in\Xn$. More precisely, see
\cite[Theorem 16]{Kach} or \cite[Theorem 4(vii)]{Gap}, for any
$\varepsilon > 0$ and  $\mu$-a.e.\ $\Lambda\in\Xn$ one has  
\eq{eq;main rate in X}{
\frac{1}{T} \int_0^T \varphi \big (g_t  \Lambda \big)
\dd t=\int_{\Xn} \varphi \dd\mu+
o(T^{-1/2}\log^{\frac32+\varepsilon} T).}
(The notation $f(T) =
  o\big(g(T)\big)$ means $\lim_{T\to\infty} \frac{|f(T)|}{g(T)} = 0$.)

\smallskip
Now let $U^+$ be  the {\em unstable horospherical subgroup of $G$ relative to\/} $D^+$, defined by
\eq{ehs}{U^+ := \{g \in G : g_{-t}
gg_t \to e \text{ as }t \to\infty\}.
}
Assume in addition that  %the one-parameter subgroup 
$D$ is diagonalizable.
%, and denote by $\nu$ a Haar measure on $U^+$. 
Then, using a local decomposition of $G$ as the product of its
unstable, neutral and stable horospherical subgroups with respect to
$D^+$, see e.g.\ \cite[\S 1.3]{km96},  it is easy to conclude that for
any $\Lambda \in \Xn$  the element  $g\Lambda$ is $\big(D^+, 
C_c(\Xn)\big)$-generic   for Haar-a.e.\ $g\in U^+ $.


Recently in \cite{s}, as a corollary of a more
general result, the second-named author obtained a similar conclusion
for some proper subgroups of $U^+$ -- namely, for the class of
so-called {\em $D^+$-expanding\/} subgroups, introduced in \cite{KW2}
(see also \cite{SW} for some related ideas).  More precisely, the
following is a special case of  \cite[Theorem 1.2]{s}:  
if $D$ is a diagonalizable one-parameter
subgroup of $G$ and $U$  is a connected $D^+$-expanding abelian subgroup of $U^+$, then
 \eq{shi}{  
 \forall\,\Lambda \in \Xn, \ 
 %\text{ one has }
  g\Lambda\text{  is $\big(D^+,
C_c(\Xn)\big)$-generic   for Haar-a.e.\ }g\in U.}

In the present paper we consider a specific family of pairs $(D,U)$,
where $D\subset G$ is  one-parameter    and $U\subset G$ is
$D^+$-expanding,  which are  important for number-theoretic
applications. Namely, take $m,n\in \N$ with $m+n = d$,  denote by  $M$
the space %$M_{n,m}(\R)$ 
 of ${\m}\times {\n}$ real matrices, and  consider %the map
\eq{defU}{U := u(M)\ \text{ where }
u: M \to G \text{ is given by }
u(\vartheta):=\left(\begin{array}{cc}
1_{\m} & \vartheta \\
0 & 1_{\n}
\end{array}\right)
}
(here and hereafter $1_k$ stands for the identity matrix of order $k$).  Also fix two `weight vectors' 
\begin{align*} \mathbf a=(a_1,\ldots, a_{\m})\in &\mathbb R^{\m}_{>0} \quad \text{and}\quad
\mathbf b=(b_1, \ldots, b_{\n})\in \mathbb R^{\n}_{>0} \\
\qquad \mbox{such that }&\sum_{i=1}^{\m} a_i =\sum_{i=1}^{\n} b_i=1.
\end{align*}
We will refer to the case %\eq{equal}
$${\ba = \bm =   (1/{\m},\dots,1/{\m}) , \  \bb = \bn =  (1/{\n},\dots,1/{\n})}$$ as
the case of {\em equal weights.} Then take 
\eq{defD}{D = \{g_t\},\  \text{ where }g_t=\mathrm{diag}(e^{a_1t},
  \ldots, e^{a_{\m}t}, e^{-b_1t},\ldots, e^{-b_{\n}t} ).  
}

It is easy to see that  %whenever $D$ is as in \equ{defD} 
for any choice of weight vectors $\ba,\bb$,  the group $U$ as in
\equ{defU}  is contained in the unstable horospherical subgroup
relative to $D^+$  (and coincides with it in the case of equal
weights). It was shown in \cite{KW2} that $U$ is  $D^+$-expanding for
any $D$ as in \equ{defD}, which,  in view of \cite{s}, implies
\equ{shi}. 
%In the next four theorems 
For the rest of the paper we will fix arbitrary weight vectors
$\ba,\bb$ and choose $D$ as in \equ{defD}. 

\smallskip

Our first main result gives an analogue of \equ{eq;main rate in X}  for almost all points on $U$-orbits: 

\begin{thm}\label{thm;main rate}
Let  $\Lambda \in \Xn$, $\varphi\in C_c^\infty(\Xn)$ and $\varepsilon>0$ be given.
Then for  almost every $\vartheta\in M$ 
\begin{align}\label{eq;main rate goal}
\frac{1}{T} \int_0^T \varphi \big (g_t u(\vartheta)\Lambda \big)
\dd t=\int_{\Xn} \varphi \dd\mu+
o(T^{-1/2}\log^{\frac32+\varepsilon} T).
\end{align}
\end{thm}

Here and hereafter `almost every $\vartheta$' means almost every with
respect to Lebesgue measure on $M \cong \R^{{\m}{\n}}$; this
corresponds to taking a Haar measure on $U$.

\smallskip
Our proof of Theorem \ref{thm;main rate} is completely
independent of \cite{s}, and, in view of the density of
$C_c^\infty(\Xn)$   in $C_c(\Xn)$, provides an alternative
demonstration \equ{shi} for the case \equ{defU}--\equ{defD}.
It is based on the following `effective double equidistribution' of
$g_t$-translates of $U$, generalizing the 
`effective equidistribution' result of \cite{km12}: 



\begin{thm}\label{thm;mixing}
There exists $\delta >0$ such that the following holds. 
Given $f\in C_c^\infty (M)$ and $\varphi,\psi \in C_c^\infty(\Xn) $
%{with $\int_{\Xn} \varphi  = \int_{\Xn}\psi = 0$}
and 
%with $\int_{\Xn} \varphi \dd \mu = \int_{\Xn} \psi \dd \mu =0$, and
%given
 a compact subset  $L$  of $\Xn$, there exists $C%=C(f, \varphi, \psi,
                                %L)
>0$ such that 
for any $\Delta, \Lambda \in L$ and $t, {w}\ge 0$  one has
\begin{align}\label{eq;mixing rate}
\left|
%{\int_{M} f(\vartheta)\varphi
%\big(g_tu(\vartheta)\Delta\big)\psi \big(g_{w} u(\vartheta)\Lambda\big)\dd  \vartheta}
I_{\Delta,\Lambda{,f,\varphi,\psi}}{(t, w)} -\int_M f \dd\vartheta  \int_{\Xn} \varphi\dd\mu \int_{\Xn}\psi\dd\mu
\right|
\le C e^{-\delta \min (t,  {w}, |{w}-t|)},
\end{align}
where
\begin{align}\label{eq;mixing goal}
I_{\Delta,\Lambda{,f,\varphi,\psi}}(t,{w}) :=
\int_{M} f(\vartheta)\varphi
\big(g_tu(\vartheta)\Delta\big)\psi \big(g_{w}u(\vartheta)\Lambda\big)\dd  \vartheta.
\end{align}
\end{thm}








Under the same assumptions one can also prove `effective $k$-fold equidistribution' of $g_t$-translates of $U$ for all
$k\in\N$. We hope to return to this topic elsewhere. 
%Our method of proof  of Theorem \ref{thm;mixing} is similar to 
See
also
\cite{BEG} and   \cite{konst} for related results on `effective $k$-mixing'.
To derive pointwise equidistribution of $g_t$-trajectories of points
on $U$-orbits with an error rate from  
effective double equidistribution of $g_t$-translates of $U$ we use a
method borrowed from Schmidt's work \cite{s60}  
and originally due to Cassels \cite{cassels}. It is also similar
  to the argument used in \cite{Gap} to derive the rate of convergence
  in Birkhoff's Theorem from the rate of decay of correlations. As a
  byproduct of the method, we show in \S\ref{sec: effective} (Remark
  \ref{alternative proof}) how the estimate  \equ{eq;main rate in X}
  for $\mu$-a.e.\ $\Lambda$ can be derived from the exponential mixing
  of the $D$-action on $\Xn$.

\smallskip

The second main theme of the present paper is establishing $(D^+,
\varphi)$-genericity with respect to some unbounded functions
$\varphi$. It follows from Birkhoff's Theorem that for any 
 $\varphi \in L^1(X,\mu)$,  $\mu$-almost every $\Lambda
\in \Xn$ is $(D^+, \varphi)$-generic; however a passage from $\mu$-a.e.\ $\Lambda
\in \Xn$ to Haar-almost all points on $U^+$-orbits requires some regularity of $\varphi$, such as 
continuity and Lipschitz property away from a compact subset. 
When $U^+$ is replaced by its proper subgroup, the situation becomes more complicated. In particular,  our method of proof of Theorem \ref{thm;main rate}, as well as the argument in \cite{s}, 
do not work for unbounded functions.
Yet we are able to prove a partial result for the setup
\equ{defU}--\equ{defD} and explore its number-theoretic consequences.  

In order to control the rate of growth at infinity of unbounded
functions on $\Xn$, following \cite{emm98},  
let us introduce a function measuring `penetration into 
the thin part of $X$'. For a lattice $\Lambda \in X$, given a subgroup
$\Lambda' \subset \Lambda$, let %$r(\Lambda_0)$ denote its rank and
%let 
$d(\Lambda')$ denote the covolume of $\Lambda'$ in
$\mathrm{span}_{\R}(\Lambda')$ (measured with respect to the standard
Euclidean structure on $\R^d$). We denote 
$$
\alpha(\Lambda) = \max \left\{d(\Lambda')^{-1
%/r(\Lambda_0)
} : \Lambda'
\text{ a subgroup of } 
\Lambda \right\}.
$$
 It is well-known that this maximum is attained, and defines a proper
map $X \to [1, \infty)$. 

Now let us denote by $C_{\alpha}(X)$  the space of  functions
$\varphi$ on $X$ satisfying the following two properties: 
\begin{itemize}
\item[($C_\alpha$-$1$)] $\varphi$ is continuous except on a set of $\mu$-measure zero;
\item[($C_\alpha$-$2$)] the growth of $\varphi$  
is majorized by  $\alpha$, namely  there exists $C>0$ 
%and $K \subset X$ compact 
such that for all $\Lambda \in X $, we have 
$$
|\varphi(\Lambda)| \leq C\alpha(\Lambda)
$$
(in particular, $\varphi$ is bounded on compact sets).
\end{itemize}

One can show that the space $C_{\alpha}(X)$ contains certain unbounded
functions which often arise in number-theoretic applications.
% in
%connection with the Siegel integral formula. 
Recall
that $f: \mathbb R^{ d }\to \mathbb R$ is said to be 
\emph{Riemann 
integrable} if $f$ is bounded with bounded support
and continuous except on  a set of Lebesgue measure zero.  
  {For such $f$, we} define a function $\widehat f$ on
$\Xn$ by 
\eq{eq: Siegel transform}{
\widehat f\left ( \Lambda \right):=\sum_{\mathbf v\in \Lambda \nz} f(\mathbf v).
}
The {\em Siegel integral formula}, established in \cite{siegel}, asserts that for any $f$ as above,
%\eq{siegel}{
$\int_{\Xn} \widehat f \dd\mu=\int_{\mathbb R^{ d }} f$.
%(\mathbf v) \dd\mathbf v.
%}
We show in Lemma \ref{lem: Blichfeldt} that for any Riemann integrable
$f$, the function $\widehat{f}$ satisfies conditions ($C_\alpha$-$1$) and 
($C_\alpha$-$2$). 
In fact, up to constants, when $f$ is non-negative and 
%not identically
  {nonzero on a set of positive measure}, the order of growth of $\widehat f$
is precisely the same as that of $\alpha$. 
As a consequence, in view of the Siegel integral formula, $C_\alpha(X)$ is
contained in $L^1(X,\mu)$. 

\smallskip

For our second main result,
we denote by $\Lambda_0$ the  
%identity coset in $X = G\/Gamma$ (identified by the 
standard lattice $\Z^d\subset \R^d$.


\begin{thm}\label{thm;main} Let $U$ and $D$ be as in \equ{defU}--\equ{defD}.
Then for almost every $\vartheta\in M$, 
$u(\vartheta)\Lambda_0$ is 
$\big(D^+, C_{\alpha}(X)
\big)$-generic. 
\end{thm}



Since $\big(D^+,C_c(\Xn)\big)$-genericity has already been proved,
the
main additional point in
the proof of Theorem \ref{thm;main} is  to obtain 
 upper bound{s} for Birkhoff averages of {restrictions of non-negative $\varphi\in %\mathcal{SR}\cup 
 C_\alpha(X)$ to complements of large compact subsets of $X$.}
To this end we
employ a lattice point counting result of Schmidt \cite{s60}. 
Lattice 
point counting  was
used for a similar purpose in \cite{ms14},
and the
connection between Schmidt's result and the action of $D^+$ was already noted in
\cite{apt} in the equal weights case.
Note that in our result we assume that the lattices are of the form
$ u(\vartheta)\Lambda_0$; 
we expect a similar result to be true if $\Lambda_0$ is replaced by
any other lattice in $\Xn$. However our proof does not yield this more
general statement. 

\smallskip


Let us now explain the number-theoretic implications of Theorem \ref{thm;main}. 
In Diophantine approximation  one interprets $\vartheta \in M$ as a
system of ${\m}$ linear forms in $n$ variables and studies how close to
integers are the values $\vartheta\bq$ of those forms at integer vectors $\bq$. 
During recent years there have been many developments in {\em \da\
  with weights\/}, which allows to treat individual  components of
$\bq$ and $\vartheta\bq$ differently.     
This is done by choosing weight vectors $\ba,\bb$ and considering
`weighted quasi-norms' introduced in  \cite{dima weights}: 
\[
\|\mathbf x\|_{\mathbf a}=\max_{1\le i\le {\m}} |x_i|^{\frac{1}{a_i}} 
\quad \left(\text{resp.}~\|\mathbf y\|_{\mathbf b}=\max_{j\le i\le {\n}}
|y_j|^{\frac{1}{b_j}} \right).
\]
It is a consequence of the general version of the Khintchine-Groshev Theorem proved by Schmidt 
\cite{s60}
 that for  a.e.\  $\vartheta \in M$ 
and any $c > 0$ there are infinitely many
solutions $(\bp,\bq)\in\Z^{\m}\times\Z^{\n}
%\smallsetminus\{0\}
$ to the
inequality 
\eq{kg1}{\|\vartheta\bq - \bp\|_{\ba} < \frac c{\|\bq\|_{\bb}}.}
More precisely, the number of integer
solutions of \equ{kg1} with $\bq$ satisfying
\eq{kg2}{1\le \|\bq\|_{\bb} < e^T}
has the same asymptotic growth
as $ 2^d c T$  (here and hereafter we say that $f(T)$ and $g(T)$ {\em have the same
 asymptotic growth\/}, denoted $f(T) \thicksim g(T)$, if $\displaystyle{\lim_{T \to \infty} 
f(T)
%}{
/g(T)
%} 
= 1}$). 
We remark that the set of nonzero integer solutions of \equ{kg1}--\equ{kg2}
 is in one-to-one correspondence with the intersection of the lattice
$${u(\vartheta) \Lambda_0 = 
\left\{ \left( \begin{matrix}\vartheta \mathbf{ q} - \mathbf{p} \\
      \mathbf{q} \end{matrix} \right) : \mathbf{p} \in \Z^{\m}, \,
  \mathbf{q} \in \Z^{\n} \right\}
}$$
with the set 
$$
E_{ T, c} :=
\left\{(\bx, \by) \in \R^{\m} \times \R^{\n}:\|\bx\|_\ba  < \frac c{\|\by\|_\bb},\ \  
1\le \|\by\|_\bb<e^{T} \right\},
$$
and an elementary computation shows that the volume 
of this set is equal to $ 2^d c T$.

\ignore{
Instead of \equ{kg}--\equ{lessthan}, we consider the system
\eq{kgweighted}{
\begin{cases}\|\vartheta\bq - \bp\|_{\ba} &< \frac c{\|\bq\|_{\bb}} \\ \qquad\ \,  \|\bq\|_{\bb} &< e^T\end{cases}}
Schmidt's work  implies that  as $T\to\infty$, the number
of integer solutions of the system \equ{kgweighted}  for a.e.\
$\vartheta$ %has the same asymptotic growth as the volume of  
grows asymptotically like . 
%(independent of the choice of $\ba,\bb$). \comdima{It looks like this si what the computations mentioned in \S\ref{sec;count} give, right?}
%$$
%E^{\ba,\bb}_{ T, c} :=
%\left\{(\bx, \by) \in \R^n \times \R^m:\|\bx\|_\ba  < \frac c{\|\by\|_\bb},\ \  
%1\le \|\by\|_\bb<e^{T} \right\}.
%$$


For
example, 
(here and hereafter $\|\cdot\|$stands for the supremum norm on $\R^n$, $\R^m$). 
%\comdima{(Or do we want to fix he supremum norm here?}
Furthermore, according to Schmidt's result \cite{s60} the number of
solutions of \equ{kg} in a ball of large radius grows as the volume of certain
family of regions in $\R^{m+n}$. 
Namely, define
$$
E_{ T, c} := \left
\{(\bx, \by) \in \R^n \times \R^m:\|\bx\|  <\left( \frac c{\|\by\|^{m}}\right)^{1/n},\ \  
1\le \|\by\|<e^{T/m} \right\};
$$
then the set of integer solutions of \equ{kg}
with  \eq{lessthan}{\|\bq\| < e^{T/m}} is in one-to-one correspondence with the intersection of the lattice
%\eq{lambdatheta}
$${u(\vartheta) \Lambda_0 = 
\left\{ \left( \begin{matrix}\vartheta \mathbf{ q} - \mathbf{p} \\
      \mathbf{q} \end{matrix} \right) : \mathbf{p} \in \Z^n, \,
  \mathbf{q} \in \Z^m \right\}
}
$$
with $E_{ T, c} $. 
A special case of Schmidt's theorem says that for every $c > 0$ and
a.e.\  $\vartheta \in M$, as $T\to\infty$ the cardinality of this intersection
  has the same asymptotic growth
as the Lebesgue measure  $
% \mathrm{Vol}(
|E_{ T, c} |$ of $E_{ T, c} $.
% \comdima{shouldn't $c$ be raised to power $n$? And now we want the norm to be supremum, right? we should mention it somewhere.}
  (we say that $f(T)$ and $g(T)$ have the same
 asymptotic growth, denoted $f(T) \thicksim g(T)$, if $\displaystyle{\lim_{T \to \infty} %\frac{
f(T)
%}{
/g(T)
%} 
= 1}$).  Note that an elementary computation shows that $
% \mathrm{Vol}(E_{ T, c})
|E_{ T, c} | = 2^d c T$.}

\smallskip
A finer question concerning  directions of  vectors
$\vartheta\bq - \bp$ and $\bq$ was addressed in a recent paper \cite{agt1} by Athreya,  Ghosh and Tseng. In the equal weights case % \equ{equal} 
 they considered integer solutions of \equ{kg1}--\equ{kg2} in addition satisfying 
 \eq{unweightedspherical}{\pi( \vartheta\bq - \bp) \in  A\text{ and }\pi( \bq) \in B ,}
 where $\pi$ stands for the radial
projection from $\R^{\m}$ and $\R^{\n}$ to the unit spheres $\mathbb
S^{{\m}-1}$ and  $\mathbb S^{{\n}-1}$ respectively, and  $A \subset \mathbb 
S^{{\m}-1}, \ B \subset \mathbb S^{{\n}-1}$ are two measurable
subsets. When  the boundaries of $A$ and $B$ are of measure zero, they
showed that  
 for almost every  $\vartheta \in M$ the 
number  of  solutions of \equ{kg1}--\equ{unweightedspherical} (with $\ba = \bm$ and $\bb = \bn$)
has the same asymptotic growth as the volume of the set 
\begin{equation*}
\left\{(\bx, \by) \in \R^{\m} \times \R^{\n} :  \begin{aligned}\|\bx\|^{\m}  <  \frac c{\|\by\|^{{\n}}}&,\ \  
1\le \|\by\|^{\n}<e^T\\ \pi( \bx) \in  A&, \ \  
\pi( \by) \in B\end{aligned} 
\right\}
\end{equation*}
(here $\|\cdot\|$ stands for the supremum norm; 
note that \cite{agt1} deals only with the case $m=1$ and uses
  Euclidean norm instead of the supremum norm, but the argument can be
  applied to an arbitrary $m$ and arbitrary norm). 

In the case of arbitrary weight vectors it no longer makes sense to project radially, since these
projections accumulate near directions corresponding to
smallest (resp., largest) weights. We remedy this as follows. Let
$F_{\mathbf a t}$ be the $\mathbf a$-weighted  flow on $\mathbb R^{\m}$
defined by  
 \[
F_{\mathbf a t}( \mathbf x):=\left(e^{a_1t}x_1, \ldots, e^{a_{\m}t}x_{\m} \right)
 \]
(note that in the case of equal weights, these are just homotheties of
$\R^{\m}$). Similarly we define the transformation $F_{\bb t}$ of $\R^{\n}$. Then 
for  nonzero $\mathbf x\in \mathbb R^{\m}$ and $\by \in \mathbb R^{\n}$  let 
\eq{defpi}
{\pi_{\ba}( \mathbf x):=\{ F_{\mathbf a t}(\mathbf x): t\in \mathbb R\}
  \cap \mathbb S^{{\m}-1}, \ \pi_{\bb}( \mathbf y):=\{ F_{\mathbf b t}(\mathbf y): t\in \mathbb R\}
  \cap \mathbb S^{{\n}-1}}
 (these intersection point are clearly unique),
replace
\equ{unweightedspherical} with \eq{weightedspherical}{\pi_\ba(
  \vartheta\bq - \bp) \in  A\text{ and }\pi_\bb( \bq) \in B,} 
  and define 
$$
E_{ T, c}(A,B) := 
\left\{(\bx, \by) \in \R^{\m} \times \R^{\n}: \begin{aligned}\|\bx\|_\ba  < \frac c{\|\by\|_\bb}&,\ \  
1\le \|\by\|_\bb<e^{T}\\ \pi_\ba( \bx) \in  A&,\ \  
\pi_\bb( \by) \in B\end{aligned}\right\}.
$$
Using  Theorem \ref{thm;main}, we obtain: 

\begin{thm}\label{thm;diophantine}
Let  $c > 0$ and measurable  subsets $A \subset \mathbb
S^{{\m}-1}, \ B \subset \mathbb S^{{\n}-1}$ with boundaries of measure zero be given.
Then for a.e.\  $\vartheta \in M$, as $T\to\infty$,  the number of
integer solutions  to \equ{kg1}, \equ{kg2} and \equ{weightedspherical}
has
the same asymptotic growth as
the volume of $E%^{\ba,\bb}
_{ T, c}(A,B)$. 
\end{thm}

Note that the above counting result does not follow from the
techniques of \cite{s60, S60a}. See also \cite{agt} for some related results. The
reduction of Theorem \ref{thm;diophantine} to Theorem \ref{thm;main} is based on
the observation that the number of
integer solutions  to \equ{kg1}, \equ{kg2} and \equ{weightedspherical} is equal
 to $\sharp \big( u(\vartheta)\Lambda_0\cap E_{T, c}(A, B)\big)$, that
 is, to $\widehat f$,  where $f$ is a certain Riemann integrable
 function on $\R^d$.  
 
 
 \smallskip
The structure of the paper is as follows: in \S\ref{sec:double} we
prove Theorem \ref{thm;mixing}, and  use it  in  \S\ref{sec:
  effective} to  prove Theorem \ref{thm;main rate}. \S\ref{sec;count}
contains a discussion of Schmidt's asymptotic formula (Theorem
\ref{thm;schmidt use}) and some auxiliary results, which are needed in
the final section of the paper, where Theorems \ref{thm;main} and
\ref{thm;diophantine} are proved.  
 \smallskip
 
\noindent{\bf Acknowledgements.}
The support of grants 
NSFC (11201388), NSFC (11271278),
 ERC starter grant DLGAPS 279893, NSF DMS-1101320 and BSF 2010428 is gratefully
 acknowledged. 
 We would like to thank Manfred Einsiedler for a discussion of
 effective double equidistribution and MSRI for its hospitality during
 Spring 2015.  
  


\section{Effective double equidistribution}\label{sec:double}
In this section we prove Theorem \ref{thm;mixing}. We first recall several facts from the paper
\cite{km12}. Its main result, i.e.\ effective equidistribution of
$g_t$-translates of $U$, can be stated as follows: 


\begin{prop}[\cite{km12}, Theorem 1.3]\label{prop;one mixing}
There exists $\delta_1>0$ such that for any $f\in C_c^\infty(M)$,
$\varphi \in C_c^\infty(\Xn)$, and 
for any compact $L\subset \Xn$ there exists $C_1=C_1(f, \varphi, L)$
such that for any $\Lambda\in L$ and  
$t\ge 0$,
\begin{align}
\left|
\int_M f(\vartheta)\varphi\big(g_t u(\vartheta)\Lambda\big)\dd \vartheta
 -\int_M f \dd \vartheta \int_{\Xn} \varphi \dd \mu
\right|
\le C_1 e^{-\delta_1 t}.
\end{align} 
\end{prop}
Let  
$$b = \min \left\{\frac{a_i}{{\n}}, \frac{b_j}{{\m}} : 1\le i\le {\m},\ 1\le j\le
  {\n}\right \},
$$ 
and let 
\begin{align}
g_t'=\left (
\begin{array}{cc}
e^{{\n}bt}1_{\m} & 0 \\
0 & e^{-{\m}bt} 1_{\n} 
\end{array}
\right),
\quad
g_t''=g_tg'_{-t}.
\end{align}
Note that $ g_t'$ (after reparametrization) corresponds to the case of
equal weights; in particular, $U$ is the unstable horospherical
subgroup relative to $\left\{g_t': t\ge 0 \right\}$.  On the other hand $ g_t''$  
is of the form
$$\mathrm{diag}\left(e^{a'_1t}, \ldots, e^{a'_{\m}t},
  e^{-b'_1t},\ldots, e^{-b'_{\n}t} \right )
$$
where $a_i'$ and $b_j'$ are nonegative.


Let $B_r^G$ be the ball of radius $r$ centered at the identity element
of $G$ with respect to the metric induced by some right invariant 
Riemannian metric. For $\Lambda\in \Xn$ we let 
$\pi_\Lambda: G\to \Xn$ be the map $g\mapsto g\Lambda$, and define
\[
\mathrm{Inj}_\varepsilon:=\left\{\Lambda\in \Xn:
  \pi_\Lambda|_{B_\varepsilon^G} \mbox{ is injective}\right \}. 
\]
Also let $B_r$ denote  the ball of radius $r$ centered at zero with respect to 
the Euclidean norm on $M\cong \mathbb R^{mn}$. 
The following quantitative non-divergence result is a combination of \cite[Corollary 3.4]{km12}  
and \cite[Proposition 3.5]{km12}:
\begin{prop}\label{prop;quantitative}
Let $L\subset \Xn$ be compact and let $r>0$. There exists 
$t_0=t_0(L, r)>0$ and 
$C_3>0$ such that for every 
$0<\varepsilon<1$, $\Lambda\in L$, $s\ge 0$ and $t\geq t_0$ one has
\[
\left| \left\{\vartheta\in B_r: g_s''g_t u(\vartheta)\Lambda\not \in
    \mathrm{Inj}_\varepsilon \right\} \right|\le 
C_3\varepsilon ^ {{1}/{d^4}} \left|B_r \right|.
\]
\end{prop}
Here and hereafter for a measurable subset $S$ of a Euclidean space we use 
$|S|$ to denote the Lebesgue measure of $S$.  
For $h\in C_c^\infty(M)$ and a nonnegative 
integer $k$ we define the $k$-th Sobolev norm of $h$ to be 
\[
\|h\|_k=\max_{|\beta|\le k} \left\|\partial ^\beta h \right\|_{L^2(M)}
\]
where $|\beta|$ is the order of the multi-index $\beta$. 


We will also need the following effective equidistribution estimate for the  $g_t'$-action:
 
\begin{prop}[\cite{km12}, Theorem 2.3]\label{prop;injective}
There exist $\delta_2, r_0 > 0$ and $k\in \mathbb N$ with the
following property:  for any $\psi\in C_c^\infty(\Xn)$ 
there exists $C_2 > 0$
such that for any $0<r<r_0$, any $ h\in C_c^\infty (M)$ with
$\mathrm{supp}(h)\subset B_r$, any $\Lambda\in 
\mathrm{Inj}_{2r}$  and any $t\ge 0$ one has
 \[
 \left|
 \int_M h(\vartheta)\psi\big(g_t'u(\vartheta)\Lambda
\big )\dd\vartheta
-\int_M h\int_{\Xn}\psi 
 \right|\le
  C_2\left(r^{-k}\|h\|_ke ^{-\delta_2 t}+
  r\int_M |h|
  \right).
 \]
\end{prop}

Note that the statement is not precisely the one which appears in
\cite{km12} but can be deduced from it by taking $k$ large enough. 



\begin{proof}[Proof of Theorem \ref{thm;mixing}] Recall that we are
  given a compact subset  $L$  of $\Xn$, $ \varphi,\psi \in
  C_c^\infty(\Xn) $ and  $f\in C_c^\infty (M)$. 
In this proof, the notation 
$y\ll x$  will mean that $y \leq Cx$ where $C$ is a
constant independent of $x$, but which could depend on $f, \varphi,
\psi$ or $L$.


 Let $\delta_0=\min(\delta_1, \delta_2)$ where $\delta_1$
and $\delta_2$ are given 
in Propositions \ref{prop;one mixing} and \ref{prop;injective} respectively. 
We also fix $k\in \mathbb N$ so that Proposition \ref{prop;injective} holds. 
Without loss of generality we assume that $k\ge mn$. Now let 
 $$a:=\min\{a_i+b_j: 1\le i\le  {\m}, 1\le j\le {\n} \}$$
and 
\eq{eq: choice of delta}{
\delta:=\min \left( \frac{\delta_0}{2(1+3k)}, \frac{a}{2}\right).
}
Let 
 $\Delta$,   $\Lambda$,   $ t$ and   ${w}$ be as in the statement of
Theorem \ref{thm;mixing}.  
Put 
$$r:=e^{-\delta |{w}-t|} \ \text{ and }  s:=\frac{t + {w}}{2}.$$
We can assume with no loss of generality
that ${w} \geq t \geq t_0$, where $t_0$ is as in Proposition
\ref{prop;quantitative}, and moreover that $r< r_0$, where $r_0$ is as in
 Proposition \ref{prop;injective}. 
According to \cite[Lemma 2.2]{km12}
there exists $h\in C_c^\infty (M)$ such that supp$(h)\subset B_r$, $h\ge 0$,
$\int_M h (\vartheta)\dd \vartheta=1$ and $\|h\|_k\ll r^{-2k}$.  
Given $\zeta\in M$, we define  $\zeta_1, \zeta_2\in M$ by the formulae
\[
u(\zeta_1): =g_{t-s}''g_{-s}u(\zeta) g_{s}g_{s-t}'', \quad
u(\zeta_2): =g_{t-s}''g_{t-s}u(\zeta) g_{s-t}g_{s-t}'',
\]
which imply 
\eq{eq: with}{
g_t u(\zeta_1) = u(\zeta_2) g_t \ \text{ and } \  g_{{w}} u(\zeta_1) =
g_{s-t}' u(\zeta) g_s g''_{s-t}. 
}


Let $I_{\Delta,\Lambda{,f,\varphi,\psi}}(t, {w})$ be as in \equ{eq;mixing goal}. Then, 
{using Fubini's theorem %in \equ{eq;q 2} 
and, for each $\zeta$, making a change of variables 
 $\vartheta\mapsto \vartheta+\zeta_1$, % in \equ{eq;mixing goal} and
we find in view of \equ{eq: with} 
that 
\begin{equation}
\begin{aligned}
&I_{\Delta,\Lambda{,f,\varphi,\psi}}(t, {w})=  \int_M  f(\vartheta) \varphi\big(g_tu(\vartheta)\Delta\big)
\psi\big(g_{w} u(\vartheta)\Lambda\big)  \dd \vartheta \int_M h (\zeta)\dd \zeta\\ =&
{\int_M \int_M} \  f(\vartheta+\zeta_1)
\varphi\big(u(\zeta_2)g_t u(\vartheta)\Delta\big)\psi\big(g_{s-t}'u(\zeta)g''_{s-t}g_s u(\vartheta)\Lambda\big)
h(\zeta)\dd \vartheta\dd \zeta.
\label{eq;q 2}
\end{aligned}
\end{equation}
}



Note  that for all $\zeta\in \mathrm{supp}(h)$ we have 
  $$\|\zeta_1\| \ll  e^{-sa} \|\zeta\| \ll
  e^{-\frac{{w}-t}{2}a} \stackrel{\equ{eq: choice of delta}}{\ll}
  e^{-\delta({w} -t)} ,$$ 
and similarly  $\|\zeta_2\| \ll e^{-\delta({w}-t)}$. 
Denote
$$
{\Psi}(\vartheta) := \int_M \psi\left(g_{s-t}'u(\zeta)g_{s-t}''g_s u(\vartheta)\Lambda\right)
h(\zeta) \dd \zeta.
$$
Then, by approximating the function $f(\vartheta+\zeta_1)
\varphi\big(u(\zeta_2)g_t u(\vartheta)\Delta\big)$ by $f(\vartheta) 
\varphi\big(g_t u(\vartheta)\Delta\big)$ in (\ref{eq;q 2}) we 
get
%can rewrite $I_{\Delta,\Lambda{,f,\varphi,\psi}}(t, {w})$ as 
\begin{align}
\left|
I_{\Delta,\Lambda{,f,\varphi,\psi}}(t, {w})
- 
\int_M 
{\Psi}(\vartheta)
 f(\vartheta) \varphi\big(g_t u(\vartheta)\Delta\big)\dd \vartheta\right| \ll
%+O\left(
e^{-\delta({w}-t)} 
%\right)
.
\label{eq;effective 1}
\end{align}


Let $r_1>0$ be such that $\mathrm{supp} \, f \subset B_{r_1}$, let 
$\varepsilon=e^{-\delta ({w}- t)d^4}$, and {denote}
 $$E:=\{\vartheta\in B_{r_1}: g_{s-t}''g_s
 u(\vartheta)\Lambda\in \mathrm{Inj}_\varepsilon\}.$$  
It follows from 
% \cite[Corollary 3.4]{km12} 
Proposition \ref{prop;quantitative} that  
 \begin{align*} 
|B_{r_1}\ssm  E|\ll e^{-\delta({w}-t)}|B_{r_1}|,
\end{align*}
and hence 
\begin{align}\label{eq;effective 11}
\left |
\int_{B_{r_1}\ssm E}{\Psi}(\vartheta)f(\vartheta)\varphi\big(g_t u(\vartheta )\Delta\big)\dd \vartheta
\right|
\ll e^{-{\delta}({w}-t)}.
\end{align}
By Proposition \ref{prop;injective} 
for $\vartheta\in E$ (with $g_{s-t}'' g_s u(\vartheta)
\Lambda$ in place of $\Lambda$) one has
$${
\left| {\Psi}(\vartheta) - \int_{\Xn} \psi % \dd \mu 
\right |%&
\ll r + r^{-k}
  \|h\|_k e^{-\delta_2(s-t)}
\ll e^{-\delta({w}- t)}+e^{-(\delta_0/2-3k\delta){({w}-t) } }
,
}
$$
{which, in view of 
\equ{eq: choice of delta} and  (\ref{eq;effective 11}), implies} 
\begin{equation}
\label{eq;effective 2}
{
 \begin{aligned}
%%%%%%%%%% 1
\left | \int_{M}{\Psi}(\vartheta)f(\vartheta)\varphi\big(g_t u(\vartheta )\Delta\big)\dd \vartheta  
 -  \int_{M}f(\vartheta)\varphi\big(g_t u(\vartheta )\Delta\big)\dd \vartheta\int_{\Xn}\psi
\right|
\\
 \ll
%%%%%%%%% 2
%= & +O\left(
e^{-{\delta({w}-t)}}+e^{-\frac{\delta_0({w}-t) } {2-3k\delta} }%\right) 
% \\
%%%%%%3
%=&  \int_{M}f(\vartheta)\varphi\big(g_t u(\vartheta )\Delta\big)\dd \vartheta\int_{\Xn}\psi
%+ O\left(
%&
\ll e^{-\delta({w}-t)}
%\right)
. \qquad\qquad\qquad
%\label
\end{aligned}
}
\end{equation}
On the other hand {from Proposition \ref{prop;one mixing} %and  \equ{zero} 
one gets}
\begin{align}\label{eq;effective 3}
\left|
\int_{M}f(\vartheta)\varphi\big(g_t u(\vartheta )\Delta\big)\dd \vartheta
-\int_M
f\int_{\Xn}\varphi
\right|\ll
%+O\left(
e^{-\delta_0 t} 
%\right)
.
\end{align}
Combining (\ref{eq;effective 1}), (\ref{eq;effective 2})   and (\ref{eq;effective 3}), one arrives at (\ref{eq;mixing rate}).
\end{proof}



\section{Pointwise equidistribution with an error rate}
\label{sec: effective}
In this section we prove Theorem \ref{thm;main rate}.  
The method works in a general framework as follows:
\begin{thm}\label{thm;effective}
Let $(Y, \nu)$  be a probability space, and let $F:Y\times \R_+\to \R$ be a bounded measurable function.
Suppose  there exist $\delta >0$ and $C>0$ such that for any $w
\ge t \ge 0$,
\begin{align}\label{eq;effective rate}
\left |
\int_Y{F(x,t)}{F(x,w)}\dd \nu( x)
\right|
\le Ce^{-\delta \min(t, {w}-t)}.
\end{align}
Then given $\varepsilon >0$ we have 
\begin{align}\label{eq;effective goal}
\frac{1}{T}\int_0^T {F(y,t)}\dd t=o
%_{\varphi, x, \varepsilon}
(T^{-1/2}\log^{{\frac32+\varepsilon} } T)
\end{align}
for $\nu$-almost every ${y}\in Y$. 
\end{thm}
%We remark here that since we use Borel-Cantelli lemma, 
%the implied constant above is not computable. 


We begin with some lemmas. In the statements below the notation and
assumptions are as in Theorem \ref{thm;effective}. 

\begin{lem}\label{lem;effective help}
 Let $[b,c]$ be a closed 
interval in $[0,\infty)$. Then 
\[
\int_Y \left(
\int_b^c{F(x,t)}\dd t
\right)^2 \dd \nu(x)
\le { 4C\delta ^{-1}(c-b)}.
\]
\end{lem}

\begin{proof} The left hand side can be written as
\begin{align}
%%%%%% 1
  & \int_Y \left(
\int_b^c{F(x,t)}\dd t\right)^2 \dd \nu(x)  \notag\\
%%%%%%%%%2
 = &\int_b^c \int _b^c  \left
 [\int_Y{F(x,t)}{F(x,w)} \dd\nu (x)\right]\dd {w} 
\dd t  &&\mbox{by Fubini}\notag\\
%%%%%%%%%%3
 \le & 2\int_b^c \int_t^c  \left |\int_Y{F(x,t)}
{F(x,w)} \dd\nu (x)\right|\dd {w} \dd t  \notag\\
%%%%%%%%%%4
 \le& 2C\int_b^c  \left [\int_t^{c}
(  e^{-\delta ({w}-t)}+
   e^{-\delta t})\dd {w}  
 \right]\dd t  &&\mbox{by (\ref{eq;effective rate})} \notag  \\
 %%%%%%5
  \le  & 2C\int_b^c \big({\delta^{-1} }+e^{-\delta
    t}(c-t )\big)\dd t \notag\\
\leq  & 2C\int_b^c \big({\delta^{-1} }+e^{-\delta
    t}(c-b )\big)\dd t
    % \notag\\
%\notag
\    \le\ % &
    {4C\delta^{-1}(c-b)}, \notag
\end{align}
and the proof is finished.
\end{proof}



For a positive integer $s$ we let $L_s$ be the set of intervals of the form 
$[2^i j, 2^i(j+1)]$
where $i, j$ are nonnegative integers and $2^i(j+1)<2^s$. 


\begin{lem}\label{lem;effective uniform}
One has 
\begin{align}
%%%%%line 1
\sum_{[b, c]\in L_s} 
\int_Y 
\left(
\int_b^c{F(x,t)}\dd t
\right)^2 
 \dd\nu(x)\le 4C \delta^{-1} s2^s.
\end{align}
\end{lem}

\begin{proof} We estimate the left hand side as
\begin{align}
&\sum_{[b, c]\in L_s} 
\int_Y 
\left(
\int_b^c{F(x,t)}\dd t
\right)^2 
 \dd\nu(x) \notag\\
 %%%%%%line 2
 \le & \sum_{i=0}^{s-1} \sum_{j=0}^{2^{s-i}-1}
 \int_Y
\left( \int_{2^ij}^{2^i(j+1)}{F(x,t)}\dd t \right)^2 \dd\nu(x)  \notag\\
%%%%%%%line 3
\le & \sum_{i=0}^{s-1} \sum_{j=0}^{2^{s-i}-1}   {4C\delta^{-1}2^i},
   && \mbox{by Lemma \ref{lem;effective help}} \notag\end{align}
   which is clearly bounded from above by $4C \delta^{-1} s2^s$.
\end{proof}




\begin{lem}\label{lem;effective cover}
Let $k, s$ be positive integers with $ k < 2^s$. Then the interval 
$[0, k]$ can be covered by at most $s$ intervals in $L_s$.
\end{lem}

{\begin{proof} These intervals can be easily constructed using binary
    expansion of $k$. See also \cite[Lemma 1]{s60}. \end{proof}} 


\begin{lem}\label{lem;effective key}
For every $\varepsilon >0$, there exists a sequence of measurable
subsets $\{Y_s\}_{s\in \N}$ of $Y$ such that: 
\begin{itemize}
\item[\rm (i)] $\nu(Y_s)\le 4C\delta ^{-1} s^{-(1+{2}\varepsilon)}$.
\item[\rm (ii)] For every positive  integer $k$ with $k<2^s$ and every ${y}\not \in Y_s$ one has 
\begin{align}\label{eq;effective k}
\left |\int_0^k {F(y,t)}\dd t\right |\le 2^{s/2} s^{{\frac32+\varepsilon} }. 
\end{align}
\end{itemize}
\end{lem}

\begin{proof}
Let 
\[
Y_s=
\left \{
{y}\in Y:\sum_{I\in L_s }    \left( \int_I {F(y,t)}\dd t 
\right )^2  > 2^s s^{2+{2}\varepsilon}
\right \}. 
\]
Assertion (i) follows from Lemma \ref{lem;effective uniform}
and %the Chebyshev i
{Markov's} Inequality. 
By  Lemma \ref{lem;effective cover} there exists a 
subset $L(k)$ of $L_s$ with cardinality at most $s$ such that 
$[0, k]=\bigcup_{I\in L(k)}I$.  For $k<2^s$ and  ${y}\not\in Y_s$ we {estimate}
\begin{align}
\notag
%%%%%%%%%% line 1
& \left(
\int_0^k {F(y,t)}\dd t
\right)^2      \\
%%%% line 2
\notag
\le & \left(\sum_{I \in L(k)} \int_I {F(y,t)} \dd t \right)^2
\\
%%%%%%%%%% line 2.5
\notag
 \le &\ s   \sum_{I\in L(k)}  \left( \int_I {F(y,t)}\dd t 
\right)^2     && \mbox{by Cauchy's inequality} \\
%%%%%% line 3
\notag
\le &\ s \sum_{I\in L_s }    \left( \int_I {F(y,t)}\dd t 
\right )^2    % \\
%%%%%%%% 4
%\notag
\le %&
\ 2^s s^{3+{2}\varepsilon}   && \mbox{since } y\not\in Y_s.\label{eq;effective line}
\end{align}
Now (\ref{eq;effective k}) follows % from (\ref{eq;effective line}) 
by taking square roots. 
\end{proof}



\begin{proof}[Proof of Theorem \ref{thm;effective}]
We fix $\varepsilon>0$ and choose a sequence of measurable subsets 
$\{Y_s\}_{s\in \N}$ as in Lemma \ref{lem;effective key}. 
Note that 
\[
\sum_{s=1}^\infty\nu(Y_s)\le \sum_{s=1}^\infty4C\delta ^{-1}s^{-(1+{2}\varepsilon)}<\infty.
\]
The Borel-Cantelli lemma implies that there exists a measurable subset ${Y(\varepsilon)}$ of $Y$
with full measure such that for every $y\in {Y(\varepsilon)}$ there exists {$s_y\in\N$ such that $y\not\in Y_s$
whenever $s\ge  s_y$}.  

We will show that  
%(\ref{eq;effective goal}) holds  
for 
every $y\in {Y(\varepsilon)}$ {one has
\eq{ll}{
 \frac{1}{T}\left|\int_0^T {F(y,t)}\dd t\right| {\ll}\ 
%C(\varphi, {y}, \varepsilon)
 T^{-{1/2}} \log^{{\frac32+\varepsilon} }(T)
}
 provided 
 %$T\ge 2$
  $T$ is large enough, where the implicit constant depends only on ${F}$.}
  %since $\varepsilon$ can be %chosen 
% arbitrarily small, %from which  
Given
%A real number 
$T > 2$,
% uniquely determines 
 let {$k = \lfloor T\rfloor$ and $s = 1 + \lfloor \log T\rfloor$, so that
 % positive integers  $k $ and $s$  with $ T-1< k\le T$  and  
 $2^{s-1}\le k \le T<k+1\le  2^s$.} 
 Suppose $T\ge 2^{s_y{-1}}$, then $s\ge s_y$ and hence $y\not\in Y_s$. 
Therefore we have
 \begin{align*}
 %%%%%% 1
 %&
 \left|\int_0^T {F(y,t)}\dd t\right| %\\
 %%%%%%%  2
 \le &\  {\|F\|_\infty} +\left|\int_0^k {F(y,t)}\dd t\right| \\
 %%%%%%% 3
 \le &\   {\|F\|_\infty} +2^{s/2}s^{{\frac32+\varepsilon} }
 && \mbox{by (\ref{eq;effective k})} \\
 \le&\  {\|F\|_\infty} +(2T)^{{1/2}}\log^{{\frac32+\varepsilon} }(2T).
  \end{align*}
{ and \equ{ll} follows.
This clearly implies (\ref{eq;effective goal})  for $y\in \cap_{k\in\N}Y(1/k)$.} 
%\comdima{Do we want to say something here about the lack of control
%for the rate of convergence due to the use of the Borel-Cantelli
%Lemma? feel free to make suggestions.}  
%for some constant $C(\varphi, {y}, \varepsilon)>0$ one has  
 %\le 
 \end{proof}


\begin{proof}[Proof of Theorem \ref{thm;main rate}] Recall that we are
  given $\Lambda \in \Xn$, $\varphi\in C_c^\infty(\Xn)$ and
  $\varepsilon>0$. 
Take $f\in C_c^\infty(M)$ with $f\ge 0$ and $\int_M f\dd \vartheta=1$.  
Let $\nu$ be the probability measure on $\Xn$ defined by 
\[
\int_{\Xn}\psi \dd \nu=\int_M f(\vartheta)\psi\big(u(\vartheta)\Lambda\big )\dd \vartheta
\]
for every $\psi\in C_c({X})$. 
{Denote $\alpha=\int_{\Xn} \varphi \dd \mu$ and for $t,w\ge 0$ write
%It 
%follows from 
%{Applying} Theorem \ref{thm;mixing} %and Proposition \ref{prop;one mixing}
%{with $\Delta = \Lambda$ and  with $\varphi - \alpha$ in place of $\varphi$ and $\psi$},
%{we conclude}  that there exists $C>0$ and $\delta>0$ such that  for any ${w} \geq
%  t \geq 0$
%\begin{align}
%%%%%%%line 1
$$%{|I_{\Lambda,\Lambda,f,\varphi-\alpha,\varphi-\alpha}(t,w)|}\  
%&
%{=}\ 
\begin{aligned}
%\left |
\int_{\Xn}\big(\varphi(g_t \Lambda )-{\alpha}\big)&\big(\varphi(g_{w}
\Lambda )-{\alpha}\big) \dd\nu =
I_{\Lambda,\Lambda,f,\varphi,\varphi}( t,{w}) -  
\int_M f \dd\vartheta \left( \int_{\Xn} \varphi\dd\mu\right)^2\\ 
&-\ 
\alpha \left(\int_{\Xn} f(\vartheta)\varphi\big(g_t u(\vartheta)\Lambda\big )\dd \vartheta \ - 
\int_M f \dd\vartheta \int_{\Xn} \varphi\dd\mu\right) \qquad  \\
&-\ 
\alpha \left(\int_{\Xn} f(\vartheta)\varphi\big(g_w u(\vartheta)\Lambda\big )\dd \vartheta - 
\int_M f \dd\vartheta \int_{\Xn} \varphi\dd\mu\right). \qquad\ 
%\right| 
%\notag\\
%%%%%%line 2
%=&\left |\int_{\Xn} \left(\varphi(g_t\Lambda )\varphi(g_{w}
%    \Lambda )-b\varphi(g_t\Lambda )-b\varphi(g_{w} \Lambda )+b^2 \right)\dd\nu
%\right| \notag\\
%%%%%%%line 3
%&
%.\notag
\end{aligned}
$$
Applying  Theorem \ref{thm;mixing}  and Proposition \ref{prop;one mixing},
%{with $\Delta = \Lambda$ and  with $\varphi - \alpha$ in place of $\varphi$ and $\psi$},
we conclude  that there exist $C>0$ and $\delta>0$ such that the estimate 
$$
\left|\int_{\Xn}\big(\varphi(g_t \Lambda )-{\alpha}\big)
  \big(\varphi(g_{w} \Lambda )-{\alpha}\big) \dd\nu\right|\le \  C
e^{-\delta\min(t,  {w}-t)} 
$$
holds  for any ${w} \geq
  t \geq 0$.
{Then we can apply} Theorem \ref{thm;effective}  with $F(x,t) = \varphi(g_tx) - \alpha$} 
%one obtains 
{and obtain} (\ref{eq;main rate goal}) for 
almost every $\vartheta\in S_f$, where $S_f:= \{\vartheta\in M: f(\vartheta)>0\}$.
Since countably many {sets of the form} $S_f$ 
exhaust $M$, we reach the desired conclusion.
\end{proof}

\begin{rem} \label{alternative proof} Arguing similarly with $\nu$
    replaced by $\mu$  and using  exponential mixing of the
    $g_t$-action, see e.g.\ \cite[Theorem 1.1]{km12}, instead of
    Theorem \ref{thm;mixing},  
%one obtains 
one can easily obtain \equ{eq;main rate in X}  for 
almost every $\Lambda\in\Xn$. \end{rem}

\section{Lattice points counting}\label{sec;count}
In this section we  recall a result of Schmidt \cite{s60} concerning a
counting problem arising from Diophantine approximation,  and relate it to the 
  $D$-action on $\Xn$.  From this we will deduce some estimates which
  will be used in the proof of 
Theorem \ref{thm;main}. 


Let  $\mathbf{a}, \mathbf{b}$ be weight vectors, let
$\pi_{\mathbf{a}}, \pi_{\mathbf{b}}$ be %the corresponding
%projections defined in \S\ref{sec;intro},
 as in \equ{defpi}, 
and let $c, A, B$ be as in Theorem
\ref{thm;diophantine}. 
For an individual vector $\bv = (\mathbf{x}, \mathbf{y}) \in \R^d$ we
write $g_t\bv = (\mathbf{x}_t, \mathbf{y}_t)$, and then we have
$$\|\mathbf{y}_t\|_{\mathbf{b}} = e^{-t} \|\mathbf{y}
\|_{\mathbf{b}}$$
and
$$\|\mathbf{x}_t\|_{\mathbf{a}} \cdot \| \mathbf{y}_t \|_{\mathbf{b}} =
\|\mathbf{x}\|_{\mathbf{a}} \cdot \| \mathbf{y}\|_{\mathbf{b}}, \
\pi_{\mathbf{a}}(\mathbf{x}_t) = \pi_{\mathbf{a}}(\mathbf{x}), \ 
\pi_{\mathbf{b}}(\mathbf{y}_t) = \pi_{\mathbf{b}}(\mathbf{y})$$ 
for all $t$. 

%To simplify 
%notation, %when there is no risk of confusion 
%we will drop the dependence on the weight vectors  $\ba,\bb$. %and will write $E_{T,c}(A,B)$
%instead of $E^{\mathbf{a}, \mathbf{b}}_{T, c}(A, B)$. 
%\comdima{There was a very high risk of confusion before, so I changed the notation in a few places, hope it got better that way.}
For $r>0$, define \eq{deff}{f_{A,B,r,c}:=\mathbbm 1_{E_{r,c}(A,B)}}  to be the characteristic function of  $E_{r,c}(A,B)$.  
%let 
%$\mathbbm 1_{E_{r,c}}$ be the characteristic function of  $E_{r,c}$.
It follows that 
$$
%\mathbbm 1_{E_{r,c}}
f_{A,B,r,c}(g_t\bv) = 
%|\{t \in [0,T] : g_t v \in E_{T,c} \}|  = 
\begin{cases} 1 & 
\begin{aligned}
 \text{if }
&e^t \leq \|\mathbf{y}\|_{\mathbf{b}} < e^{t+r}, \|\bx\|_\ba\cdot \|\by\|_\bb<c
\\
&\text{ and }
(\pi_{\mathbf{a}}\big(\mathbf{x}), \pi_{\mathbf{b}}(\mathbf{y})\big) \in A \times B
\end{aligned}
\\
0 & \text{otherwise.}
\end{cases} 
$$
Therefore 
$$
\bv \in E_{T,c}(A,B) \ssm E_{r,c}(A,B) \implies |\{ t \in [0,T]: g_t\bv \in
E_{r,c}(A,B) \}| = r
$$
and
$$
g_t\bv \in E_{r,c}(A,B) \text{ for some } t \in [0,T] \implies \bv \in E_{r+T,c}(A,B).
$$
Using \equ{eq: Siegel transform}, and changing the order
of summation and integration, it follows that for any
$\Lambda \in \Xn$ and any $T>r$ we have 
\eq{eq;key link}
{\begin{aligned}%\label
\sharp\Big (\Lambda \cap \big(E_{T,c}(A,B)\ssm E_{r,c}(A,B)\big)\Big)
&\le \frac{1}{r}\int_0^T\widehat{
%\mathbbm{1}}_{E_{r,c}}
f}_{A,B,r,c}(g_t \Lambda)  \dd t \\ &\le
\sharp \big( E_{ r+T,c}(A,B)\cap \Lambda\big).
\end{aligned}}


Let $\mathbb S^{{\m}-1}_+:=\{\bx\in \mathbb S^{{\m}-1}: x_i\ge 0\}$ and
$\mathbb S^{{\n}-1}_+:=\{\by\in \mathbb S^{{\n}-1}: y_j\ge 0\}$. Also for brevity let us denote by $\Lambda_\vartheta$ the lattice $u(\vartheta)\Lambda_0$.
{Our main tool for proving Theorem \ref{thm;main}} is the following result 
of Schmidt.
\begin{thm}[\cite{s60}{,} Theorem{s} 1 and 2]\label{thm;schmidt use}
For almost every $\vartheta\in M$, 
\begin{align}\label{eq;count s60}
\sharp \left(E_{T, c}(\mathbb S^{{\m}-1}_+,\mathbb S^{{\n}-1}_+)\cap \Lambda_{\vartheta}
\right)\thicksim  cT \qquad \mbox{as }T\to \infty.
\end{align}
\end{thm}
The case ${\bb}=\bn$ of this result follows from \cite[Thm.\ 1 and {2} ]{s60} 
(setting $h=e^T, \, \psi_i(q) =
\big(\frac{c}{q}\big)^{a_i} $ in Schmidt's notation),
and in fact Schmidt also obtains an
error estimate $o\left(T^{1/2} \log^{\tau+\varepsilon} (T) \right)$ where $\tau=2$
if $n=1$ and $\tau=1.5$ otherwise. The
case ${\bb}\neq \bn$ does not appear in \cite{s60}, 
but the argument given there works for arbitrary $\bb$. 


 
\begin{cor}\label{cor;count equi}
Let $r, c>0$.
%,  $A=\mathbb S^{n-1}$ and $B=\mathbb S^{m-1}$. \comdima{Why not use the notation from the previous theorem?}
Then for almost every $\vartheta\in M$, 
\begin{align}\label{eq;count equi}
\lim_{T\to \infty}\frac{1}{T}\int_0^T \widehat{
%\mathbbm 1}
%_{E_{r,c}
f}_{\mathbb S^{{\m}-1},\mathbb S^{{\n}-1},r,c}
(g_t \Lambda_{\vartheta} )\dd t =|E_{r,c}%(\mathbb S^{n-1},\mathbb S^{m-1})
|.
\end{align}
\end{cor} 
\begin{proof}
Let $N_{T,c}( \vartheta)$
be 
the number of %simultaneous 
solutions
 $(\mathbf p, \mathbf q)\in \mathbb Z^{{\m}}\times \mathbb Z^{\n}_{\ge 0}$ of the system 
\begin{align*}
\begin{cases}
& 0\le (\vartheta\mathbf q)_i-p_i< c^{a_i}\|\mathbf q\|_{\mathbf b}^{-a_i} \qquad (i=1, \ldots, {\m}) 
%\label{eq;rational bad}
\\
&1\le \|\mathbf q\|_{\mathbf b}< e^T.
\end{cases}
\end{align*}
It follows directly from the definitions that 
\begin{align}\label{eq;count equi n}
N_{T,c}(\vartheta)=\sharp\left (E_{T, c}(\mathbb S^{{\m}-1}_+, \mathbb S^{{\n}-1}_+)\cap \Lambda_{\vartheta}\right).
\end{align}

Let $\{\mathbf e_i:1\le i\le {\m}\}$ be the standard basis of $\R^{\m}$.
 For $I\subset \{1, \ldots, {\m}\}$ 
 we let $\zeta_I : \R^{\m} \to \R^{\m}$ be the linear transformation defined
 by 
$$\zeta_I\mathbf e_i= \begin{cases} -\mathbf e_i & i\in I \\ 
\mathbf e_i & i \notin I. \end{cases}$$ 
Analogously we define $\eta_J: \R^{\n} \to
 \R^{\n}$  for $J\subset \{1, \ldots, {\n}\}$. 
It follows from Theorem \ref{thm;schmidt use} that for almost every $\vartheta\in M$
\begin{align}\label{eq;count number}
N_{T,c}( \zeta_I\vartheta \eta_J)\thicksim c  T \qquad \mbox{as } T \to \infty.  
\end{align}
On the other hand, it is easy to see that 
$
N_{T,c}( \zeta_I\vartheta \eta_J)
$
is the number of %simultaneous 
solutions
$(\mathbf p, \mathbf q)\in \mathbb Z^{ d }$ of the system 
\begin{align*}
%\begin{cases}
&c^{a_i}\|\mathbf q\|_{\mathbf b}^{-a_i}<  (\vartheta\mathbf q)_i-p_i\le 0 &(i \in I)\notag\\
& 0\le (\vartheta\mathbf q)_i-p_i< c^{a_i}\|\mathbf q\|_{\mathbf b}^{-a_i} &(i\not \in I) \notag\\
&1\le \|\mathbf q\|_{\mathbf b}< e^T  & \notag \\
& q_j\le 0 &(j \in J)  \notag\\
& q_j\ge 0 &(j\not \in J) \notag
%\end{cases}
\end{align*}


Now let $\widetilde N_{T,c}( \vartheta)$ be the number of %simultaneous 
solutions $(\mathbf p, \mathbf q )\in \mathbb Z^{ d }$
of the system \equ{kg1}--\equ{kg2}.
%\begin{align*}
%\begin{cases}
%& |(\vartheta\mathbf q)_i-p_i|< c^{a_i}\|\mathbf q\|_{\mathbf b}^{-a_i} \qquad (i=1, \ldots, n) \\
%&1\le \|\mathbf q\|_{\mathbf b}< e^T.
%\end{cases}
%\end{align*}
%\comdima{This system already appeared in the introduction as \equ{kg1}--\equ{kg2}, maybe simply give a reference?}
Then  for almost every $\vartheta\in M$ (i.e.\ for those $\vartheta$ for
which (\ref{eq;count number}) holds for all $I$ 
and 
%(\ref{eq;rational bad})
$(\vartheta\mathbf q)_i-p_i$ is not equal to zero for any $\mathbf{p}, \mathbf{q}$ with $\bq\neq 0$ and any $i$), one has 
\begin{align}\label{eq;diophantine claim}
\widetilde N_{T,c}( \vartheta)\thicksim  2^{ d } cT\qquad\mbox{as }T\to \infty.
\end{align}
%Similarly to (\ref{eq;count equi n}) one has
%\begin{align}\label{eq;count equi s}
%\widetilde N_{T,c}(\vartheta)=\sharp\left (E_{T, c}\cap \Lambda_{\vartheta}\right).
%\end{align}
%\comdima{Now $A,B$ are different? confusing again.}
As was mentioned in the introduction, % direct calculation shows that \comdima{(this was already mentioned in \S\ref{sec;intro})}
\begin{align}\label{eq;count equi erc}
\widetilde N_{T,c}(\vartheta)=\sharp\left (E_{T, c}\cap \Lambda_{\vartheta}\right)\text{ \  and  \ }|E_{T, c}|=2^dcT .
\end{align}
%\comdima{In the presence of weights, $E_{T, c}$ is a rather complicated set, so, as I said before, an explanation of why there is no dependence on weights would be nice.}
It follows from %(\ref{eq;count equi s}), 
(\ref{eq;count equi erc})
and (\ref{eq;key link})
% Lemma \ref{lem;very useful}
that  (\ref{eq;count equi}) holds for those $\vartheta$ which satisfy
(\ref{eq;diophantine claim}).
\end{proof}

\begin{cor}\label{cor;count equi f}
For $r, c>0$ we let
\begin{align}\label{eq;f r c}
F_{r, c}:=
\left\{(\mathbf x, \mathbf y)\in \mathbb R^d: \|\bx\|_{\mathbf a}\|\mathbf
y\|_{\mathbf b}< c, \, 1\le  
\|\mathbf x\|_{\mathbf a}<e^r \right\}.
\end{align}
Then for almost every $\vartheta\in M$
\begin{align*}%\label{eq;count equi f}
\lim_{T\to \infty}\frac{1}{T}\int_0^T \widehat{\mathbbm 1} _{F_{r,c}}(g_t \Lambda_{\vartheta})\dd t =|F_{r,c}|.
\end{align*}
\end{cor}
\begin{proof}
Let %$E_{r, c}$ and 
$\widetilde {N}_{T, c}$ be as in Corollary \ref{cor;count equi}. 
%\comdima{But $E_{r, c}$ is not defined in Corollary \ref{cor;count equi}.}
For every  $T> |\log c|$   and $\vartheta\in M$, similarly to (\ref{eq;key link}) one has
\[
\frac{1}{r}\int_0^T \widehat{\mathbbm 1}_{F_{r,c}} (g_t\Lambda_{\vartheta})\, dt\le
  \widetilde{ N}_{T+\log c}(\vartheta)+\sharp (\widetilde F\cap \Lambda_{\vartheta})
\]
where 
\[
\widetilde F:=\{(\mathbf x, \mathbf y)\in \mathbb R^d: 
\|\mathbf x\|_{\mathbf a}<e^r, 
\|\mathbf y\|_{\mathbf b}<c\}.
\]
For every $\vartheta$, $\sharp (\widetilde F\cap \Lambda_{\vartheta})$
is a number independent of $T$ and $|F_{r,c}|=|E_{r,c}|$. So for  $\vartheta\in M$
satisfying (\ref{eq;diophantine claim}) one has
\[
\limsup_{T\to \infty}
\frac{1}{T}\int_0^T \widehat{\mathbbm 1}_{F_{r,c}}
(g_t\Lambda_{\vartheta})\dd t\le |F_{r, c}|.
\]
On the other hand by \cite[Corollary 1.3]{s}, there is a conull subset
of $\vartheta \in M$ for which $\Lambda_{\vartheta}$ is $\big(D^+,
C_c(\Xn)\big)$-generic. For any $\vre>0$, since $\mathbbm{1}_{F_{r,c}} $
is a   {Riemann integrable} non-negative function, there is $\varphi_1 \in
C_c(\R^d)$ such that 
%\eq{eq: first inequality}{
$$
\mathbbm{1}_{F_{r,c}}(\bv) \geq \varphi_1(\bv) \text{ for all } \bv \text{ and
} \int_{\R^d} \varphi_1 > |F_{r,c}|-\vre.
$$
%}
Now $\widehat{\varphi}_1$ is a continuous non-negative %function
integrable function on $\Xn$, and therefore there exists $\varphi_2 \in
C_c(\Xn)$ such that 
\eq{eq: second inequality1}{
%$$
\varphi_2(\Lambda) \leq \widehat{\varphi}_1(\Lambda) \leq \widehat{\mathbbm{1}}_{F_{r,c}}(\Lambda)
\text{ for all } \Lambda
}
%$$
and 
\eq{eq: second inequality}{
\int_{\Xn} \varphi_2 \dd \mu \geq \int_{\Xn} \widehat{\varphi}_1 \dd
\mu -\vre  = \int_{\R^d} \varphi_1 - \vre \geq |F_{r,c}| -2\vre.
}
%and  
%, and the Portmanteau theorem (cf. \cite[Theorem
%1.3.4(iv)]{v}), for 
Since $\vre>0$ was arbitrary, \equ{eq: second inequality1} and
\equ{eq: second inequality} imply that for almost every $\vartheta\in M$
\[
\liminf_{T\to \infty}
\frac{1}{T}\int_0^T \widehat{\mathbbm 1}_{F_{r,c}}
(g_t\Lambda_{\vartheta})\dd t\ge  |F_{r, c}|.
\]
This completes the proof. 
\end{proof}


\section{Pointwise equidistribution with respect to unbounded functions}\label{sec:unbdd}
In this section we prove 
Theorems \ref{thm;main} and \ref{thm;diophantine}. We first show that
for  Riemann integrable $f,$   {the function} $\widehat{f}$   {as in \equ{eq: Siegel transform}} belongs to the class $C_\alpha(X)$.


\begin{lem}\label{lem: Blichfeldt}
For any $d$ and all sufficiently large $r$ there are constants $c_1,
c_2$ such that %for all sufficiently large $r$, 
if $\mathbbm 1_{B_r}$  is the characteristic function of the open ball 
of radius $r$ centered at origin, 
 then for all $\Lambda \in X $, 
\begin{align}\label{eq;upper lower}
c_1 \alpha(\Lambda) \leq \widehat{\mathbbm{1}}_{B_r}(\Lambda)
\leq c_2 \alpha(\Lambda).
\end{align}
In particular, for any Riemann   {integrable} $f: \R^d \to \R$,
$\widehat{f} \in C_\alpha(X)$. 
\end{lem}
\begin{proof}
For any discrete subgroup $  {\Delta} \subset \R^d$, the $i$-th
Minkowski 
successive minimum of $  {\Delta}$ with respect to $B_r$ is defined to be 
$$
\lambda_i\left(  {\Delta}\right) := \inf\left\{t>0: \dim \big( \spa\,
 (tB_r \cap   {\Delta})\big) \geq i \right\}.
$$
%(with the convention that $  {\lambda_i\left(  {\Delta}\right)}= \infty$ 
%is  $\infty$ 
%when the set in the right
%hand side is empty). 
%Let $\lambda_i(\new{\Delta}), i =1, \ldots,d$ be the Minkowski successive minima
%of the discrete group $\new{\Delta}$, with respect to the ball $B_r$.
%, and let
%$\lambda'_i(\new{\Delta})$ be the successive minima with respect to the
%unit Euclidean ball. Then we have 
%\eq{eq: then we have}{
%\lambda'_i(\Lambda') =
%r \lambda_i(\Lambda') \text{ for each } i.
%}
Let $r_0$ be large enough (depending on $d$) so that for any $r \geq
r_0$ and any $\Lambda
\in X$, $\lambda_1( \Lambda) <1$. 
The notation $x\asymp y$ will mean that $x$ and
$y$ are functions of discrete subgroups of  $\R^d$ and there are positive
constants $C_1, C_2$, depending on $d$ and $r$, such that
$\displaystyle{C_1 \leq x/y \leq C_2}$.  By Minkowski's second theorem, see e.g.\ 
%Cassels 
\cite[\S VIII.2]{ca}, 
$$\lambda_1\left(  {\Delta} \right) \cdots \lambda_\ell\left(  {\Delta}\right)\asymp d\left(  {\Delta} \right),$$ where $d
\left(  {\Delta} \right)$
is the covolume of $  {\Delta}$ in $\spa \left(  {\Delta} \right)$ and $\ell$ is the rank
of $  {\Delta}$.   {Now for $\Lambda
\in X$ define $\Delta $ to be the subgroup of
$\Lambda$ generated by $\Lambda \cap \overline{B_r}$. Then}
%This implies that 
$$\alpha(\Lambda) \asymp \big(\lambda_1(\Lambda) 
\cdots \lambda_j(\Lambda)\big)^{-1} =
\big(\lambda_1(  {\Delta}) 
\cdots \lambda_j(  {\Delta})\big)^{-1},$$ where $j$ is   {the} index for which $\lambda_j(\Lambda) \leq 1
< \lambda_{j+1}(\Lambda)$.
% and $\Lambda' $ is the subgroup of
%$\Lambda$ generated by $\Lambda \cap \overline{B_r}. $ 

%Using \equ{eq: then we have}, we find that for fixed $r>r_0$, 
%$\alpha(\Lambda) \asymp (\lambda_1(\Lambda') \cdots \lambda_{j}(\Lambda'))^{-1}.$
It follows from
\cite[Prop.\ 2.1 and Cor.\ 2.1]{Henk} (applied to $K=B_r, \, d=j$ and $\mathbb{L}=  {\Delta}$) that 
$$
  {\sharp}(  {\Delta}\cap B_r)\asymp
\big(\lambda_1 (  {\Delta} )\cdots \lambda_{j}(  {\Delta})\big)^{-1}.
$$
Since $\widehat{ \mathbbm{1}}_{B_r} (\Lambda) =  {\sharp}(\Lambda \cap B_r)   {-1}=
  {\sharp}(  {\Delta}\cap B_r)  {-1}$, \equ{eq;upper lower} follows. 

For the second assertion, note that for any Riemann   {integrable} $f:\R^d \to \R$ there are positive $r$ and $C$ so that $|f| \leq
C\cdot \mathbbm{1}_{B_r}$ and hence condition ($C_\alpha$-$2$) follows from
\equ{eq;upper lower}. To prove ($C_\alpha$-$1$), let $S$ be the set of
discontinuities of $f$ in $\R^d \sm \{0\}$, so that $|S|=0$. From
\equ{eq: Siegel transform} it follows that the set
$S'$ of discontinuities of
$\widehat{f}$ is contained in $$S'':=\{\Lambda: \Lambda \cap S \neq
\varnothing\}.$$ For each $\bv \in \Z^d \sm \{0\}$, the set of
$g \in G$ such that $g\bv \in S$ has Haar measure zero in $G$, and hence
$S''$ is a countable union of sets of $\mu$-measure zero. In
particular $\mu(S')=0$. 
\end{proof}




We now derive some general properties  from  equidistribution of measures.

\begin{lem}\label{lem;truncation}
Let $\psi \in C_\alpha(X)$ and
let $\{\mu_i\}$ be a sequence of probability measures on $\Xn$
such that $\mu_i \to \mu$, 
with respect to the weak-$*$ topology. 
Then for any non-negative $\varphi \in C_c( \Xn)$ one has
\begin{equation}\label{eq;compact}
\lim_{i\to \infty}\int_{\Xn}\varphi
%\widehat f
\psi\dd\mu_i =\int_{\Xn} \varphi
%\widehat f
\psi \dd\mu.
\end{equation}
\end{lem}
\begin{proof} 
Using assumption $(C_\alpha$-$1)$ we see that the function $\varphi\psi$ is bounded,  compactly 
supported and continuous except on a set of measure zero. {By using a partition of unity, without loss of generality one can assume that $\varphi$ is supported on a coordinate chart. 
Applying Lebesgue's criterion for Riemann integrability to $\varphi\psi$, one can write $\int_X \varphi\psi\dd\mu$ as the limit of  upper and lower Riemann sums.  From this it  easily follows that}
 for any $\varepsilon>0$ there exist $h_1,h_2\in C_c(X)$  
such that $h_1\le \varphi\psi\le h_2$ and 
\begin{align}
\int_{X}(h_2-h_1)\dd\mu \leq\varepsilon.
%\|\varphi\|_{\infty} 
 \label{eq;truncation 4}
\end{align}
Thus we have
\begin{align}
%%%%%%%%%1
&\limsup_{ i\to \infty}\int_{\Xn}\varphi
\psi\dd\mu_i \le \int_{\Xn} 
h_2 \dd\mu
 \label{eq;truncation 1}\\
 %%%%%%%%%2
&\liminf_{ i\to \infty}\int_{\Xn}\varphi
\psi\dd\mu_i \ge \int_{\Xn} 
h_1 \dd\mu 
 \label{eq;truncation 2}\\
 %%%%%%%%3
& \int_{\Xn}%\widehat f
h_1\dd\mu\le \int_{\Xn} \varphi%\widehat f
\psi\dd\mu 
\le \int_{\Xn}%\widehat f
h_2 \dd\mu.  \label{eq;truncation 3}
\end{align}
Therefore by taking $\varepsilon\to 0$, (\ref{eq;compact}) follows from 
 (\ref{eq;truncation 1})--(\ref{eq;truncation 3})
 %,  (\ref{eq;truncation 2}), 
 and (\ref{eq;truncation 4}).
\end{proof}

\begin{cor}\label{cor;tail}
Let the notation be as in Lemma \ref{lem;truncation} and assume that
\begin{align}\label{eq;equi siegel}
\lim_{i\to \infty}\int_{\Xn} 
\psi\dd\mu_i=\int_{\Xn} 
\psi\dd\mu.
\end{align}
Then for any $\varepsilon >0$ there exists $i_0>0$ and
$\varphi \in C_c(\Xn)$ with $0 \leq \varphi \leq 1$ such that
\begin{align}\label{eq;con tail}
\left|\int_{\Xn} (1-\varphi)
\psi \dd\mu_i\right |< \varepsilon
\end{align}
for any $i\ge i_0$.
\end{cor}
\begin{proof}
Since $%\widehat f
\psi \in L^1(\Xn,\mu)$,
%. {Hence} 
there exists a compactly supported continuous function 
 $\varphi: \Xn\to [0, 1]$ such that 
 \begin{align}\label{eq;tail 1}
\left|\int_{\Xn} (1-\varphi)%\widehat f
\psi\dd\mu\right|<\frac{\varepsilon}{3}.
 \end{align}
 By Lemma \ref{lem;truncation} and (\ref{eq;equi siegel}), there
 exists $i_0>0$ such that for $i\ge i_0$ 
 \begin{align}
\left | \int_{\Xn} \varphi
%i(x) 
%\widehat f
\psi
%(x)
\dd\mu_i- \int_{\Xn} \varphi
%(x) 
%\widehat f
\psi
%(x)
\dd\mu\right|&<\frac{\varepsilon}{3}
\label{eq;tail 2}\\
\left | \int_{\Xn}%\widehat f
\psi
%(x)
\dd\mu_i- \int_{\Xn}%\widehat f
\psi
%(x)
\dd\mu\right|&<\frac{\varepsilon}{3}
\label{eq;tail 3}.
 \end{align}
 It follows from (\ref{eq;tail 1}),
 (\ref{eq;tail 2}) and (\ref{eq;tail 3}) that  (\ref{eq;con tail}) holds if $i\ge i_0$.
\end{proof}

\begin{cor}\label{cor;dominated}
Let the notation be as in Lemma \ref{lem;truncation}.
 Assume  that
there exists a non-negative Riemann 
integrable  function 
$f_0$ on $\mathbb R^{ d }$ such that  $|\psi |\le \widehat{f_0}$ and 
\begin{align}\label{eq;dominated assume}
\lim_{i\to \infty}\int_{\Xn} \widehat {f_0} \dd\mu_i=\int_{\Xn}
\widehat {f_0} \dd\mu. 
\end{align}
Then
\[
\lim_{i\to \infty}\int_{\Xn} \psi\dd\mu_i=\int_{\Xn}  \psi \dd\mu.
\]
\end{cor}
\begin{proof}
By Lemma \ref{lem;truncation}, Corollary \ref{cor;tail} and
(\ref{eq;dominated assume}), 
for any $\varepsilon>0$ there exists $i_0>0$
and a continuous compactly supported function $\varphi: \Xn\to [0, 1]$
such that for $i\ge i_0$ one has
\begin{align*}
\left| \int_{\Xn}\varphi \psi \dd\mu_i-\int_{\Xn}\varphi \psi \dd\mu\right|
&<\frac{ \varepsilon}{3}, \\
\int_{\Xn}(1-\varphi) {\widehat f_0 }\dd\mu_i&< \frac{\varepsilon}{3}, \\
\int_{\Xn}(1-\varphi) {\widehat f_0 }\dd\mu &<\frac{ \varepsilon}{3}.
\end{align*}
Using the assumption $|\psi \, |\le \widehat{f_0}$ and these
inequalities, one finds
\[
\left|\int_{\Xn} \psi \dd\mu_i-\int_{\Xn}  \psi \dd\mu\right|<\varepsilon
\]
for $i\ge i_0$.
This completes the proof.
\end{proof}


%For a Riemann integrable function  $f$ we now 
Now, given $\psi\in C_\alpha(X)$, we are going to apply the results of
\S\ref{sec;count} to construct a function $f_0$  satisfying the
assumption of Corollary \ref{cor;dominated}. 

 



\begin{lem}\label{lem;diophantine}
For $r> d $ let  $A_r$ be the annular region defined by 
$$A_r:=\{\mathbf v\in \mathbb R^{ d }:  d <\|\mathbf v\|<r\}.$$
Then for almost every $\vartheta\in M$, 
\begin{align}\label{eq;annular}
\lim_{T\to \infty}\frac{1}{T}\int_0^T 
\widehat {\mathbbm 1}_{A_r}\big(g_t
%u(\vartheta)
\Lambda_\vartheta\big)\dd t=|A_r|.
\end{align}
\end{lem}





\begin{proof}
For ${r'},c>0$ let
%$E_{{r'}, c}=E_{{r'}, c}(\mathbb S^{n-1}, \mathbb S^{m-1})$ be
%as defined in Theorem \ref{thm;diophantine} 
%which
% is defined in (\ref{eq;count region}) 
%and let 
$F_{{r'}, c}$ be as in (\ref{eq;f r c}).
Since  $r>  d $, there exist $c,{r'}> 1$ such that 
$A_r\subset E_{{r'}, c}\cup F_{{r'}, c}$.  It follows that 
\[
{\mathbbm 1}_{A_{r}}\le 
{\mathbbm 1}_{E_{{r'},c}}+{\mathbbm 1}_{F_{{r'},c}}.
\]
It follows from Corollaries \ref{cor;count equi} and \ref{cor;count equi f} that (\ref{eq;dominated assume}) holds
for $$f_0={\mathbbm 1}_{E_{{r'},c}}+{\mathbbm 1}_{F_{{r'},c}}.$$
Hence the conclusion of the lemma follows  from Corollary \ref{cor;dominated}.
\end{proof}

\begin{lem}\label{lem;key}
Let $B_r$ be the open Euclidean ball of radius $r>0$ centered at the origin in $\mathbb R^{ d }$. 
Then 
\begin{align*}%\label{eq;ball}
\lim_{T\to \infty}\frac{1}{T}\int_0^T \widehat {\mathbbm{1}}_{B_r}(g_t
\Lambda_{\vartheta})\dd t=
|B_r|
\end{align*}
for almost every $\vartheta\in M$. 
\end{lem}
\begin{proof}
We first observe that any annular region of width $d$ centered at the
origin contains a lattice point of  
 any unimodular lattice in  $ \mathbb R^{ d }$. 
 Also for any unimodular lattice $\Lambda$ in $\R^d$ and any $\bv\in
 \Lambda$ one has 
 $$\sharp (B_r\cap \Lambda)=\sharp \big((B_r+\bv)\cap \Lambda\big)\,.$$
 It follows from these two observations that for any lattice $\Lambda
 \in X$, the number of lattice points in $B_r$ is at least the number
 of lattice points in $A_{r'}$, where $r' = 2d+2r$. That is, 
 $
\widehat{  {\mathbbm{1}}}_{B_r}\le  \widehat{\mathbbm{1}}_{A_{r'}}.
 $
The conclusion now  follows from Lemma \ref{lem;diophantine} and Corollary \ref{cor;dominated}.
 \end{proof}
 
 \begin{proof}[Proof of Theorem \ref{thm;main}] 
 It follows from Lemma \ref{lem;key} and the Siegel integral formula 
 that there is a Borel  subset $M'$ of $M$ 
  of full measure such that for 
 any $\vartheta\in M'$ 
 and  $r\in \mathbb \N$  one has
 \[
 \lim_{T\to \infty}\frac{1}{T} \int_0^T \widehat{\mathbbm 1}_{B_r}(g_t \Lambda_\vartheta)\dd t
 =\int_X \widehat{\mathbbm 1}_{B_r}\dd\mu.
 \]
 Moreover we assume that $\Lambda_\vartheta$ is $\big(D^+, C_c(X)\big)$-generic for  every $\vartheta\in M'$. 
 In view of (\ref{eq;upper lower}) for any $\psi\in %\mathcal{SR}\cup 
 C_\alpha(X)$ there is $c>0$ 
 and $r\in \N$ such that 
 \[
 |\psi(\Lambda)|\le c\widehat{\mathbbm 1}_{B_r}(\Lambda)
 \]
 for every $\Lambda \in X$. 
 It follows from Corollary \ref{cor;dominated} that 
 $\Lambda_\vartheta$ is $(D^+,\psi)$-generic for every $\vartheta\in M'$,
 % the element
 and 
 the proof is complete.  
 \end{proof}




\begin{proof}[Proof of Theorem \ref{thm;diophantine}] Our argument is similar to the one used in \cite{apt}. 
%We write $E_{r, c}$ for $E_{r, c}(A, B)$ to simplify the notation. 
The assumption on the boundaries of $A$ and $B$ 
in $\mathbb S^{{\m}-1}$ and $\mathbb S^{{\n}-1}$ respectively implies that the boundary of  
$E_{r, c}(A, B)$ in $\R^d$ has zero Lebesgue measure.   
With the notation $f_{A,B,r,c}$ as in \equ{deff}, it follows from
\equ{eq;key link} and Theorem \ref{thm;main} that for almost every
$\vartheta\in M$ 
\begin{equation}\begin{aligned}\label{eq;application}
\sharp\big(E_{T,c}(A, B) \cap \Lambda_{\vartheta}\big) \cdot r  & \thicksim
\int_0^T\widehat{f}_{A,B,r,c}
%\mathbbm{1}}_{E_{r,c}}
(g_t \Lambda_{\vartheta})\dd t\\ &\thicksim |E_{r,c}(A, B)|\cdot T \quad \mbox{as }T \to 
\infty.
\end{aligned}\end{equation}

In order to conclude the proof,
%of  (\ref{eq;diophantine asym}),
 it suffices to show that 
\begin{align}\label{eq;compare}
|E_{r,c}(A, B)|\cdot T =|E_{T,c}(A, B)| \cdot r.
\end{align}
Indeed, when $T = kr $ for some $k \in \N$ we have $$E_{T,c}(A, B) =
\bigcup_{j=0}^{k-1} g_{-jr} \big(E_{r,c}(A, B)\big)$$ (a disjoint union), and \equ{eq;compare} follows from the fact
that the $g_t$-action preserves Lebesgue measure. From this one
deduces \equ{eq;compare} when  $T/r\in\Q$, and finally one gets
\equ{eq;compare} for arbitrary $T,r>0$ by continuity. 
%\comdima{Somewhere we should credit \cite{apt} since the method comes from there, doesn't it?}
\end{proof}




 %%%%%%%comment below
\begin{comment}
 \begin{cor}\label{cor;counting}
 For almost every $\vartheta\in M$, 
\begin{align}\label{eq;counting}
\sharp (E_{T, c}(\mathbb S^{n-1}, \mathbb S^{m-1})\cap u( \vartheta)
\Z^d)\thicksim c T \qquad \mbox{as }T\to \infty.
\end{align}
 \end{cor}
For $c, T\in (0, \infty)$, we 
let $N_{T, c}( \vartheta)$ 
\begin{lem}\label{lem;calculus}
\begin{align*}
|E_{r,c}(A, B)|_{\mathbb R^{ d }}&=|B_{\ba}|_{\mathbb R^n}\, |B_{\bb}|_{\mathbb R^m}\,
ac^{a}(r-s) \cdot\mu_{\ba}(A)\mu_{\bb}(B).
\end{align*}
\end{lem}
Now we assume Theorem \ref{thm;main} and give a 
\begin{proof}[Proof of Theorem \ref{thm;diophantine}]
It follows directly from the definition that the counting number
 $N(c,T, \vartheta; A, B)$ in Theorem \ref{thm;diophantine} is equal
to  
\[
\sharp (E_{T, c}(A, B)\cap u(\vartheta)\mathbb Z^{ d } ).
\]

\end{proof}



and show 
in Theorem \ref{thm;diophantine}
that
 Diophantine approximation vectors are random.  
Let $\mu_{\mathbf a}$ be the probability measure on 
$\mathbb S^{n-1}$  defined by 
\[
\mu_{\mathbf a}(A)=\frac{|\{ F_{\mathbf a t}(\mathbf x):\mathbf x\in A: t\le 0 \} |}
{| B_1|}
\]
where $B_{1}$ is the unit ball in $\R^n$ under Euclidean norm. 
It follows that the Lebesgue measure of $\mathbb R^n$ is equal  to 
\begin{align}\label{eq;lebesgue}
|B_{\ba}|_{\mathbb R^n}\, ar^{{a}-1} dr d\mu_{\mathbf a}  \quad \mbox{via the map } 
\mathbf x\to (\|\mathbf x\|_{\mathbf a}, \widehat{ \mathbf x}).
\end{align}
We remark here 
  the measure  $\mu_{\mathbf a}$ is absolutely continuous with respect to the volume measure on 
  $\mathbb S^{n-1}$. 
  Similar to (\ref{eq;lebesgue}) the Lesbesgue measure of $\mathbb R^m$
 is equal to 
 \begin{align}\label{eq;lebesgueb}
2^m\, ar^{{a}-1} dr d\mu_{\mathbf b}  \quad \mbox{via the map } 
\mathbf y\to (\|\mathbf y\|_{\mathbf b}, \widehat{ \mathbf y}).
\end{align}
we 
let $ N( \vartheta, T;c, A, B)$ be the number of simultaneous solutions
 $(\mathbf p, \bq)\in \mathbb Z^{n}\times \mathbb Z^m$ of the system 
\begin{align}
& \| \vartheta\bq +\mathbf p\|_{\mathbf a} \cdot\|\bq\|_{\bb}< c, \label{eq;diophantine a}\\
&\widehat{q\mathbf x+\mathbf p}\in A \quad \mbox{and}\quad \widehat \bq\in B \label{eq;diophantine b}, \\
&1\le  \|\bq\|_{\bb}< e^T, \label{eq;lower bound q}.
\end{align}
Then for almost every $\vartheta\in M$, 

\end{comment}





%%%%%%%%%%%%%%Appendix not needed





%%%%%%%%%%%%%%%%%Apendix above not needed

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\end{document}


\ignore{
Let $D=\{g_t: t\in\R\}$ be a one-parameter subgroup of $G$, and let $\varphi$
be a real-valued integrable  function on $\Xn$. Denote $D^+ := \{g_t: t\ \ge 0\}$. We say that $\Lambda
\in \Xn$ is {\em $(D^+, \varphi)$-generic} if 
\begin{align}\label{eq;generic}
\lim_{T\to \infty}\frac{1}{T}\int_0^T \varphi(g_t\Lambda)  \dd t= \int_{\Xn}
\varphi \dd\mu, 
\end{align}
and for a collection $\mathcal{S}$ of functions, that $\Lambda$ is
{\em $(D^+, \mathcal{S})$-generic}  if it is $(D^+, \varphi)$-generic
for every $\varphi \in \mathcal{S}$. Let $C_c$ denote the
continuous compactly supported functions on $\Xn$.  It is a well-known consequence of
the Moore ergodicity theorem and the Birkhoff ergodic theorem, that
for any unbounded $D$, $\mu$-almost every $\Lambda$ is $(D^+,
C_c)$-generic. 

In this paper we discuss some 
strengthenings of this fact for diagonal subgroups, and their
number-theoretic consequences. 


To this end, we fix two `weight vectors' 
\begin{align*}
&\mathbf a=(a_1,\ldots, a_n)\in \mathbb R^n_{>0} \\
&\mathbf b=(b_1, \ldots, b_m)\in \mathbb R^m_{>0}\qquad \mbox{such that } \\
&\sum_{i=1}^n a_i =\sum_{i=1}^m b_i=1,
\end{align*}
and refer to the case $a_1 = \cdots = a_n=1/n, \, b_1 = \cdots =b_m=1/m$ as
the case of {\em equal weights.} 
Let % $D$ be the one parameter subgroup $\{g_t: t\in \mathbb R\}$ where 
\[g_t=\mathrm{diag}(e^{a_1t}, \ldots, e^{a_nt}, e^{-b_1t},\ldots, e^{-b_mt} )\in G. 
\]


%Suppose that  $\varphi: \Xn\to \mathbb R$ is  a measurable function with $\int_{\Xn}| \varphi|\, d\mu<\infty$.
%We say $x\in \Xn$ is $(D^+, \varphi)$ generic if 
%
%
%Given a collection $\mathcal S$ of measurable functions on $\Xn$,  we
%say that $x$ is $(D^+, \mathcal S)$ generic
%if (\ref{eq;generic}) holds for every $\varphi\in \mathcal S$. 
Let $1_k$ be the identity matrix of order $k$, and let 
\[u: M \to G, \ 
u(\vartheta)=\left(\begin{array}{cc}
1_n & \vartheta \\
0 & 1_m
\end{array}\right). 
\]
%and for $g \in G, \, \vartheta \in M$, let 
%$$
%\Lambda_g := g\Z^d, \ \ \Lambda_{\vartheta} := \Lambda_{u(\vartheta)}.
%$$
%Let $C_c$  (resp. $C_c^\infty(\Xn)$)
% be the set of compactly supported continuous (resp. 
% smooth) real valued functions on $\Xn$. 
In the case of equal weights,
 the group $\{u(\vartheta): \vartheta \in M\}$ is the so-called
 `unstable horospherical subgroup' for $\{g_t\}$ and a standard
 argument implies that
\eq{eq: standard argument}{\forall \Lambda \in \Xn, \text{ for
    almost every } \vartheta \in M,\, u(\vartheta) \Lambda \text{ is } 
 (D^+, C_c)\text{-generic}
}  
(here and hereafter `almost every $\vartheta$' means almost every with
respect to Lebesgue measure on $M \cong \R^{nm}$). 
In \cite{s}, as a corollary of a more
general result, the second-named author generalized this to the case of non-equal
weights. Namely he showed that \equ{eq: standard argument} holds for
all choices of the weight vectors. 
The goal of this paper is to strengthen this result and obtain some number theoretic applications. 

Our first result provides equidistribution with an error bound for
smooth compactly supported functions.  We fix a right-invariant
Riemannian metric on $G$. This induces a right invariant metric on any
closed subgroup of  $G$ and a metric on $\Xn$.  
%All the implied
%constants in the following result may depend on this choice of metric. 
\begin{thm}\label{thm;main rate}
Let  $\Lambda \in \Xn,\varphi\in C_c^\infty(\Xn)$ and $\varepsilon>0$.
Then for  almost every $\vartheta\in M$ and every $T>2$
\begin{align}\label{eq;main rate goal}
\frac{1}{T} \int_0^T \varphi \left (g_t u(\vartheta)\Lambda \right)
\dd t=\int_{\Xn} \varphi \dd\mu+
O_{\varphi, \Lambda, \varepsilon, \vartheta}(T^{-1/2}\log^{1.5+\varepsilon} T).
\end{align}
\end{thm}
\combarak{ Try to rewrite the error in terms of
  something depending on $\varphi$ and something depending on the rest.}

There are two main ingredients in  the proof of Theorem \ref{thm;main rate}. 
One is a method which makes it possible to
derive pointwise equidistribution with an error rate from 
effective mixing and 2-mixing (according to \cite{s60} this method is
due to Cassels). We will state and prove a general result in 
Theorem \ref{thm;effective} which works in a broader context than we require. 
The other is the following effective 2-mixing result, generalizing an
`effective mixing' result of \cite{km12}: 



\begin{thm}\label{thm;mixing}
There exists $\delta >0$ such that the following holds. 
Given $f\in C_c^\infty (M)$ and $\varphi,\psi \in C_c^\infty(\Xn) $
{with $\int_{\Xn} \varphi  = \int_{\Xn}\psi = 0$}
and 
%with $\int_{\Xn} \varphi \dd \mu = \int_{\Xn} \psi \dd \mu =0$, and
%given
 a compact subset  $L$  of $\Xn$, there exists $C%=C(f, \varphi, \psi,
                                %L)
>0$ such that 
for any $\Delta, \Lambda \in L$ and $t, {w}\ge 0$  one has
\begin{align}\label{eq;mixing rate}
\left|
{\int_{M} f(\vartheta)\varphi
\left(g_tu(\vartheta)\Delta\right)\psi \left(g_{w} u(\vartheta)\Lambda\right)\dd  \vartheta}
%I( t, {w}) -\int_M f \int_{\Xn} \varphi \int_{\Xn}\psi
\right|
\le C e^{-\delta \min (t,  {w}, |{w}-t|)}
\end{align}
where
\begin{align}\label{eq;mixing goal}
I( t, {w}):=
%I(f, \varphi, \psi, x,z;t, {w}):=
\int_{M} f(\vartheta)\varphi
\left(g_tu(\vartheta)\Delta\right)\psi \left(g_{w} u(\vartheta)\Lambda\right)\dd  \vartheta.
\end{align}
\end{thm}

Under the same assumptions we can prove effective $k$-mixing for all
$k$. We hope to return to this topic elsewhere. See also
\cite{BEG}. 

\smallskip

Now let 
\eq{eq: defn Lambda theta}{
\Lambda_0 = \Z^d, \ \Lambda_{\vartheta} = u(\vartheta) \Z^d = 
\left\{ \left( \begin{matrix}\vartheta \mathbf{ q} - \mathbf{p} \\
      \mathbf{q} \end{matrix} \right) : \mathbf{p} \in \Z^n, \,
  \mathbf{q} \in \Z^m \right\}. 
}
Our second main result is that for for almost
 every $\vartheta$, $\Lambda_{\vartheta}$ is $(D^+,
 \varphi)$-generic for certain unbounded functions $\varphi$ 
on $\Xn$.
Recall that $f: \mathbb R^{ d }\to \mathbb R$ is said to be Riemann
integrable if $f$ is  bounded with bounded support
and it is continuous except on  a set of Lebesgue measure zero.  
The {\em Siegel transform} of $f$ is the function $\widehat f$ on
$\Xn$ defined by %the Siegel summation formula: 
\[
\widehat f\left ( \Lambda \right)=\sum_{\mathbf v\in \Lambda\nz} f(g\mathbf v).
\]
%The main result of this paper is:
It was shown by Siegel that 
%\[
$
\int_{\Xn} \widehat f \dd\mu=\int_{\mathbb R^{ d }} f(\mathbf v) \dd\mathbf v.
$
%\]
\begin{thm}\label{thm;main}
Let $\mathcal{SR}$ be the class of all functions on $\Xn$ which are
Siegel transforms of Riemann integrable functions $\R^d \to \R$. 
Then for almost every $\vartheta\in M$, %with respect to additive Haar measure
$\Lambda_{\vartheta}$ is $(D^+, \mathcal{SR})$-generic. 
\end{thm}


%\begin{thm}\label{thm;main}
%Let $f: \mathbb R^{ d }\to \mathbb R$ be a Riemann integrable function. 
%Then for almost every $\vartheta\in M$, %with respect to additive Haar measure
%\begin{align}\label{eq;main}
%\lim_{T\to \infty}\frac{1}{T}\int_0^T \widehat f(g_t
%\Lambda_{\vartheta})\dd t = \int_{\Xn} \widehat f\dd\mu
%%\int_{\mathbb R^{ d }} f(\mathbf
%%v)\, \dd \mathbf v.
%\end{align}
%\end{thm}
%\comronggang{I don't think I can prove  almost every  theta works for any Riemann integrable $f$.
%The difficulty is if two functions are pointwise close we don't know whether their siegel transform are pointwise
%close.}

%\combarak{If we can prove the more general result, rewrite with the
%  terminology `$(D^+, \mathcal{S})$-generic' where $\mathcal{S}$ is
%  the collection of Siegel transforms of Riemann integrable functions.}



%Suppose $f$ as in Theorem \ref{thm;main} is a nonnegative function. 
%Since we already know 
% equidistribution (i.e.~$(D^+,C_c)$ genericity),
%  to prove (\ref{eq;main}) it suffices to get the correct  upper bound for the left hand side of (\ref{eq;main}). 
% In this sense Theorem \ref{thm;main} is quite similar to \cite[Theorem 3.4]{emm98}. But we feel that 
% the methods in \cite{emm98} do not apply in our case since the key exponent ${w}$ in 
%  \cite[Lemma 5.1]{emm98} is always $1$ for the natural representation $G$ on $\wedge_{1}^{ d -1}\mathbb R^{ d }$. 
% %But it is not clear to us whether the new ideas in their later paper \cite{emm05} are helpful in our situation or not. 
% Here we use another approach:
Since $(D^+,C_c)$-genericity was already proved in \cite{s}, the
main additional point  in
the proof of Theorem \ref{thm;main} is to obtain the correct upper
bound for Birkhoff averages of Siegel transforms of non-negative functions. 
%the right hand side of \eqref{eq;main}, in case $f$ is non-negative. 
To this end we
employ a lattice point counting result of Schmidt \cite{s60}. Lattice
point counting  was
used for a similar purpose in \cite{ms14} and \cite{agt} \comdima{(Is this a correct reference?)}, and the
connection between Schmidt's result and the action of $D^+$ was already noted in
\cite{apt}.
Note that in our result we assume that the lattices are of the form
$\Lambda_{\vartheta} = u(\vartheta)\Lambda_0$; 
we expect a similar result to be true if $\Lambda_0$ is replaced by
any other lattice in $\Xn$. However our proof does not yield this more
general statement. 

Armed with Theorem \ref{thm;main} (and the dynamical input \cite{s}
used in its proof) we can reverse the logic and derive
a result on lattice point counting.
Following \cite{dima weights}, we define `weighted quasi-norms' as follows: 
\[
\|\mathbf x\|_{\mathbf a}=\max_{1\le i\le n}\left
  \{|x_i|^{\frac{1}{a_i}} \right\}
\quad \left(\text{resp.}~\|\mathbf y\|_{\mathbf b}=\max_{1\le i\le m}
\left\{|y_i|^{\frac{1}{b_i}} \right\}\right).
\]
  Let $F_{\mathbf a t}$ be the $\mathbf a$-weighted  flow on $\mathbb R^n$ defined by 
 \[
F_{\mathbf a t}( \mathbf x)=(e^{a_1t}x_1, \ldots, e^{a_nt}x_n)
 \]
(note that in the case of equal weights, these are just homotheties of
$\R^n$). 
 For every $\mathbf x\in \mathbb R^n\nz$  let $\widehat{ \mathbf x}$ 
 be the unique intersection point of $\{ F_{\mathbf a t}(\mathbf x): t\in \mathbb R\}$
  with  the unit sphere $\mathbb S^{n-1}$  under the usual  Euclidean norm. 
Similarly, for  $\by \in \mathbb R^m\nz$  we define 
%$\bb$-weighted flow $F_{\mathbf b t}$ and  
$\widehat \by \in \mathbb S^{m-1}$. 
\begin{thm}\label{thm;diophantine}
For  $c, T\in (0, \infty)$ and  measurable  $A \subset \mathbb
S^{n-1}, \ B \subset \mathbb S^{m-1}$, 
we let $E_{ T, c}( A, B)$ denote 
$$
%\begin{align}\label{eq;count region}
%E_{ T, c}( A, B)=
\{(\bx, \by) \in \R^n \times \R^m: \|\bx\|_\ba\cdot \|\by\|_\bb<c, \,
1\le \|\by\|_\bb<e^T,\,  \widehat \bx\in  A, \, \widehat \by\in B\}.
%\end{align}
$$
Suppose that with respect to the volume measure on $\mathbb S^{n-1}$ and 
$\mathbb S^{m-1}$, the measures of $A$ and $B$ are positive and  the boundaries of   
$A$ and $B$ have measure zero. 
Then for almost every $\vartheta\in M$
\begin{align}\label{eq;diophantine asym}
\sharp(E_{ T,c}(A, B) \cap \Lambda_{\vartheta})\thicksim |E_{ T,c}( A, B)|\quad \mbox{as } T\to \infty
\end{align}
(where on the right-hand side, $| \cdot |$ denotes the Lebesgue measure on $\R^n \times
\R^m$). 
\end{thm}

To put this result in context, note that a standard application of Minkowski's convex body
theorem shows that for any $\Lambda \in \Xn$, there are infintely many
solutions $(\mathbf{x}, \mathbf{y})\in \Lambda \nz$ to the inequality
\eq{eq: inequality}{
\|\mathbf{x}\|_{\mathbf{a}} \cdot \|\mathbf{y}\|_{\mathbf{b}} \leq 1. 
}
Thus in the special case $A = \mathbb S^{n-1}, \ B = \mathbb
S^{m-1}, \ c=1$, \equ{eq;diophantine asym} provides an asymptotic count 
for the number of solutions of \equ{eq: inequality} with a bound on 
$\|\mathbf{y}\|_{\mathbf{b}}$. The formula \equ{eq;diophantine
  asym} extends the asymptotic
count to general Riemann integrable $A, B$ and general $c$. The validity of 
\equ{eq;diophantine asym} for $\mu$-almost every $\Lambda$ was
established in \cite{S60a}, see also \cite{agt1}. Our result treats $\Lambda_{\vartheta}$
for almost every $\vartheta$, which is a set of
interest in diophantine approximation. In the case of unequal weights
it cannot be handled by the techniques of \cite{s60, S60a}.
}
\ignore{generalized this to the case of non-equal
weights. Namely he showed  \cite[Corollary 1.3]{s} that \equ{eq: standard argument} holds for
all choices of the weight vectors. 

In this paper we %consider the case of diagonal
%subgroups $D$, 
discuss some strengthenings of 
%ways to strengthen
this fact and derive some number-theoretic consequences. 
To this end, we
%Suppose that  $\varphi: \Xn\to \mathbb R$ is  a measurable function with $\int_{\Xn}| \varphi|\, d\mu<\infty$.
%We say $x\in \Xn$ is $(D^+, \varphi)$ generic if 
%
%
%Given a collection $\mathcal S$ of measurable functions on $\Xn$,  we
%say that $x$ is $(D^+, \mathcal S)$ generic
%if (\ref{eq;generic}) holds for every $\varphi\in \mathcal S$. 
Let 
%and for $g \in G, \, \vartheta \in M$, let 
%$$
%\Lambda_g := g\Z^d, \ \ \Lambda_{\vartheta} := \Lambda_{u(\vartheta)}.
%$$
%Let $C_c(\Xn)$  (resp. $C_c^\infty(\Xn)$)
% be the set of compactly supported continuous (resp. 
% smooth) real valued functions on $\Xn$. 
In the case of equal weights,
 the group $\{u(\vartheta): \vartheta \in M\}$ is the so-called
 `unstable horospherical subgroup' for $\{g_t\}$ and a standard
 argument implies that
\eq{eq: standard argument}{\forall \Lambda \in \Xn, \text{ for
    almost every } \vartheta \in M,\, u(\vartheta) \Lambda \text{ is } 
 (D^+, C_c(\Xn))\text{-generic}
}  
(here and hereafter `almost every $\vartheta$' means almost every with
respect to Lebesgue measure on $M \cong \R^{nm}$). }


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