# Quantitative Diophantine approximation with congruence conditions

Mahbub Alam  Mahbub Alam
School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India
400005
https://sites.google.com/view/mahbubweb<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Anish Ghosh  Anish Ghosh
School of Mathematics
Tata Institute of Fundamental Research, Mumbai, India 400005
<EMAIL_ADDRESS>and  Shucheng Yu  Shucheng Yu
Department of Mathematics, Uppsala University, Box 480, SE-75106, Uppsala,
SWEDEN<EMAIL_ADDRESS>
###### Abstract.

In this short paper we prove a quantitative version of the Khintchine-Groshev
Theorem with congruence conditions. Our argument relies on a classical
argument of Schmidt on counting generic lattice points, which in turn relies
on a certain variance bound on the space of lattices.

A.G. gratefully acknowledges support from a grant from the Indo-French Centre
for the Promotion of Advanced Research, a Department of Science and
Technology, Government of India Swarnajayanti fellowship and a MATRICS grant
from the Science and Engineering Research Board. M. A. and A. G. acknowledge
support of the Department of Atomic Energy, Government of India, under project
$12-R\&D-TFR-5.01-0500$. This work received support from a grant from the
Infosys foundation. S.Y. acknowledges the support of the Knut and Alice
Wallenberg Foundation and ISF grant number 871/17. S.Y. acknowledges that this
project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation program (grant
agreement No. 754475).

##  1\. Introduction

Let $\psi:[1,\infty)\to(0,\infty)$ be a continuous and non-increasing
function. Let $m,n,d$ be positive integers satisfying $d=m+n$ and let
$\vartheta\in\mathrm{M}_{m\times n}(\mathbb{R})$. Consider the following
system of inequalities

(1.1)
$\left\|\vartheta\bm{q}+\bm{p}\right\|^{m}<\psi(\left\|\bm{q}\right\|^{n}),$

with $(\bm{p},\bm{q})\in\mathbb{Z}^{m}\times\mathbb{Z}^{n}$. Here
$\left\|\cdot\right\|$ denotes the supremum norm on the corresponding
Euclidean spaces. The classical Khintchine-Groshev Theorem gives a criterion
on when (1.1) has infinitely many integer solutions in
$(\bm{p},\bm{q})\in\mathbb{Z}^{m}\times\mathbb{Z}^{n}$ for generic (with
respect to the Lebesgue measure) $\vartheta\in\textrm{M}_{m\times
n}(\mathbb{R})$:

######  Theorem  (Khintchine-Groshev).

For almost every (respectively almost no) $\vartheta\in\mathrm{M}_{m\times
n}(\mathbb{R})$ there are infinitely many solutions
$(\bm{p},\bm{q})\in\mathbb{Z}^{m}\times\mathbb{Z}^{n}$ to (1.1) if and only if
the series $\sum_{t=1}^{\infty}\psi(t)$ diverges (respectively converges).

In [NRS20] Nesharim-Rühr-Shi refined the Khintchine-Groshev Theorem by
requiring certain congruence conditions:

######  Theorem  ([NRS20, Theorem 1.2]).

Let $(\bm{v},N)\in\mathbb{Z}^{d}\times\mathbb{N}$. Then for almost every
$\vartheta\in\mathrm{M}_{m\times n}(\mathbb{R})$ there are infinitely many
$(\bm{p},\bm{q})\in\mathbb{Z}^{m}\times\mathbb{Z}^{n}$ satisfying
$(\bm{p},\bm{q})\equiv\bm{v}\ (\mathrm{mod}\ N)$ and (1.1) if and only if the
series $\sum_{t=1}^{\infty}\psi(t)$ diverges.

Schmidt’s theorem [Sch60a] on Diophantine approximations refined the
Khintchine-Groshev Theorem in another respect, by giving an asymptotic formula
for the number of solutions to (1.1) as $\|\bm{q}\|$ grows:

######  Theorem  ([Sch60a, Theorem 1]).

Let $\psi:[1,\infty)\to(0,\infty)$ be as above and let

$\Psi(T):=\sum_{1\leq t<T}\psi(t)\quad\text{for}\ T\geq 1.$

Then for almost every $\bm{\vartheta}\in\mathbb{R}^{m}$, the number of integer
solutions in $(\bm{p},q)\in\mathbb{Z}^{m}\times\mathbb{Z}$ satisfying (1.1)
(with $n=1$) and $1\leq|q|<T$ is

(1.2) $\displaystyle
2^{d}\Psi(T)+O_{\epsilon}\left(\Psi(T)^{1/2+\epsilon}\right).$

######  Remark 1.3.

Schmidt actually studied the more refined one-sided Diophantine approximation
problem by requiring all entries of the vector $(\bm{\vartheta}q+\bm{p},q)$ to
be positive. With this extra requirement one then needs to drop the factor
$2^{d}$ in the above asymptotic formula.

In this paper we refine both [NRS20, Theorem 1.2] and [Sch60a, Theorem 1] by
proving a Schmidt-type asymptotic formula for the system of Diophantine
inequalities (1.1) with extra congruence conditions.

For the remaining of this paper we fix $d,m,n\in\mathbb{N}$ with $d=m+n$ and
we assume $d\geq 3$. We also fix $\psi:[1,\infty)\to(0,\infty)$ a continuous
and non-increasing function satisfying that
$\sum_{t=1}^{\infty}\psi(t)=\infty$.

We now state our main result.

######  Theorem 1.1.

Let $d,m,n\in\mathbb{N}$ and $\psi$ be fixed as above. Let
$(\bm{v},N)\in\mathbb{Z}^{d}\times\mathbb{N}$. Let $\nu_{1}$ and $\nu_{2}$ be
two norms on $\mathbb{R}^{m}$ and $\mathbb{R}^{n}$ respectively, with
$\nu_{2}$ normalized such that

(1.4)
$\displaystyle\min_{\bm{w}\in\mathbb{Z}^{n}\smallsetminus\\{\bm{0}\\}}\nu_{2}(\bm{w})=1.$

For any $\vartheta\in\mathrm{M}_{m\times n}(\mathbb{R})$ and for any $T>1$,
let $\mathcal{N}(\vartheta,T)$ denote the number of integer solutions in
$(\bm{p},\bm{q})\in\mathbb{Z}^{m}\times\mathbb{Z}^{n}$ satisfying

(1.5)
$\displaystyle\nu_{1}(\vartheta\bm{q}+\bm{p})^{m}<\psi(\nu_{2}\left(\bm{q}\right)^{n}),\
(\bm{p},\bm{q})\equiv\bm{v}\ (\mathrm{mod}\ N),\ 1\leq\nu_{2}(\bm{q})^{n}<T.$

Then for almost every $\vartheta\in\mathrm{M}_{m\times n}(\mathbb{R})$,

(1.6) $\displaystyle\mathcal{N}(\vartheta,T)\ \sim\
N^{-d}c_{\nu_{1}}c_{\nu_{2}}\sum_{1\leq t<T}\psi(t)\quad\textrm{as
$T\to\infty$},$

where for $i=1,2$, $c_{\nu_{i}}$ is the volume of the unit ball with respect
to $\nu_{i}$.

######  Remark 1.7.

Let $\nu$ be a norm on $\mathbb{R}^{\ell}$ for some $\ell\in\mathbb{N}$. If
$\nu$ is the $L^{p}$-norm for some $p\geq 1$, then
$c_{\nu}=\tfrac{2^{\ell}\Gamma(1+1/p)^{\ell}}{\Gamma(1+\ell/p)}$; if $\nu$ is
the supremum norm, then $c_{\nu}=2^{\ell}$. In particular, if $N=n=1$ and both
$\nu_{1}$ and $\nu_{2}$ are the supremum norms, then Theorem 1.1 recovers the
main term in Schmdt’s asymptotic formula (1.2) when $m\geq 2$.

Our proof of Theorem 1.1 consists of two steps: first we prove a counting
result regarding the number of points of a generic lattice in an increasing
family of Borel sets determined by the inequality system (1.5). The second
step is then an argument reducing the counting result on generic lattices in
the space of lattices to generic lattices in a much smaller sub-manifold of
positive co-dimension. We note that this was also the strategy used in [APT16,
AG20a, AG20b] where similar quantitative results were proved in various
settings for the special approximating function $\psi_{0}(t)=\frac{c}{t}$ with
$c>0$. The arguments used in [APT16, AG20a, AG20b] for the counting result in
step one use ergodic theory and are special to $\psi_{0}$. Here, instead of
using this ergodic argument, we use a more soft counting argument of Schmidt
[Sch60b] which only relies on a variance bound and works for arbitrary
increasing family of Borel sets. To incorporate the congruence condition, we
use a recently proved variance bound [GKY20] on a certain congruence cover of
the space of lattices.

######  Remark 1.8.

We note that the assumption that $d\geq 3$ in Theorem 1.1 is due to the lack
of such variance bounds on congruence covers of the space of rank two
lattices. However, when $d=2$ (so that $m=n=1$), $\psi=\psi_{0}$ and $\nu_{1}$
and $\nu_{2}$ are both supremum norms, one can still use the ergodic argument
to prove the desired counting result in step one and hence deduce an analogous
asymptotic formula in this setting. For simplicity of the presentation, we
omit the details here.

### Notation and conventions

Throughout the paper, $f(T)\sim g(T)$ means that $\frac{f(T)}{g(T)}\to 1$ as
$T\to\infty$. For two positive quantities $A$ and $B$, we will use the
notation $A\ll B$ or $A=O(B)$ to mean that there is a constant $c>0$ such that
$A\leq cB$, and we will use subscripts to indicate the dependence of the
constant on parameters. We will write $A\asymp B$ for $A\ll B\ll A$. For any
Borel subset $S$ in a Euclidean space, we use the notation $|S|$ to mean its
volume with respect to the usual Lebesgue measure. All vectors in this paper
are column vectors even though we will write them as row vectors.

##  2\. Preliminaries

###  2.1. Siegel transforms and variance estimates

Let $d\geq 3$ be an integer. Let $G=\operatorname{SL}_{d}(\mathbb{R})$ and
$\Gamma=\operatorname{SL}_{d}(\mathbb{Z})$. It is well known that the
homogeneous space $G/\Gamma$ parameterizes $X$, the space of unimodular
lattices in $\mathbb{R}^{d}$ via $g\Gamma\leftrightarrow g\mathbb{Z}^{d}$.
More generally, let $(\bm{v},N)\in\mathbb{Z}^{d}\times\mathbb{N}$ such that
$\gcd(\bm{v},N)=1$. Let $X_{N,\bm{v}}$ be the space of affine lattices of the
form $g(\mathbb{Z}^{n}+\tfrac{\bm{v}}{N})$ with $g\in G$. Similar as for $X$,
the space $X_{N,\bm{v}}$ can be identified with the homogeneous space
$G/\Gamma_{N,\bm{v}}$ via $g\Gamma_{N,\bm{v}}\leftrightarrow
g\left(\mathbb{Z}^{d}+\tfrac{\bm{v}}{N}\right)$, see e.g. [GKY20, Lemma 3.1].
Here

$\Gamma_{N,\bm{v}}:=\left\\{\gamma\in\Gamma:\gamma\bm{v}\equiv\bm{v}\
(\mathrm{mod}\ N)\right\\}$

is the stabilizer in $\Gamma$ of the affine lattice
$\mathbb{Z}^{d}+\tfrac{\bm{v}}{N}$. Note that $\Gamma_{N,\bm{v}}$ is a
congruence subgroup (since it contains the principal congruence subgroup
$\Gamma(N)$) and $\Gamma_{N,\bm{v}}=\Gamma$ if $N=1$. Generalizing the
classical Siegel transform defined on $X$, for any bounded and compactly
supported function $f:\mathbb{R}^{d}\to\mathbb{C}$ we define its Siegel
transform on $X_{N,\bm{v}}\cong G/\Gamma_{N,\bm{v}}$ by

$\widehat{f}\left(g\right):=\sum_{\bm{w}\in\left(\mathbb{Z}^{d}+\tfrac{\bm{v}}{N}\right)\smallsetminus\\{\bm{0}\\}}f(g\bm{w}).$

It was shown in [MS10, Proposition 7.1] that for any bounded and compactly
supported $f$,

(2.1)
$\int_{X_{N,\bm{v}}}\widehat{f}(g)d\mu(g)=\int_{\mathbb{R}^{d}}f(\bm{x})d\bm{x},$

where $\mu$ is the Haar measure of $G$ normalized so that
$\mu(X_{N,\bm{v}})=1$. Note that when $f=\chi_{A}$ is the indicator function
of some bounded Borel subset $A\subset\mathbb{R}^{d}$ not containing $\bm{0}$,
then

$\widehat{f}(g)=\\#(g\left(\mathbb{Z}^{d}+\tfrac{\bm{v}}{N}\right)\cap A)$

counts the number of points of the affine lattice
$g\left(\mathbb{Z}^{d}+\tfrac{\bm{v}}{N}\right)$ inside $A$. Hence the
integration formula (2.1) implies that on average, the counting function
$\\#(g\left(\mathbb{Z}^{d}+\tfrac{\bm{v}}{N}\right)\cap A)$ is $|A|$, the
volume of $A$. Using (2.1) together with a second moment formula [GKY20,
Equation (3.1)] the following variance bound was proved in [GKY20, Corollary
3.4]: for any bounded Borel set
$A\subset\mathbb{R}^{d}\smallsetminus\\{\bm{0}\\}$

(2.2)
$\int_{X_{N,\bm{v}}}\left|\\#(g\left(\mathbb{Z}^{d}+\tfrac{\bm{v}}{N}\right)\cap
A)-|A|\right|^{2}d\mu(g)\ll_{d,N}|A|.$

###  2.2. Schmidt’s counting results for generic lattices

It is a classical result by Schmidt [Sch60b, Theorem 1] that given any
increasing family of finite-volume Borel sets
$\\{A_{T}\\}_{T>0}\subset\mathbb{R}^{d}\smallsetminus\\{\bm{0}\\}$, (i.e.,
$A_{T_{1}}\subset A_{T_{2}}$ whenever $T_{1}<T_{2}$), for $\mu$-a.e.
unimodular lattice $\Lambda\in X$,

(2.3) $\\#(\Lambda\cap A_{T})=|A_{T}|+O_{\epsilon}(|A_{T}|^{1/2+\epsilon}).$

The main technical tool for Schmidt’s arguments is a variance bound in the
setting of unimodular lattices, more precisely, the estimate (2.2) when $N=1$.
In particular, applying Schmidt’s arguments and the variance bound (2.2) one
can get the following counting result for generic lattices in $X_{N,\bm{v}}$,
analogous to (2.3).

######  Proposition 2.1.

Let $(\bm{v},N)\in\mathbb{Z}^{d}\times\mathbb{N}$ satisfying that
$\gcd(\bm{v},N)=1$. Let
$\\{A_{T}\\}_{T>0}\subset\mathbb{R}^{d}\smallsetminus\\{\bm{0}\\}$ be an
increasing family of bounded finite-volume Borel subsets. Then for any
$\epsilon>0$ and for $\mu$-a.e. $g\in G$, there exists some $T_{g}>0$ such
that for all $T>T_{g}$

$\displaystyle\\#\left(g(\mathbb{Z}^{d}+\tfrac{\bm{v}}{N})\cap
A_{T}\right)=|A_{T}|+O_{\epsilon}\left(|A_{T}|^{1/2+\epsilon}\right).$

Using a simple scaling argument we have the following counting result which we
will use later.

######  Corollary 2.2.

Let $(\bm{v},N)\in\mathbb{Z}^{d}\times\mathbb{N}$ and let $\\{A_{T}\\}_{T>0}$
be as in Proposition 2.1. We further assume that
$\lim\limits_{T\to\infty}|A_{T}|=\infty$. Then for $\mu$-a.e. $g\in G$,

$\\#(g(N\mathbb{Z}^{d}+\bm{v})\cap A_{T})\ \sim\ N^{-d}|A_{T}|\quad\textrm{as
$T\to\infty$}.$

###### Proof.

Let $l=\gcd(\bm{v},N)$ and let $\bm{v}^{\prime}=\bm{v}/l$ and $N^{\prime}=N/l$
so that $\tfrac{\bm{v}}{N}=\tfrac{\bm{v}^{\prime}}{N^{\prime}}$ and
$\gcd(\bm{v}^{\prime},N^{\prime})=1$. Applying Proposition 2.1 to the space
$X_{N^{\prime},\bm{v}^{\prime}}$ and the family
$\left\\{N^{-1}A_{T}\right\\}_{T>0}$, (here we can apply Proposition 2.1 since
$\lim\limits_{T\to\infty}|N^{-1}A_{T}|=\lim\limits_{T\to\infty}N^{-d}|A_{T}|=\infty$)
to get for $\mu$-a.e. $g\in G$,

$\displaystyle\\#\left(g(\mathbb{Z}^{d}+\tfrac{\bm{v}^{\prime}}{N^{\prime}})\cap
N^{-1}A_{T}\right)\ \sim\ |N^{-1}A_{T}|=N^{-d}|A_{T}|\quad\textrm{as
$T\to\infty$}.$

We can thus finish the proof by noting that

$\\#\left(g(\mathbb{Z}^{d}+\tfrac{\bm{v}^{\prime}}{N^{\prime}})\cap
N^{-1}A_{T}\right)=\\#\left(g(\mathbb{Z}^{d}+\tfrac{\bm{v}}{N})\cap
N^{-1}A_{T}\right)=\\#\left(g\left(N\mathbb{Z}^{d}+\bm{v}\right)\cap
A_{T}\right).\qed$

###  2.3. Relating to counting lattice points

Let $U:=u(\mathrm{M}_{m\times n}(\mathbb{R}))<G$, where $u:\mathrm{M}_{m\times
n}(\mathbb{R})\to G$ is defined as

$u(\vartheta):=\begin{pmatrix}1_{m}&\vartheta\\\ 0&1_{n}\end{pmatrix}.$

Here $1_{\ell}$ denotes the $\ell\times\ell$ identity matrix and $0$ is the
zero matrix in $\textrm{M}_{n\times m}(\mathbb{R})$. Note that

$N\mathbb{Z}^{d}+\bm{v}=\left\\{(\bm{p},\bm{q})\in\mathbb{Z}^{m}\times\mathbb{Z}^{n}:(\bm{p},\bm{q})\equiv\bm{v}\
(\mathrm{mod}\ N)\right\\}$

and any lattice point of $u(\vartheta)\left(N\mathbb{Z}^{d}+\bm{v}\right)$ is
of the form

$u(\vartheta)(\bm{p},\bm{q})=(\vartheta\bm{q}+\bm{p},\bm{q})$

for some $(\bm{p},\bm{q})\in N\mathbb{Z}^{d}+\bm{v}$. This, together with the
normalization (1.4) implies that

(2.4)
$\displaystyle\mathcal{N}(\vartheta,T)=\\#\left(u(\vartheta)\left(N\mathbb{Z}^{d}+\bm{v}\right)\cap
E_{T})\right),$

where for any $T>1$

(2.5) $\displaystyle
E_{T}=E_{\psi,\nu_{1},\nu_{2},T}:=\left\\{(\bm{x},\bm{y})\in\mathbb{R}^{m}\times\mathbb{R}^{n}:\nu_{1}(\bm{x})^{m}<\psi\left(\nu_{2}(\bm{y})^{n}\right),\
1\leq\nu_{2}(\bm{y})^{n}<T\right\\}.$

###  2.4. Volume calculation

In this subsection we give a quick computation for the volume of the sets
$E_{T}$ defined in (2.5). We first record a simple volume formula for balls
with respect to a norm $\nu$ on $\mathbb{R}^{\ell}$. For all $r>0$ let
$B_{\nu}(r):=\\{\bm{x}\in\mathbb{R}^{\ell}:\nu(\bm{x})<r\\}$ be the open
radius $r$-ball with respect to $\nu$, centered at the origin. Since $\nu$ is
positive homogeneous, i.e., $\nu(r\bm{x})=r\nu(\bm{x})$ for any $r>0$ and
$\bm{x}\in\mathbb{R}^{\ell}$, we have

(2.6)
$\displaystyle|B_{\nu}(r)|=|rB_{\nu}(1)|=|B_{\nu}(1)|r^{\ell}=c_{\nu}r^{\ell}.$

Now using the identity (2.6) we have

$\displaystyle|E_{T}|$
$\displaystyle=\int_{\\{\bm{y}\in\mathbb{R}^{n}:1\leq\nu_{2}(\bm{y})^{n}<T\\}}\int_{\\{\bm{x}\in\mathbb{R}^{m}:\nu_{1}(\bm{x})^{m}<\psi(\nu_{2}(\bm{y})^{n})\\}}d\bm{x}d\bm{y}$
$\displaystyle=c_{\nu_{1}}\int_{\\{\bm{y}\in\mathbb{R}^{n}:1\leq\nu_{2}(\bm{y})^{n}<T\\}}\psi(\nu_{2}(\bm{y})^{n})d\bm{y}=:c_{\nu_{1}}F(T).$

It is not hard to check that (again using (2.6)) for any $T>1$,
$F^{\prime}(T)=c_{\nu_{2}}\psi(T)$. This, together with the fact that
$F(1)=0$, implies that $F(T)=c_{\nu_{2}}\int_{1}^{T}\psi(r)dr$. Hence

(2.7)
$\displaystyle|E_{T}|=c_{\nu_{1}}c_{\nu_{2}}\int_{1}^{T}\psi(r)dr=c_{\nu_{1}}c_{\nu_{2}}\sum_{1\leq
t<T}\psi(t)+O_{\psi}(1).$

###  2.5. Decomposition of the Haar measure

Let $H<G$ be the parabolic subgroup such that

$\displaystyle H:=\left\\{h=\begin{pmatrix}\alpha&0\\\
\beta&\gamma\end{pmatrix}\in G:\alpha\in\operatorname{GL}_{m}(\mathbb{R}),\
\gamma\in\operatorname{GL}_{n}(\mathbb{R}),\ \beta\in\textrm{M}_{n\times
m}(\mathbb{R})\right\\}.$

Here $0$ denotes the zero matrix in $\textrm{M}_{m\times n}(\mathbb{R})$. We
note that there is a Zariski dense subset of $G$ such that any $g$ in this
subset can be written uniquely as the product $g=hu(\vartheta)$ with $h\in H$
and $u(\vartheta)\in U$. We note that under this decomposition, the Haar
measure $\mu$ decomposes as (up to scalars) $d\mu(g)=dhd\vartheta$, where $dh$
is a left $H$-invariant Haar measure of $H$ and $d\vartheta$ is the usual
Lebesgue measure on $U(\cong\textrm{M}_{m\times
n}(\mathbb{R})\cong\mathbb{R}^{mn})$. In view of this measure decomposition
and Fubini’s theorem we can restate Corollary 2.2:

######  Proposition 2.3.

Keep the assumptions as in Corollary 2.2. Then for a.e. $h\in H$ (with respect
to $dh$) and for a.e. $\vartheta\in\textrm{M}_{m\times n}(\mathbb{R})$ (with
respect to $d\vartheta$)

$\\#(hu(\vartheta)(N\mathbb{Z}^{d}+\bm{v})\cap A_{T})\ \sim\
N^{-d}|A_{T}|\quad\textrm{as $T\to\infty$}.$

##  3\. Proof of Theorem 1.1

In this section we give the proof of Theorem 1.1. The following lemma is the
key step for our reduction argument. It says that the sets $E_{T}$ are stable
under small perturbations of elements in $H$ close to the identity matrix.

######  Lemma 3.1.

There exists $0<c_{0}<\frac{1}{2}$ such that for any $\epsilon\in(0,c_{0})$
there exists an open neighborhood $H_{\epsilon}\subset H$ of the identity
element such that for all $h\in H_{\epsilon}$

$E_{T,\epsilon}^{-}\subset hE_{T}\subset E_{T,\epsilon}^{+},\ \forall\ T>10,$

where $E_{T}$ is as given in (2.5) and

$E_{T,\epsilon}^{-}:=\left\\{(\bm{x},\bm{y})\in\mathbb{R}^{m}\times\mathbb{R}^{n}:\nu_{1}(\bm{x})^{m}<(1+\epsilon)^{-1}\psi\left((1+\epsilon)\nu_{2}(\bm{y})^{n}\right),\frac{3}{2}\leq\nu_{2}(\bm{y})^{n}<(1+\epsilon)^{-1}T\right\\},$

and $E_{T,\epsilon}^{+}:=E^{\prime}_{T,\epsilon}\cup C_{0}$ with

$E_{T,\epsilon}^{\prime}:=\left\\{(\bm{x},\bm{y})\in\mathbb{R}^{m}\times\mathbb{R}^{n}:\nu_{1}(\bm{x})^{m}<(1+\epsilon)\psi\left((1+\epsilon)^{-1}\nu_{2}(\bm{y})^{n}\right),\frac{3}{2}\leq\nu_{2}(\bm{y})^{n}<(1+\epsilon)T\right\\},$

and

$C_{0}:=\left\\{(\bm{x},\bm{y})\in\mathbb{R}^{m}\times\mathbb{R}^{n}:\nu_{1}(\bm{x})^{m}<2\psi(1),\
\frac{1}{2}<\nu_{2}(\bm{y})^{n}\leq\frac{3}{2}\right\\}.$

###### Proof.

View $(\mathbb{R}^{m},\nu_{1})$ and $(\mathbb{R}^{n},\nu_{2})$ as two normed
spaces, and for any $\alpha\in\operatorname{GL}_{m}(\mathbb{R})$,
$\beta\in\textrm{M}_{m\times n}(\mathbb{R})$ and
$\gamma\in\operatorname{GL}_{n}(\mathbb{R})$, we denote by
$\|\alpha\|_{\nu_{1}}$, $\|\beta\|_{\nu_{1},\nu_{2}}$ and
$\|\gamma\|_{\nu_{2}}$ their corresponding operator norms. For each
$\epsilon>0$ let

$\displaystyle\tilde{H}_{\epsilon}:=\left\\{h=\begin{pmatrix}\alpha&0\\\
\beta&\gamma\end{pmatrix}\in
G:\max\\{\|\alpha\|^{m}_{\nu_{1}},\|\gamma\|^{n}_{\nu_{2}}\\}<1+\frac{\epsilon}{2},\
\|\beta\|_{\nu_{1},\nu_{2}}<\frac{\epsilon}{4n\psi(1)^{1/m}}\right\\}.$

and define $H_{\epsilon}:=\tilde{H}_{\epsilon}\cap\tilde{H}_{\epsilon}^{-1}$.
Then clearly $H_{\epsilon}\subset H$ is an open neighborhood of the identity
element. We need to prove the above inclusion relations for all $h\in
H_{\epsilon}$.

Fix $h=\left(\begin{smallmatrix}\alpha&0\\\
\beta&\gamma\end{smallmatrix}\right)\in H_{\epsilon}$ and $T>10$. We first
prove the relation $hE_{T}\subset
E_{T,\epsilon}^{+}=E_{T,\epsilon}^{\prime}\cup C_{0}$. Let $(\bm{x},\bm{y})\in
E_{T}$. First note that
$\nu_{1}(\bm{x})^{m}<\psi(\nu_{2}(\bm{y})^{n})\leq\psi(1)$, where for the
second inequality we used the assumption that $\nu_{2}(\bm{y})\geq 1$ and that
$\psi$ is non-increasing. Now using the definition of $\tilde{H}_{\epsilon}$
and the fact that $h\in H_{\epsilon}\subset\tilde{H}_{\epsilon}$ we have

$h(\bm{x},\bm{y})=(\alpha\bm{x},\beta\bm{x}+\gamma\bm{y})$

with

$\displaystyle\nu_{1}(\alpha\bm{x})^{m}\leq\left(1+\tfrac{\epsilon}{2})\nu_{1}(\bm{x}\right)^{m}$

and (using the triangle inequality and the inequalities that
$\nu_{2}(\bm{x})^{m}<\psi(1)$ and $\nu_{2}(\bm{y})>1$)

$\displaystyle\nu_{2}(\beta\bm{x}+\gamma\bm{y})\leq\nu_{2}(\gamma\bm{y})+\nu_{2}(\beta\bm{x})\leq(1+\tfrac{\epsilon}{2})^{1/n}\nu_{2}(\bm{y})+\frac{\epsilon}{4n\psi(1)^{1/m}}\nu_{1}(\bm{x})<(1+\epsilon)^{1/n}\nu_{2}(\bm{y}).$

Here for the last inequality we used the inequality that
$(1+\tfrac{\epsilon}{2})^{1/n}+\frac{\epsilon}{4n}<(1+\epsilon)^{1/n}$ which
can be guaranteed for all $\epsilon\in(0,c_{0})$ by taking $c_{0}>0$
sufficiently small.

Now if $\nu_{2}(\beta\bm{x}+\gamma\bm{y})^{n}>\frac{3}{2}$, then

$\nu_{2}(\beta\bm{x}+\gamma\bm{y})^{n}<(1+\epsilon)\nu_{2}(\bm{y})^{n}<(1+\epsilon)T,$

and

$\displaystyle\nu_{1}(\alpha\bm{x})^{m}\leq(1+\tfrac{\epsilon}{2})\nu_{1}(\bm{x})^{m}<(1+\tfrac{\epsilon}{2})\psi(\nu_{2}(\bm{y})^{n})<(1+\epsilon)\psi\left(\left(1+\epsilon)^{-1}\nu_{2}(\beta\bm{x}+\gamma\bm{y})^{n}\right)\right).$

This implies that $h(\bm{x},\bm{y})\in E^{\prime}_{T,\epsilon}\subset
E_{T,\epsilon}^{+}$. If $\nu_{2}(\beta\bm{x}+\gamma\bm{y})\leq\frac{3}{2}$,
then

$\displaystyle\nu_{1}(\alpha\bm{x})^{m}\leq(1+\tfrac{\epsilon}{2})\nu_{1}(\bm{x})^{m}<2\psi(1),$

and (since $h^{-1}=\left(\begin{smallmatrix}\alpha^{-1}&0\\\
-\gamma^{-1}\beta\alpha^{-1}&\gamma^{-1}\end{smallmatrix}\right)\in\tilde{H}_{\epsilon}$)

$\displaystyle\nu_{2}(\beta\bm{x}+\gamma\bm{y})\geq\nu_{2}(\gamma\bm{y})-\nu_{2}(\beta\bm{x})\geq(1+\tfrac{\epsilon}{2})^{-1/n}\nu_{2}(\bm{y})-\frac{\epsilon}{4n}\geq(1+\tfrac{\epsilon}{2})^{-1/n}-\frac{\epsilon}{4n}>\frac{1}{2^{1/n}},$

where the last inequality again can be guaranteed by taking $c_{0}$
sufficiently small. Thus in this case we have $h(\bm{x},\bm{y})\in
C_{0}\subset E_{T,\epsilon}^{+}$. This finishes the proof of the relation
$hE_{T}\subset E_{T,\epsilon}^{+}$. Similarly, using the fact that
$h^{-1},h\in\tilde{H}_{\epsilon}$, one can show
$h^{-1}E_{T,\epsilon}^{-}\subset E_{T}$, or equivalently,
$E_{T,\epsilon}^{-}\subset hE_{T}$. ∎

######  Remark 3.1.

Using similar volume calculations as in §2.4 one can easily see that for all
$T>10$ and $\epsilon\in(0,c_{0})$

$\displaystyle|E_{T,\epsilon}^{\pm}|$ $\displaystyle=(1+\epsilon)^{\pm
1}c_{\nu_{1}}c_{\nu_{2}}\int_{\frac{3}{2}}^{(1+\epsilon)^{\pm
1}T}\psi((1+\epsilon)^{\mp 1}r)dr+O(1)$ (3.2) $\displaystyle=(1+\epsilon)^{\pm
2}c_{\nu_{1}}c_{\nu_{2}}\int_{1}^{T}\psi(r)dr+O_{\psi,\nu_{1},\nu_{2}}(1).$

###### Proof of Theorem 1.1.

For simplicity of notation, let us denote
$\Lambda_{0}:=N\mathbb{Z}^{d}+\bm{v}$. Fix a sequence of positive numbers
$\\{\epsilon_{\ell}\\}_{\ell\in\mathbb{N}}\subset(0,c_{0})$ with
$\epsilon_{\ell}\to 0$ as $\ell\to\infty$. Here $c_{0}$ is the constant as in
Lemma 3.1. Note that for each $\ell\in\mathbb{N}$, both
$\\{E_{T,\epsilon_{\ell}}^{\pm}\\}_{T>10}$ are increasing families of bounded
Borel subsets not containing $\bm{0}$. Moreover, using the volume formulas
(2.7) and (3.1) we have

(3.3)
$\displaystyle\frac{|E_{T,\epsilon_{\ell}}^{+}|}{|E_{T}|}\leq(1+\epsilon_{\ell})^{2}+O_{\psi,\nu_{1},\nu_{2}}(|E_{T}|^{-1})\quad\textrm{and}\quad\frac{|E_{T,\epsilon_{\ell}}^{-}|}{|E_{T}|}\geq(1+\epsilon_{\ell})^{-2}+O_{\psi,\nu_{1},\nu_{2}}(|E_{T}|^{-1}).$

In particular, the above second estimate, together with the relation
$E_{T}\subset E_{T,\epsilon_{\ell}}^{+}$ and the fact that

$|E_{T}|\asymp\sum_{1\leq t<T}\psi(t)\to\infty\quad\textrm{as $T\to\infty$},$

implies that $\lim\limits_{T\to\infty}|E_{T,\epsilon_{\ell}}^{\pm}|=\infty$.
Thus we can apply Proposition 2.3 for the two families
$\\{E_{T,\epsilon_{\ell}}^{\pm}\\}_{T>10}$. In fact, combining Proposition 2.3
and Lemma 3.1 and using the fact that a finite intersection of full measure
sets is still of full measure (hence intersecting any open set nontrivially)
we can find, for each $\ell\in\mathbb{N}$, $h_{\ell}\in H$ such that

(3.4) $\displaystyle E_{T,\epsilon_{\ell}}^{-}\subset h_{\ell}E_{T}\subset
E_{T,\epsilon_{\ell}}^{+},\ \forall\ T>10,$

and that there exists a full measure subset $U_{\ell}\subset U$ such that for
all $u(\vartheta)\in U_{\ell}$

(3.5) $\displaystyle\\#(h_{\ell}u(\vartheta)\Lambda_{0}\cap
E_{T,\epsilon_{\ell}}^{\pm})\ \sim\
N^{-d}|E_{T,\epsilon_{\ell}}^{\pm}|\quad\textrm{as $T\to\infty$}.$

Now let $U_{\infty}=\cap_{\ell\in\mathbb{N}}U_{\ell}$ which is still of full
measure. In view of the relation (2.4) and the volume calculation (2.7), it
suffices to show that for all $\vartheta\in U_{\infty}$

$\displaystyle\\#\left(u(\vartheta)\Lambda_{0}\cap E_{T})\right)\ \sim\
N^{-d}|E_{T}|\quad\textrm{as $T\to\infty$}.$

Now for any $u(\vartheta)\in U_{\infty}$ using the relation (3.4) we have for
each $\ell\in\mathbb{N}$

$\displaystyle\\#(h_{\ell}u(\vartheta)\Lambda_{0}\cap
E_{T,\epsilon_{\ell}}^{-})\leq\\#(h_{\ell}u(\vartheta)\Lambda_{0}\cap
h_{\ell}E_{T})\leq\\#(h_{\ell}u(\vartheta)\Lambda_{0}\cap
E_{T,\epsilon_{\ell}}^{+}),$

or equivalently,

$\displaystyle\frac{\\#(h_{\ell}u(\vartheta)\Lambda_{0}\cap
E_{T,\epsilon_{\ell}}^{-})}{N^{-d}|E_{T}|}\leq\frac{\\#(u(\vartheta)\Lambda_{0}\cap
E_{T})}{N^{-d}|E_{T}|}\leq\frac{\\#(h_{\ell}u(\vartheta)\Lambda_{0}\cap
E_{T,\epsilon_{\ell}}^{+})}{N^{-d}|E_{T}|}.$

Now applying (3.5) and (3.3) we get

$\displaystyle\mathop{\overline{\lim}}_{T\to\infty}\frac{\\#(u(\vartheta)\Lambda_{0}\cap
E_{T})}{N^{-d}|E_{T}|}\leq\mathop{\underline{\lim}}_{T\to\infty}\frac{|E_{T,\epsilon_{\ell}}^{+}|}{|E_{T}|}\leq(1+\epsilon_{\ell})^{2},$

and

$\displaystyle\mathop{\underline{\lim}}_{T\to\infty}\frac{\\#(u(\vartheta)\Lambda_{0}\cap
E_{T})}{N^{-d}|E_{T}|}\geq\mathop{\overline{\lim}}_{T\to\infty}\frac{|E_{T,\epsilon_{\ell}}^{-}|}{|E_{T}|}\geq(1+\epsilon_{\ell})^{-2}.$

Finally taking $\ell\to\infty$ finishes the proof. ∎

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