An application of lattice points counting to shrinking target problems

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.


1.
Introduction. Let (X, µ) be a probability space and T : X → X a measurepreserving transformation. For a sequence of measurable sets B n ⊂ X, consider the set lim sup of points x ∈ X such that T n x ∈ B n for infinitely many n ∈ N. The Borel-Cantelli Lemma implies that if ∞ n=1 µ(B n ) is finite, then µ(lim sup 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 lim sup 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 ∈ 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 = ∞), 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 [13]. He considered nested balls in hyperbolic manifolds (quotients of the n-dimensional hyperbolic space 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 , π : T 1 V → V the canonical projection, (φ t ) t∈R the geodesic flow on T 1 V , µ the Liouville measure on T 1 V , and d the Riemannian distance on V . Let (B t ) t≥0 be a decreasing family of closed balls in V (with respect to the metric d) of radius (r t ) t ≥ 0. Then for µ-almost every v in T 1 V , the set 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≥0 by a sequence (B t ) t∈N , and instead of the continuous geodesic flow considers the h-step discrete geodesic flow (ϕ ht ) t∈N for fixed h ∈ 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 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 H n . We use the following notation throughout the paper: for two non-negative functions f and g, the notation f (x) Here is a special case of our main result: Let V be as in Theorem 1.1, and let (B t ) t∈N be a decreasing family of closed balls in V centered at p 0 ∈ V of radius r t . Fix h > 0 and let (φ ht ) t∈N be the h-step discrete geodesic flow. Then for µ-almost every v ∈ T 1 V , the set is finite provided the sum converges. Also, if one assumes that (1.3) diverges and, in addition, that − ln r t r t t for large enough t, 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 when t is large enough, therefore (1.4) is satisfied. Note that in the 'critical exponent' case α = 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 We derive Theorem 1.2 form 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∈N and h be as in Theorem 1.2. then for µ-almost every v ∈ 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.

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 → 0 when t → ∞: indeed, if (r t ) is bounded from below by a positive constant, then the ergodicity of the geodesic flow implies that Furthermore, for a fixed R > 0 we can assume that r t ≤ R for all t ∈ N. Indeed, if the theorem is proved under that assumption, then applying it to the family {B t : t ≥ t 0 } where t 0 is such that r t ≤ R when t ≥ t 0 , we still recover condition (2.1). This R will be fixed later, see (3.4).
Our proof follows Maucourant's approach in [13]. Let us first introduce some terminology. Let F = (f t ) t∈N be a family of measurable functions on a probability space (X, µ). We call F decreasing if f s (x) ≤ f t (x) for any x ∈ X whenever s ≥ t. Also let us write We are going to use the following proposition from Maucourant's paper: We note that the above proposition was stated in [13] 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 It is decreasing because the family of balls B t is nested, and clearly I T [F] is equivalent, up to a multiplicative constant, to T t=1 r n t . 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 , which is the subject of the following theorem: 3. Proof of Theorem 2.2. To prove Theorem 2.2, following the same methodology as in [13], 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 = Γ\G, where Γ is a lattice in G = SO(n, 1), the isometry group of V = H n . Choose a liftp 0 ∈ H n of p 0 and for r > 0 and i ∈ N, let us denotê Then An estimate for #Γ i (r) would follow from a reasonable estimate for the error term in the asymptotics of the size of Γ ∩ D t for large t. Such estimates are due to Huber [10] for n = 2 and to Selberg for the general case, see [12], and also [3,7] for more recent results of this flavor. Denote by m G the Haar measure on G which locally projects onto µ. The following is a consequence of [12, Theorem 1]: Theorem 3.1. There exist constants 0 < q < 1 and t 1 , c 1 > 0 such that 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: There exist t 2 , c 2 , c 3 > 0 such that: (i) For any ε < 1 and t > t 2 , we have that From the two statements above one can easily derive the following estimate: There exist constants c 4 , c 5 with the following property: if 0 < r < 1 and i ∈ N are such that Proof of Theorem 3.3. Applying Theorem 3.1 for all i with hi − r > t 0 , we get that Therefore, by (3.1) and (3.2), we have: ≤ re (n−1)hi) 1 + e −(n−1)(1−q)hi r e (n−1)qr + e −(n−1)qr .

3.2.
The space of discrete geodesics on H n . In this section we will state measure estimates for spaces of geodesics on H n .
Definition 3.4. We will write G as the space of oriented, unpointed continuous geodesics on H n . Using the fact that T 1 H n can be written as G × R, we can define a measure ν on G byμ = ν × dt, whereμ is the Liouville measure on T 1 H n .
Then we will describe a similar definition for discrete geodesic flows. Namely: In addition, since we can write T 1 H n = G h × Zh, then we can define the measure m on G h by m = ν ⊗λ, where ν is the measure on G defined above and λ the Lebesgue measure on S h . Furthermore, the measureμ on the unit tangent bundle T 1 H n becomes the product of the measure m on G h with the counting measure on Zh.
In [13], 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. [13, Lemma 4] There exists a constant c 6 > 0 such that, for any two balls in 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 ν-measure of continuous geodesics meeting those two balls is less than c 6 r n−1 Here is a similar estimate for discrete geodesics on T 1 H n : Theorem 3.7. Consider two balls in 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 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 |d − kh| > 2 max(r 1 , r 2 ); (3.3) 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 G × S h with measure m = ν ⊗ dh, one can notice that the measure of such geodesics is bounded by 2c 6 min(r 1 , r 2 )r n−1 1 r n−1 2 e −(n−1)d .

A bound for the
Recall that for t ∈ N we defined f t to be the characteristic function of B t , which is a ball centered at p 0 ∈ V = Γ\H n of radius r t , see (2.3), and considered the family of functions F = (f t ) t∈N on T 1 V . Also we have chosen a liftp 0 ∈ H n of p 0 . Now defineB t to be a ball in H n centered atp 0 of radius r t , and let g t be the characteristic function ofB t , Thus, the liftf t of f t to T 1Ṽ satisfiesf t = γ∈Γ g t • γ.

LATTICE POINTS COUNTING AND SHRINKING TARGETS 161
Fix a fundamental domain D of H n for Γ containingp 0 . and define Theorem 3.8. Let D ⊂ H n be a fundamental domain for Γ such that D contains the ball of centerp 0 and of radius 3R. Then for all T ∈ N, Proof. For fixed T ∈ N and v ∈ T 1 V , we know that Now we can integrate S T [F](v) 2 over T 1 V and make a change of variable w = φ hs v. Since φ hs preserves the measure, we have the following: By the fact thatf t is the lift of f t , we obtain that Sincef t = γ∈Γ g t • γ, we can write Recall that D is the fundamental domain of H n for Γ. This insures that for all w in T 1 D, in the sum γ∈Γ g s (γw), all terms but the one corresponding to γ = id are zero. So we have 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

DMITRY KLEINBOCK AND XI ZHAO
Since we know that 2r t +2r s < 4R, we can conclude that g s (v)g t (γφ h(s−t) v) vanishes when t is outside of the interval Therefore, for any v ∈ T 1 V and any s ∈ N, In particular, we see that the quantity g By the above fact and the fact that the union of all Γ i is Γ, we have

Now let us define
c R = 6R + 2 h and split the estimate of Theorem 3.8 into two parts: (3.6)

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 Proof. Observing that ∪ c R i=3 Γ i is a finite set, we write N as its cardinal. Moreover, using (3.5), we get that In addition, we notice the following facts: • if g s (v) = 0, then g s (v)g t (γφ s−t v) in the left side vanishes; • if g s (v) = 1, then g s (v)g t (γφ s−t v) is at most 1. Therefore, (3.7) is equivalent to the following: This allows us to write Since T 1 D g s (v)dμ(v) is equivalent to r n s , up to a multiplicative constant, there exists some positive constant c 7 , depending only on R and h, such that

3.5.
A bound on the second part of (3.6). where c 4 is as in Theorem 3.3.
Proof. Let us fix s and produce an upper bound on . This requires the following observations: 1. (3.7) tells us that h for any s and v. 2. We know that |d(p 0 , γ −1p Hence, we know that the distance between the centers of B(p 0 , r s ) and B(γ −1 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(p 0 , r s ) and B(γ −1p 0 , r t ) is bounded by 2c 6 r n−1 s r n−1 t e −(n−1)(hi−1) r s . 3. Moreover, D contains the ball of centerp 0 with radius 3R. So we know that for fixed v, #{z ∈ Zh : g t (φ hz v) > 0} ≤ 3R h . 4. In addition, notice that g s (v)g t (γφ h(s−t) v) is not zero only if |hi − h(s − t)| < 6R. This implies that Now since (r t ) is decreasing, for all i ≥ c R = 6R+2 h , we have that Therefore, for all i ≥ c R , we obtain that where N i is the number of elements of Γ i such that the integrated function is not zero. Now we can consider the sum over all s and i ≥ c R : Our goal now is to estimate N i . Recall Theorem 3.3, which allows us to estimate #Γ i (r) when hi ≥ max(−c 4 ln r, r + t 0 ) for some constants c 4 , t 0 > 0. We will take r = r s−i− 6R h −1 . Indeed, since (r t ) is decreasing and we have assumed that r t ≤ R < 1, it follows that − ln r s ≥ − ln r s−i− 6R h −1 and r s−i− 6R h −1 < R. Now let us define Then i ≥ V s implies that hi ≥ max − c 4 ln r s−i− 6R h −1 , r s−i− 6R h −1 + t 0 . Meanwhile, we also know that g s (v)g t (γφ h(s−t) v) is not zero only if , where i is such that γ −1 ∈ Γ i . Therefore