An application of lattice points counting to shrinking target problems

TL;DR

This paper uses advanced lattice points counting techniques to address a shrinking target problem within the context of geodesic flows on finite-volume hyperbolic manifolds, linking geometric analysis with dynamical systems.

Contribution

It introduces a novel application of lattice points counting results to solve a specific problem in hyperbolic dynamical systems.

Findings

Successfully applied lattice points counting to shrinking target problems

Extended methods to geodesic flows on hyperbolic manifolds

Provided new insights into the distribution of geodesic trajectories

Abstract

We apply lattice points counting results of Gorodnik and Nevo to solve a shrinking target problem in the setting of geodesic flows on hyperbolic manifolds of finite volume.

Equations

$$\underset{n}{\mathrm{limsup}}{T}^{-n}{B}_n={\cap }_{m=1}^{\infty }{\cup }_{n=m}^{\infty }{T}^{-n}{B}_n$$

$${\int }_0^{\infty }{r}_t^{n-1}dt$$

$$\{t\in \mathbb{N}:\pi ({\varphi }^{ht}v)\in B_t\}$$

$$\sum _{t\in \mathbb{N}}{r}_t^n$$

$$\frac{-\ln r_t}{r_t}\ll t\text{ for large enough }t,$$

$$\mu (B_t)\gg \frac{\ln t}{t}.$$

$$\frac{-\ln r_t}{r_t}=\frac{-\ln \frac{C}{t^{\alpha }}}{\frac{C}{t^{\alpha }}}=\frac{1}{C}(-\ln C+\alpha \ln t)t^{\alpha }\ll t$$

$$r_t=\frac{C}{t^{1/n}(\ln t{)}^{\beta }}$$

$$\frac{-\ln r_t}{r_t}=\frac{-\ln \frac{C}{t^{1/n}(\ln t{)}^{\beta }}}{\frac{C}{t^{1/n}(\ln t{)}^{\beta }}}=\frac{1}{C}{t}^{1/n}(\ln t{)}^{\beta }\left(\frac{1}{n}\ln t+\beta \ln (\ln t)-\ln C\right)\ll t$$

$$\sum _{t=[s+C_1\ln r_s-C_2]}^s{r}_t^{n-1}\ll \sum _{t=1}^s{r}_t^n\text{when s is large enough},$$

$$\pi ({\phi }^{ht}v)\in B_t\text{ infinitely often for μ-a.e. }v\in T^{1}V.$$

$$S_T[\mathcal{F}](x)\stackrel{\mathrm{def}}{=}\sum _{t=1}^T{f}_t({\varphi }^{ht}x),I_T[\mathcal{F}]\stackrel{\mathrm{def}}{=}\sum _{t=1}^T{\int }_X{f}_{t}d\mu .$$

$$\underset{T\to +\infty }{\mathrm{limsup}}\frac{S_T[\mathcal{F}](x)}{I_T[\mathcal{F}]}\ge 1.$$

$${f_{t}(v)=\left\{\begin{array}[]{ll}1&\hbox{if $d(\pi(v),p_{0})\leq r_{t}$}\\ 0&\hbox{otherwise.}\end{array}\right.}$$

$$\hat{\Gamma}_{i}(r){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\big{\{}\gamma\in\Gamma:d(\tilde{p_{0}},\gamma\tilde{p_{0}})\in(hi-r,hi+r]\big{\}}.$$

$$\#{\hat{\Gamma }}_i(r)=\#(\Gamma \cap D_{hi+r})-\#(\Gamma \cap D_{hi-r}),$$

$$D_t=\{g\in G:d(g\widetilde{p_0},\widetilde{p_0})\le t\}.$$

$$|\#(\Gamma \cap D_t)-m_G(D_t)|\le c_1{m}_G(D_t{)}^q,$$

$$m_G(D_{t+\epsilon })-m_G(D_{t-\epsilon })\le c_2\epsilon m_G(D_{t-\epsilon }).$$

$$m_G(D_t)\le c_3{e}^{(n-1)t}.$$

$$hi\ge \max (-c_4\ln r,r+t_0),$$

$$\#{\hat{\Gamma }}_i(r)\le c_{5}r{e}^{(n-1)hi}.$$

$$\#(\Gamma \cap D_{hi+r})\le m_G(D_{hi+r})+c_1{m}_G(D_{hi+r}{)}^q$$

$$\#(\Gamma \cap D_{hi-r})>m_G(D_{hi-r})-c_1{m}_G(D_{hi-r}{)}^q.$$

$$\#\{\gamma \in \Gamma :d(\widetilde{p_0},\gamma \widetilde{p_0})\in [hi-r,hi+r]\}$$

$$e^{(n-1)qr}+e^{-(n-1)qr}<2e^{n-1},$$

$$hi\ge \max \left(-\frac{1}{(1-q)(n-1)}\ln r,r+t_0\right),$$

$$c_6{r}_1^{n-1}{r}_2^{n-1}{e}^{-(n-1)d(o_1,o_2)}.$$

$$\{\begin{array}{ll}2c_6{r}_1^{n-1}{r}_2^{n-1}{e}^{-(n-1)d}\min (r_1,r_2)& \text{if }\mathrm{dist}(d,h\mathbb{Z})\le 2\max (r_1,r_2)\\ 0& \text{otherwise,}\end{array}$$

$$|d-kh|>2\max (r_1,r_2);$$

Full text

1. Introduction

Let $(X,\mu)$ be a probability space and $T:X\mapsto X$ a measure-preserving transformation. For a sequence of measurable sets $B_{n}\subset X$, consider the set

[TABLE]

of points $x\in X$ such that $T^{n}x\in B_{n}$ for infinitely many $n\in\mathbb{N}$. The Borel-Cantelli Lemma implies that if $\sum_{n=1}^{\infty}\mu(B_{n})$ is finite, then $\mu(\limsup_{n}T^{-n}B_{n})=0$. The (converse) divergence case requires additional assumptions on the sets $B_{n}$. The classical Borel-Cantelli Lemma would imply that the measure of $\limsup_{n}T^{-n}B_{n}$ is full if the sets $T^{-n}B_{n}$ are pairwise independent, an assumption which is hard to establish for deterministic dynamical systems.

In many cases however a milder version of independence can be verified, still implying the full measure of the limsup set. Such results are usually referred to as dynamical Borel-Cantelli Lemmas. In many applications the family of sets $\{B_{n}\}$ is nested, and thus can be viewed as a ‘shrinking target’, hence the terminology ‘Shrinking Target Problems’. For example, if $\{B_{n}\}$ are shrinking balls centered at a point $p\in X$, a dynamical Borel-Cantelli Lemma can be thought of as a quantitative way to express density of trajectories of a generic point of $X$ at this fixed point $p$. Starting from the work of Phillip [15], there have been many results of this flavor. For example Sullivan [17] proved a Borel-Cantelli type theorem for cusp neighborhoods in hyperbolic manifolds of finite volume (here $p=\infty$), and the first named author with Margulis [11] extended the result of Sullivan to non-compact Riemannian symmetric spaces. See also [2, 5, 6, 8, 9] for more references, and [1] for a nice survey of the area.

One particular example of a shrinking target property can be found in a paper by Maucourant [12]. He considered nested balls in hyperbolic manifolds (quotients of the $n$-dimensional hyperbolic space $\mathbb{H}^{n}$) of finite volume, and proved the following theorem:

Theorem 1.1**.**

Let $V$ be a finite volume hyperbolic manifold of real dimension $n$, $T^{1}V$ the unit tangent bundle of $V$, $\pi:T^{1}V\to V$ the canonical projection, $(\phi_{t})_{t\in\mathbb{R}}$ the geodesic flow on $T^{1}V$, $\mu$ the Liouville measure on $T^{1}V$, and $d$ the Riemannian distance on $V$. Let $(B_{t})_{t\geq 0}$ be a decreasing family of closed balls in $V$ (with respect to the metric $d$) of radius $(r_{t}){{}_{t}\geq 0}$. Then for $\mu$-almost every $v$ in $T^{1}V$, the set $\{t\geq 0:\pi(\varphi^{t}v)\in B_{t}\}$ is bounded provided

[TABLE]

converges, and is unbounded if (1.1) diverges.

Note that Maucourant’s theorem holds for the continuous-time geodesic flow on $T^{1}V$. Now suppose that one replaces the continuous family $(B_{t})_{t\geq 0}$ by a sequence $(B_{t})_{t\in\mathbb{N}}$, and instead of the continuous geodesic flow considers the $h$-step discrete geodesic flow $(\varphi^{ht})_{t\in\mathbb{N}}$ for fixed $h\in\mathbb{R}_{+}$. The goal of this work is to provide additional argument needed to prove the Borel-Cantelli property, assuming some restrictions on the sequence $(B_{t})$.

One of the ingredients in Maucourant’s proof is a counting result for the number of lattice points inside balls in $\mathbb{H}^{n}$. To address a discrete time analogue of Theorem 1.1 we use more refined lattice point counting results, namely an error term estimate for the number of lattice points in large balls in $\mathbb{H}^{n}$.

We use the following notation throughout the paper: for two non-negative functions $f$ and $g$, the notation $f(x)\ll g(x)$ means $f(x)\leq Cg(x)$ where $C>0$ is a constant independent of $x$.

Here is a special case of our main result:

Theorem 1.2**.**

Let $V$ be as in Theorem 1.1, and let $(B_{t})_{t\in\mathbb{N}}$ be a decreasing family of closed balls in $V$ centered at $p_{0}\in V$ of radius $r_{t}$. Fix $h>0$ and let $(\phi^{ht})_{t\in\mathbb{N}}$ be the $h$-step discrete geodesic flow. Then for $\mu$-almost every $v\in T^{1}V$, the set

[TABLE]

is finite provided the sum

[TABLE]

converges. Also, if one assumes that (1.3) diverges and, in addition, that

[TABLE]

then for $\mu$-almost every $v$ in $T^{1}V$, the set (1.2) is infinite.

That is, in the terminology of [4], the sequence $(B_{t})$ is a Borel-Cantelli sequence. Note that the difference in exponents in (1.1) and (1.3) is due to the fact that Theorem 1.1, unlike Theorem 1.2, deals with a continuous time setting.

It is well known that the geodesic flow on $T^{1}V$ as above has exponential decay of correlations, see e.g. [14, 16]. For systems with exponential mixing similar dynamical Borel-Cantelli Lemmas have been established before. For example, it follows from [9, Theorem 4.1] that the set (1.2) will be infinite provided

[TABLE]

or, equivalently, $r_{t}\gg\left(\frac{\ln t}{t}\right)^{1/n}$. This shows that the restriction (1.4) is weaker than the one coming from [9, Theorem 4.1]. For example, take $r_{t}=\frac{C}{t^{\alpha}}$, where $\alpha\leq\frac{1}{n}$. Then (1.3) diverges, and one can write

[TABLE]

when $t$ is large enough, therefore (1.4) is satisfied. Note that in the ‘critical exponent’ case $\alpha=1/n$ condition (1.5) fails to hold, thus the methods of [9] are not powerful enough to treat this case. The same also works for

[TABLE]

where $0<\beta\leq 1$: one has

[TABLE]

for large enough $t$.

We derive Theorem 1.2 from a more general statement, Theorem 1.3, which involves a technical condition (1.6) weaker than (1.4):

Theorem 1.3**.**

Let $V$ be as above, and let $(B_{t})_{t\in\mathbb{N}}$ and $h$ be as in Theorem 1.2. Then for $\mu$-almost every $v\in T^{1}V$, the set (1.2) is finite provided the sum (1.3) converges. Also there exist $C_{1},C_{2}>0$ such that if (1.3) diverges and, in addition, that

[TABLE]

then for $\mu$-almost every $v\in T^{1}V$, the set (1.2) is infinite.

In the next section we will reduce Theorem 1.3 to a certain $L^{2}$ bound, Theorem 2.2, which will be verified in §3, and in §4 we will deduce Theorem 1.2 from Theorem 1.3.

Acknowledgments

The authors want to thank Dubi Kelmer, Keith Merrill, Amos Nevo, Hee Oh and the anonymous referee for useful comments.

2. Reduction to Theorem 2.2

First note that for the divergence case of Theorem 1.3 without loss of generality one can assume that $r_{t}\to 0$ when $t\to\infty$: indeed, if $(r_{t})$ is bounded from below by a positive constant, then the ergodicity of the geodesic flow implies that

[TABLE]

Furthermore, for a fixed $R>0$ we can assume that $r_{t}\leq R$ for all $t\in\mathbb{N}$. Indeed, if the theorem is proved under that assumption, then applying it to the family $\{B_{t}:t\geq t_{0}\}$ where $t_{0}$ is such that $r_{t}\leq R$ when $t\geq t_{0}$, we still recover condition (2.1). This $R$ will be fixed later, see (3.4).

Our proof follows Maucourant’s approach in [12]. Let us first introduce some terminology. Let $\mathcal{F}=(f_{t})_{t\in\mathbb{N}}$ be a family of measurable functions on a probability space $(X,\mu)$. We call $\mathcal{F}$ decreasing if $f_{s}(x)\leq f_{t}(x)$ for any $x\in X$ whenever $s\geq t$. Also let us write

[TABLE]

We are going to use the following proposition from Maucourant’s paper:

Proposition 2.1**.**

([12, Proposition 1])* Let $\mathcal{F}=(f_{t})_{t\in\mathbb{N}}$ be a decreasing family of non-negative measurable functions on $X$ such that $f_{t}\in L^{2}(X,\mu)$ for all $t$. Assume that $\lim\limits_{T\to\infty}I_{T}[\mathcal{F}]=\infty$, and that $S_{T}[\mathcal{F}]/I_{T}[\mathcal{F}]$ is bounded in $L^{2}$-norm as $T\to\infty$. Then, as $T\to\infty$, $S_{T}[\mathcal{F}]/I_{T}[\mathcal{F}]$ converges to $1$ weakly in $L^{2}(X,\mu)$, and for $\mu$-almost every $x$ in $X$ one has*

[TABLE]

We note that the above proposition was stated in [12] for the case of a continuous family of functions, but it is immediate to deduce a discrete version. To prove Theorem 1.3, we will apply Proposition 2.1 to the family of characteristic functions of $B_{t}$, i.e. take

[TABLE]

It is decreasing because the family of balls $B_{t}$ is nested, and clearly $I_{T}[\mathcal{F}]$ is equivalent, up to a multiplicative constant, to $\sum_{t=1}^{T}r_{t}^{n}$. Also it is clear that the conclusion (2.2) of Proposition 2.1 implies that the set (1.2) is infinite. Since the convergence case of Theorems 1.2 and 1.3 immediately follows from the Borel-Cantelli Lemma, we can see that Theorem 1.3 can be reduced to proving a uniform $L^{2}$ bound for $S_{T}[\mathcal{F}]/I_{T}[\mathcal{F}]$, which is the subject of the following theorem:

Theorem 2.2**.**

Let $\mathcal{F}=(f_{t})_{t\in\mathbb{N}}$ be as in (2.3). Then there exist $C_{1},C_{2}>0$ such that if $I_{T}[\mathcal{F}]$ diverges when $T$ goes to $\infty$ and condition (1.6) holds, then the $L^{2}$-norm of $S_{T}[\mathcal{F}]/I_{T}[\mathcal{F}]$ is bounded for all $T\geq 1$.

3. Proof of Theorem 2.2

To prove Theorem 2.2, following the same methodology as in [12], we will apply a result on counting lattice points stated below (Theorem 3.3) together with a measure estimate for the space of discrete geodesics (Theorem 3.7).

3.1. Counting lattice points

Write $T^{1}V=\Gamma\backslash G$, where $\Gamma$ is a lattice in $G=\operatorname{SO}(n,1)$, the isometry group of $\widetilde{V}=\mathbb{H}^{n}$. Choose a lift $\tilde{p}_{0}\in\mathbb{H}^{n}$ of $p_{0}$ and for $r>0$ and $i\in\mathbb{N}$, let us denote

[TABLE]

Then

[TABLE]

where

[TABLE]

An estimate for $\#\hat{\Gamma}_{i}(r)$ would follow from a reasonable estimate for the error term in the asymptotics of the size of $\Gamma\cap D_{t}$ for large $t$. Such estimates are due to Huber [10] for $n=2$ and to Selberg for the general case, see [13], and also [3, 7] for more recent results of this flavor. Denote by $m_{G}$ the Haar measure on $G$ which locally projects onto $\mu$. The following is a consequence of [13, Theorem 1]:

Theorem 3.1**.**

There exist constants $0<q<1$ and $t_{1},c_{{1}}>0$ such that

[TABLE]

for all $t>t_{1}$.

An important property of the family $\{D_{t}\}$ is so-called Hölder well-roundedness, see [7]. In particular the following is true:

Proposition 3.2**.**

There exist $t_{2},c_{{2}},c_{{3}}>0$ such that:

• (i)

For any $\varepsilon<1$ and $t>t_{2}$, we have that

[TABLE]

• (ii)

For any $t>0$,

[TABLE]

From the two statements above one can easily derive the following estimate:

Theorem 3.3**.**

There exist constants $c_{{4}},c_{5}$ with the following property: if $0<r<1$ and $i\in\mathbb{N}$ are such that

[TABLE]

*where $t_{0}=\max(t_{1},t_{2})$, then *

[TABLE]

Proof of Theorem 3.3.

Applying Theorem 3.1 for all $i$ with $hi-r>t_{0}$, we get that

[TABLE]

and

[TABLE]

Therefore, by (3.1) and (3.2), we have:

[TABLE]

Since $q<1$ and $r<1$, we have

[TABLE]

and clearly $\frac{1}{r}{e^{-(n-1){(1-q)}hi}}\leq 1$ whenever $hi\geq-\frac{\ln r}{(1-q)(n-1)}\,$. Summarizing the above, if

[TABLE]

then $\#\hat{\Gamma}_{i}(r)\ll re^{(n-1)hi}$. ∎

3.2. The space of discrete geodesics on $\mathbb{H}^{n}$

In this section we will state measure estimates for spaces of geodesics on $\mathbb{H}^{n}$.

Definition 3.4**.**

We will write $\mathscr{G}$ as the space of oriented, unpointed continuous geodesics on $\mathbb{H}^{n}$. Using the fact that $T^{1}\mathbb{H}^{n}$ can be written as $\mathscr{G}\times\mathbb{R}$, we can define a measure $\nu$ on $\mathscr{G}$ by $\tilde{\mu}=\nu\times dt$, where $\tilde{\mu}$ is the Liouville measure on $T^{1}\mathbb{H}^{n}$.

Then we will describe a similar definition for discrete geodesic flows. Namely:

Definition 3.5**.**

For fixed $h>0$, $\mathscr{G}_{h}$ is the space of all $h$-step discrete geodesic trajectories: $\{\varphi^{ht}:t\in\mathbb{Z}\}$. That is $\mathscr{G}_{h}=\mathscr{G}\times S_{h}$ where $S_{h}$ is $[0,h]$ with [math] and $h$ identified. In addition, since we can write $T^{1}\mathbb{H}^{n}=\mathscr{G}_{h}\times\mathbb{Z}h$, then we can define the measure $m$ on $\mathscr{G}_{h}$ by $m=\nu\otimes\lambda$, where $\nu$ is the measure on $\mathscr{G}$ defined above and $\lambda$ the Lebesgue measure on $S_{h}$. Furthermore, the measure $\tilde{\mu}$ on the unit tangent bundle $T^{1}\mathbb{H}^{n}$ becomes the product of the measure $m$ on $\mathscr{G}_{h}$ with the counting measure on $\mathbb{Z}h$.

In [12], Maucourant considered the space of continuous geodesics, and estimated the probability that a random geodesic visits two fixed balls in $V$ as follows:

Theorem 3.6**.**

[12, Lemma 4]* There exists a constant $c_{{6}}>0$ such that, for any two balls in $\mathbb{H}^{n}$ of respective centers and radii $(o_{1},r_{1}),(o_{2},r_{2})$ that satisfy $r_{1}$, $r_{2}<1$, and $d(o_{1},o_{2})>2$, the $\nu$-measure of continuous geodesics meeting those two balls is less than*

[TABLE]

Here is a similar estimate for discrete geodesics on $T^{1}\mathbb{H}^{n}$:

Theorem 3.7**.**

Consider two balls in $\mathbb{H}^{n}$ with respective centers and radii $(o_{1},r_{1})$, $(o_{2},r_{2})$ that satisfy $r_{1}<1$, $r_{2}<1$, and $d=d(o_{1},o_{2})>2$. Also assume that $h>2\min(r_{1},r_{2})$. Then the $m$-measure of the $h$-step geodesics which intersect those two balls is less than

[TABLE]

where $c_{{6}}$ is as in Theorem 3.6.

Proof.

An $h$-step geodesic will fail to intersect both balls if for any $k$ we have

[TABLE]

in this case the measure we are to estimate is zero. So only if there is an integer $k$ such that (3.3) fails, can the $h$-step geodesic meet those balls. Using Theorem 3.6 and the fact that the space of discrete geodesics is $\mathscr{G}\times S_{h}$ with measure $m=\nu\otimes dh$, one can notice that the measure of such geodesics is bounded by $2c_{{6}}\min(r_{1},r_{2})r_{1}^{n-1}r_{2}^{n-1}e^{-(n-1)d}$. ∎

3.3. A bound for the $L^{2}$-norm of $S_{T}[\mathcal{F}]$

Recall that for $t\in\mathbb{N}$ we defined $f_{t}$ to be the characteristic function of $B_{t}$, which is a ball centered at $p_{0}\in V=\Gamma\backslash\mathbb{H}^{n}$ of radius $r_{t}$, see (2.3), and considered the family of functions $\mathcal{F}=(f_{t})_{t\in\mathbb{N}}$ on $T^{1}V$. Also we have chosen a lift $\tilde{p}_{0}\in\mathbb{H}^{n}$ of $p_{0}$. Now define $\tilde{B}_{t}$ to be a ball in $\mathbb{H}^{n}$ centered at $\tilde{p}_{0}$ of radius $r_{t}$, and let $g_{t}$ be the characteristic function of $\tilde{B}_{t}$, Thus, the lift $\tilde{f}_{t}$ of $f_{t}$ to $T^{1}\tilde{V}$ satisfies

[TABLE]

Fix a fundamental domain $D$ of $\mathbb{H}^{n}$ for $\Gamma$ containing $\tilde{p_{0}}$. and define

[TABLE]

Also define

[TABLE]

and, for $i\in\mathbb{Z}_{+}$,

[TABLE]

Theorem 3.8**.**

Let $D\subset\mathbb{H}^{n}$ be a fundamental domain for $\Gamma$ such that $D$ contains the ball of center $\tilde{p_{0}}$ and of radius $3R$. Then for all $T\in\mathbb{N}$,

[TABLE]

Proof.

For fixed $T\in\mathbb{N}$ and $v\in T^{1}V$, we know that

[TABLE]

Now we can integrate $S_{T}[\mathcal{F}](v)^{2}$ over $T^{1}V$ and make a change of variable $w=\phi^{hs}v$. Since $\phi^{hs}$ preserves the measure, we have the following:

[TABLE]

By the fact that $\tilde{f}_{t}$ is the lift of $f_{t}$, we obtain that

[TABLE]

Since $\tilde{f}_{t}=\sum_{\gamma\in\Gamma}g_{t}\circ\gamma$, we can write

[TABLE]

Recall that $D$ is the fundamental domain of $\mathbb{H}^{n}$ for $\Gamma$. This insures that for all $w$ in $T^{1}D$, in the sum $\sum\limits_{\gamma\in\Gamma}g_{s}(\gamma w)$, all terms but the one corresponding to $\gamma=id$ are zero. So we have

[TABLE]

Making another change of variables $v=-w$, where $-w$ means the point in $T^{1}D$ with the same projection as $w$ and the tangent vector pointing in the opposite direction, we deduce that

[TABLE]

For fixed $v\in T^{1}D$, we know that $g_{s}(v)g_{t}(\gamma\phi^{h(s-t)}v)$ is zero when $\pi(v)\notin\tilde{B}_{s}$ or $\pi(\phi^{h(s-t)}v)\notin\gamma^{-1}\tilde{B}_{t}$, which implies that $g_{s}(v)g_{t}(\gamma\phi^{h(s-t)}v)$ vanishes when

[TABLE]

Since we know that $2r_{t}+2r_{s}<4R$, we can conclude that $g_{s}(v)g_{t}(\gamma\phi^{h(s-t)}v)$ vanishes when $t$ is outside of the interval

[TABLE]

Therefore, for any $v\in T^{1}V$ and any $s\in\mathbb{N}$,

[TABLE]

Furthermore, the integral is zero if $\left|d(\tilde{p_{0}},\gamma^{-1}\tilde{p_{0}})-hi\right|>2R>r_{t}+r_{s}$ for all $i$. Hence this integral vanishes when $|hi-h(s-t)|>6R$, i.e. when

[TABLE]

In particular, we see that the quantity $g_{s}(v)g_{t}(\gamma\phi^{h(s-t)}v)$ is zero if

[TABLE]

By the above fact and the fact that the union of all $\Gamma_{i}$ is $\Gamma$, we have

[TABLE]

∎

Now let us define

[TABLE]

and split the estimate of Theorem 3.8 into two parts:

[TABLE]

3.4. A bound on the first part of (3.6)

It is not hard to estimate the first part.

Theorem 3.9**.**

There is constant $c_{{7}}$, only depending on $R$ and $h$, such that for all $T\in\mathbb{N}$

[TABLE]

Proof.

Observing that $\cup^{c_{R}}_{i=3}\Gamma_{i}$ is a finite set, we write $N$ as its cardinal. Moreover, using (3.5), we get that

[TABLE]

In addition, we notice the following facts:

• •

if $g_{s}(v)=0$, then $g_{s}(v)g_{t}(\gamma\phi^{s-t}v)$ in the left side vanishes;

• •

if $g_{s}(v)=1$, then $g_{s}(v)g_{t}(\gamma\phi^{s-t}v)$ is at most $1$.

Therefore, (3.7) is equivalent to the following:

[TABLE]

This allows us to write

[TABLE]

Since $\int_{T^{1}D}g_{s}(v)d\tilde{\mu}(v)$ is equivalent to $r_{s}^{n}$, up to a multiplicative constant, there exists some positive constant $c_{{7}}$, depending only on $R$ and $h$, such that

[TABLE]

∎

3.5. A bound on the second part of (3.6)

Theorem 3.10**.**

There exist constants $c_{{8}}$ and $c_{{9}}$, only depending on $R$, such that

[TABLE]

where $c_{{4}}$ is as in Theorem 3.3.

Proof.

Let us fix $s$ and produce an upper bound on $\int\limits_{T^{1}D}\sum_{t=1}^{s}g_{s}(v)g_{t}(\gamma\phi^{h(s-t)}v)\,d\tilde{\mu}(v)$. This requires the following observations:

1. (1)

(3.7) tells us that $\sum_{t=1}^{s}g_{s}(v)g_{t}(\gamma\phi^{h(s-t)}v)\leq\frac{8R}{h}$ for any $s$ and $v$. 2. (2)

We know that $|d(\tilde{p_{0}},\gamma^{-1}\tilde{p_{0}})-hi|<2R$, i.e.

[TABLE]

Therefore, $i\geq c_{R}\geq\frac{6R+2}{h}$ implies that

[TABLE]

Hence, we know that the distance between the centers of $B(\tilde{p_{0}},r_{s})$ and $B(\gamma^{-1}\tilde{p_{0}},r_{t})$ is greater than 2. Thus by Theorem 3.7, the measure $m$ of the set of discrete geodesics intersecting both $B(\tilde{p_{0}},r_{s})$ and $B(\gamma^{-1}\tilde{p_{0}},r_{t})$ is bounded by $2c_{{6}}r_{s}^{n-1}r^{n-1}_{t}e^{-(n-1)(hi-1)}r_{s}$. 3. (3)

Moreover, $D$ contains the ball of center $\tilde{p_{0}}$ with radius $3R$. So we know that for fixed $v$, $\#\{z\in\mathbb{Z}h:g_{t}(\phi^{hz}v)>0\}\leq\frac{3R}{h}$. 4. (4)

In addition, notice that $g_{s}(v)g_{t}(\gamma\phi^{h(s-t)}v)$ is not zero only if

[TABLE]

This implies that

[TABLE]

Now since $(r_{t})$ is decreasing, for all $i\geq c_{R}=\frac{6R+2}{h}$, we have that

[TABLE]

Therefore, for all $i\geq c_{R}$, we obtain that

[TABLE]

where $N_{i}$ is the number of elements of $\Gamma_{i}$ such that the integrated function is not zero. Now we can consider the sum over all $s$ and $i\geq c_{R}$:

[TABLE]

Our goal now is to estimate $N_{i}$. Recall Theorem 3.3, which allows us to estimate $\#\hat{\Gamma}_{i}(r)$ when $hi\geq\max(-c_{{4}}\ln r,r+t_{0})$ for some constants $c_{{4}},t_{0}>0$. We will take $r=r_{s-i-\frac{6R}{h}-1}$. Indeed, since $(r_{t})$ is decreasing and we have assumed that $r_{t}\leq R<1$, it follows that

[TABLE]

and $r_{s-i-\frac{6R}{h}-1}<R$.

Now let us define

[TABLE]

Then $i\geq V_{s}$ implies that

[TABLE]

Meanwhile, we also know that $g_{s}(v)g_{t}(\gamma\phi^{h(s-t)}v)$ is not zero only if

[TABLE]

where $i$ is such that $\gamma^{-1}\in\Gamma_{i}$. Therefore

[TABLE]

when $i\geq V_{s}$.

By applying the fact that $(r_{t})$ is decreasing, we have the following:

[TABLE]

When $i<V_{s}$, we will use the counting lattice point estimate (Theorem 3.1) to conclude that $N_{i}\ll e^{(n-1)hi}$. Recalling the definition of $V_{s}$, see (3.8), we know that the assumption $i<-\frac{c_{{4}}}{h}\ln r_{s}$ implies that $i<V_{s}$. Meanwhile, since $(r_{t})$ is decreasing, we have that

[TABLE]

Putting it all together, we conclude that

[TABLE]

∎

3.6. Completion of the proof of Theorem 2.2

Proof.

Recall that so far we have

[TABLE]

Now let us take $C_{1}=c_{{4}}/h$, $C_{2}=2+6R/h$, and let us assume (1.6), i.e. that there exist $C,s_{0}$ such that

[TABLE]

Then we can write

[TABLE]

Since $\int_{T^{1}V}I_{T}[\mathcal{F}]^{2}d{\mu}(v)$ is equivalent, up to a multiplicative constant, to $\sum\limits_{s=1}^{T}r_{s}^{n}\sum\limits_{t=1}^{s}r_{t}^{n}$, and with the assumption that $\int_{T^{1}V}I_{T}[\mathcal{F}]^{2}d{\mu}(v)\to\infty$, $T\to\infty$, one can easily conclude that $\frac{S_{T}[\mathcal{F}]}{I_{T}[\mathcal{F}]}$ is bounded in $L^{2}$ -norm. ∎

4. Proof of Theorem 1.2

Recall that we are given a non-increasing sequence $r_{t}$ which tends to [math] as $t\to\infty$ and such that $\sum\limits_{0}^{\infty}r_{t}^{n}=\infty$ and in addition satisfying (1.4), that is, for some $C_{0},s_{0}>0$ it holds that

[TABLE]

We need to show that this sequence satisfies condition (1.6). This will be an easy consequence of the following lemma:

Lemma 4.1**.**

Under the above assumptions, for any $C_{1},C_{2}>0$ there exist $C_{3},T>0$ such that

[TABLE]

Proof.

By (4.1),

[TABLE]

Take $s_{1}$ such that $-\ln r_{s}\geq\frac{C_{2}}{C_{1}}$ when $s>s_{1}$. This and (4.2) imply that

[TABLE]

and

[TABLE]

Since $r_{s}$ is non-increasing, (4.3) implies that

[TABLE]

when $s>\max(s_{0},s_{1})$. Due to the fact that $0<r_{s}<r_{[s+C_{1}\ln r_{s}-C_{2}]}<1$, we have that

[TABLE]

thus

[TABLE]

Therefore, when $s>\max(s_{0},s_{1})$, we obtain that

[TABLE]

Now take $s_{2}>0$ such that $r_{s}<\frac{1}{4C_{1}^{2}C_{0}^{2}}$ when $s>s_{2}$, and let $T{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\max(s_{0},s_{1},\ s_{2})$. Then (4.3) implies that, when $s>T$,

[TABLE]

Thus, by adding $2C_{1}C_{0}s+(2C_{1}C_{0}+1)(C_{1}\ln r_{s}-C_{2})$ to both sides, we conclude that, when $s>T$

[TABLE]

Now let us define $C_{3}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}2C_{1}C_{0}+1$.

Then we have that, when $s>T$

[TABLE]

which, in view of (4.4), implies

[TABLE]

Since $r_{[s+C_{1}\ln r_{s}-C_{2}]}>0$, the above inequality implies that, when $s\geq T$,

[TABLE]

On the other hand, since $r_{s}$ is non-increasing, one will notice that

[TABLE]

and

[TABLE]

Therefore, by (4.5), we have that, when $s\geq T$,

[TABLE]

and hence

[TABLE]

∎

This shows that (1.4) implies (1.6), and finishes the proof of Theorem 1.2. ∎