An application of lattice points counting to shrinking target problems
aa r X i v : . [ m a t h . D S ] J u l AN APPLICATION OF LATTICE POINTS COUNTINGTO SHRINKING TARGET PROBLEMS
DMITRY KLEINBOCK AND XI ZHAO
Abstract.
We apply lattice points counting results to solve a shrinkingtarget problem in the setting of discrete time geodesic flows on hyper-bolic manifolds of finite volume. Introduction
Let (
X, µ ) be a probability space and T : X X a measure-preservingtransformation. For a sequence of measurable sets B n ⊂ X , consider the setlim sup n T − n B n = ∩ ∞ m =1 ∪ ∞ n = m T − n B n of points x ∈ X such that T n x ∈ B n for infinitely many n ∈ N . The Borel-Cantelli Lemma implies that if P ∞ 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 thesets B n . The classical Borel-Cantelli Lemma would imply that the measureof lim sup n T − n B n is full if the sets T − n B n are pairwise independent, anassumption 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 usuallyreferred to as dynamical Borel-Cantelli Lemmas. In many applications thefamily 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-CantelliLemma can be thought of as a quantitative way to express density of trajec-tories of a generic point of X at this fixed point p . Starting from the workof Phillip [15], there have been many results of this flavor. For exampleSullivan [17] proved a Borel-Cantelli type theorem for cusp neighborhoodsin hyperbolic manifolds of finite volume (here p = ∞ ), and the first namedauthor with Margulis [11] extended the result of Sullivan to non-compactRiemannian symmetric spaces. See also [2, 5, 6, 8, 9] for more references,and [1] for a nice survey of the area. Date : November 15, 2016The first-named author was supported by NSF grants DMS-1101320 and DMS-1600814.1991
Mathematics Subject Classification.
Primary: 37D40; Secondary: 53D25, 37A25.
Key words and phrases.
Shrinking target problems, hyperbolic geometry, geodesicflows, counting of lattice points.
One particular example of a shrinking target property can be found in apaper by Maucourant [12]. He considered nested balls in hyperbolic mani-folds (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 dimen-sion n , T V the unit tangent bundle of V , π : T V → V the canonicalprojection, ( φ t ) t ∈ R the geodesic flow on T V , µ the Liouville measure on T V , and d the Riemannian distance on V . Let ( B t ) t ≥ be a decreasingfamily of closed balls in V (with respect to the metric d ) of radius ( r t ) t ≥ .Then for µ -almost every v in T V , the set { t ≥ π ( ϕ t v ) ∈ B t } is boundedprovided Z ∞ r n − t dt (1.1) converges, and is unbounded if (1.1) diverges. Note that Maucourant’s theorem holds for the continuous-time geodesicflow on T V . Now suppose that one replaces the continuous family ( B t ) t ≥ by a sequence ( B t ) t ∈ N , and instead of the continuous geodesic flow considersthe h -step discrete geodesic flow ( ϕ ht ) t ∈ N for fixed h ∈ R + . The goal of thiswork is to provide additional argument needed to prove the Borel-Cantelliproperty, assuming some restrictions on the sequence ( B t ).One of the ingredients in Maucourant’s proof is a counting result for thenumber of lattice points inside balls in H n . To address a discrete timeanalogue 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 ballsin H n .We use the following notation throughout the paper: for two non-negativefunctions f and g , the notation f ( x ) ≪ g ( x ) means f ( x ) ≤ Cg ( x ) where C > 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 ∈ N be a decreasingfamily of closed balls in V centered at p ∈ V of radius r t . Fix h > andlet ( φ ht ) t ∈ N be the h -step discrete geodesic flow. Then for µ -almost every v ∈ T V , the set { t ∈ N : π ( φ ht v ) ∈ B t } (1.2) is finite provided the sum X t ∈ N r nt (1.3) converges. Also, if one assumes that (1.3) diverges and, in addition, that − ln r t r t ≪ t for large enough t, (1.4) then for µ -almost every v in T V , the set (1.2) is infinite. ATTICE POINTS COUNTING AND SHRINKING TARGETS 3
That is, in the terminology of [4], the sequence ( B t ) is a Borel-Cantellisequence. Note that the difference in exponents in (1.1) and (1.3) is dueto the fact that Theorem 1.1, unlike Theorem 1.2, deals with a continuoustime setting.It is well known that the geodesic flow on T V as above has exponentialdecay of correlations, see e.g. [14, 16]. For systems with exponential mixingsimilar dynamical Borel-Cantelli Lemmas have been established before. Forexample, it follows from [9, Theorem 4.1] that the set (1.2) will be infiniteprovided µ ( B t ) ≫ ln tt . (1.5)or, equivalently, r t ≫ (cid:0) ln tt (cid:1) /n . This shows that the restriction (1.4) isweaker than the one coming from [9, Theorem 4.1]. For example, take r t = Ct α , where α ≤ n . Then (1.3) diverges, and one can write − ln r t r t = − ln Ct α Ct α = 1 C ( − ln C + α ln t ) t α ≪ t when t is large enough, therefore (1.4) is satisfied. Note that in the ‘criticalexponent’ 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 r t = Ct /n (ln t ) β where 0 < β ≤
1: one has − ln r t r t = − ln Ct /n (ln t ) β Ct /n (ln t ) β = 1 C t /n (ln t ) β (cid:18) n ln t + β ln(ln t ) − ln C (cid:19) ≪ t 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 ∈ N and h be as in Theorem1.2. Then for µ -almost every v ∈ T V , the set (1.2) is finite provided thesum (1.3) converges. Also there exist C , C > such that if (1.3) divergesand, in addition, that s X t =[ s + C ln r s − C ] r n − t ≪ s X t =1 r nt when s is large enough , (1.6) then for µ -almost every v ∈ T V , the set (1.2) is infinite. In the next section we will reduce Theorem 1.3 to a certain L bound,Theorem 2.2, which will be verified in §
3, and in § DMITRY KLEINBOCK AND XI ZHAO
Acknowledgments
The authors want to thank Dubi Kelmer, Keith Merrill, Amos Nevo, HeeOh 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 gen-erality one can assume that r t → t → ∞ : indeed, if ( r t ) is boundedfrom below by a positive constant, then the ergodicity of the geodesic flowimplies that π ( ϕ ht v ) ∈ B t infinitely often for µ -a.e. v ∈ T V. (2.1)Furthermore, for a fixed R > r t ≤ R for all t ∈ N .Indeed, if the theorem is proved under that assumption, then applying it tothe family { B t : t ≥ t } where t is such that r t ≤ R when t ≥ t , we stillrecover condition (2.1). This R will be fixed later, see (3.4).Our proof follows Maucourant’s approach in [12]. Let us first introducesome terminology. Let F = ( f t ) t ∈ N be a family of measurable functions ona 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 S T [ F ]( x ) def = T X t =1 f t ( φ ht x ) , I T [ F ] def = T X t =1 Z X f t dµ. We are going to use the following proposition from Maucourant’s paper:
Proposition 2.1. ([12, Proposition 1])
Let F = ( f t ) t ∈ N be a decreasingfamily of non-negative measurable functions on X such that f t ∈ L ( X, µ ) for all t . Assume that lim T →∞ I T [ F ] = ∞ , and that S T [ F ] /I T [ F ] is bounded in L -norm as T → ∞ . Then, as T → ∞ , S T [ F ] /I T [ F ] converges to weaklyin L ( X, µ ) , and for µ -almost every x in X one has lim sup T → + ∞ S T [ F ]( x ) I T [ F ] ≥ . (2.2)We note that the above proposition was stated in [12] for the case ofa continuous family of functions, but it is immediate to deduce a discreteversion. To prove Theorem 1.3, we will apply Proposition 2.1 to the familyof characteristic functions of B t , i.e. take f t ( v ) = (cid:26) d ( π ( v ) , p ) ≤ r t B t is nested, and clearly I T [ F ] isequivalent, up to a multiplicative constant, to P Tt =1 r nt . Also it is clear thatthe 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 fromthe Borel-Cantelli Lemma, we can see that Theorem 1.3 can be reduced to ATTICE POINTS COUNTING AND SHRINKING TARGETS 5 proving a uniform L bound for S T [ F ] /I T [ F ], which is the subject of thefollowing theorem: Theorem 2.2.
Let F = ( f t ) t ∈ N be as in (2.3) . Then there exist C , C > such that if I T [ F ] diverges when T goes to ∞ and condition (1.6) holds, thenthe L -norm of S T [ F ] /I T [ F ] is bounded for all T ≥ . Proof of Theorem 2.2
To prove Theorem 2.2, following the same methodology as in [12], we willapply a result on counting lattice points stated below (Theorem 3.3) togetherwith a measure estimate for the space of discrete geodesics (Theorem 3.7).3.1.
Counting lattice points.
Write T V = Γ \ G , where Γ is a lattice in G = SO( n, e V = H n . Choose a lift ˜ p ∈ H n of p and for r > i ∈ N , let us denoteˆΓ i ( r ) def = (cid:8) γ ∈ Γ : d ( ˜ p , γ ˜ p ) ∈ ( hi − r, hi + r ] (cid:9) . Then i ( r ) = ∩ D hi + r ) − ∩ D hi − r ) , where D t = { g ∈ G : d ( g ˜ p , ˜ p ) ≤ t } . An estimate for i ( r ) would follow from a reasonable estimate for theerror term in the asymptotics of the size of Γ ∩ D t for large t . Such estimatesare 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 Haarmeasure on G which locally projects onto µ . The following is a consequenceof [13, Theorem 1]: Theorem 3.1.
There exist constants < q < and t , c > such that | ∩ D t ) − m G ( D t ) | ≤ c m G ( D t ) q , for all t > t . An important property of the family { D t } is so-called H¨older well-roundedness,see [7]. In particular the following is true: Proposition 3.2.
There exist t , c , c > such that: (i) For any ε < and t > t , we have that m G ( D t + ε ) − m G ( D t − ε ) ≤ c εm G ( D t − ε ) . (3.1)(ii) For any t > , m G ( D t ) ≤ c e ( n − t . (3.2)From the two statements above one can easily derive the following esti-mate: DMITRY KLEINBOCK AND XI ZHAO
Theorem 3.3.
There exist constants c , c with the following property: if < r < and i ∈ N are such that hi ≥ max( − c ln r, r + t ) , where t = max( t , t ) , then i ( r ) ≤ c re ( n − hi . Proof of Theorem 3.3.
Applying Theorem 3.1 for all i with hi − r > t , weget that ∩ D hi + r ) ≤ m G ( D hi + r ) + c m G ( D hi + r ) q and ∩ D hi − r ) > m G ( D hi − r ) − c m G ( D hi − r ) q . Therefore, by (3.1) and (3.2), we have: { γ ∈ Γ : d ( ˜ p , γ ˜ p ) ∈ [ hi − r, hi + r ] } = ∩ D hi + r ) − ∩ D hi − r ) ≤ m G ( D hi + r ) + c m G ( D hi + r ) q − m G ( D hi − r ) + c m G ( D hi − r ) q ≤ m G ( D hi + r ) − m G ( D hi − r ) + c (cid:0) m G ( D hi + r ) q + m G ( D hi − r ) q (cid:1) ≪ rm G ( D hi − r ) + (cid:0) m G ( D hi + r ) q + m G ( D hi − r ) q (cid:1) ≪ re ( n − hi − r ) + e ( n − q ( hi + r ) + e ( n − q ( hi − r ) ≤ re ( n − hi ) e − ( n − − q ) hi r (cid:16) e ( n − qr + e − ( n − qr (cid:17)! . Since q < r <
1, we have e ( n − qr + e − ( n − qr < e n − , and clearly r e − ( n − − q ) hi ≤ hi ≥ − ln r (1 − q )( n − . Summarizingthe above, if hi ≥ max (cid:18) − − q )( n −
1) ln r, r + t (cid:19) , then i ( r ) ≪ re ( n − hi . (cid:3) The space of discrete geodesics on H n . In this section we will statemeasure estimates for spaces of geodesics on H n . Definition 3.4.
We will write G as the space of oriented, unpointed con-tinuous geodesics on H n . Using the fact that T H n can be written as G × R ,we can define a measure ν on G by ˜ µ = ν × dt , where ˜ µ is the Liouvillemeasure on T H n . Then we will describe a similar definition for discrete geodesic flows.Namely:
ATTICE POINTS COUNTING AND SHRINKING TARGETS 7
Definition 3.5.
For fixed h > , G h is the space of all h -step discretegeodesic trajectories: { ϕ ht : t ∈ Z } . That is G h = G × S h where S h is [0 , h ] with and h identified. In addition, since we can write T H n = G h × Z h ,then we can define the measure m on G h by m = ν ⊗ λ , where ν is the measureon G defined above and λ the Lebesgue measure on S h . Furthermore, themeasure ˜ µ on the unit tangent bundle T H n becomes the product of themeasure m on G h with the counting measure on Z h . In [12], Maucourant considered the space of continuous geodesics, andestimated 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 > such that,for any two balls in H n of respective centers and radii ( o , r ) , ( o , r ) thatsatisfy r , r < , and d ( o , o ) > , the ν -measure of continuous geodesicsmeeting those two balls is less than c r n − r n − e − ( n − d ( o ,o ) . Here is a similar estimate for discrete geodesics on T H n : Theorem 3.7.
Consider two balls in H n with respective centers and radii ( o , r ) , ( o , r ) that satisfy r < , r < , and d = d ( o , o ) > . Alsoassume that h > r , r ) . Then the m -measure of the h -step geodesicswhich intersect those two balls is less than ( c r n − r n − e − ( n − d min( r , r ) if dist( d, h Z ) ≤ r , r )0 otherwise,where c is as in Theorem 3.6.Proof. An h -step geodesic will fail to intersect both balls if for any k wehave | d − kh | > r , r ); (3.3)in this case the measure we are to estimate is zero. So only if there isan 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 geodesicsis bounded by 2 c min( r , r ) r n − r n − e − ( n − d . (cid:3) A bound for the L -norm of S T [ F ] . Recall that for t ∈ N we defined f t to be the characteristic function of B t , which is a ball centered at p ∈ V = Γ \ H n of radius r t , see (2.3), and considered the family of functions F = ( f t ) t ∈ N on T V . Also we have chosen a lift ˜ p ∈ H n of p . Nowdefine ˜ B t to be a ball in H n centered at ˜ p of radius r t , and let g t be thecharacteristic function of ˜ B t , Thus, the lift ˜ f t of f t to T ˜ V satisfies˜ f t = X γ ∈ Γ g t ◦ γ. DMITRY KLEINBOCK AND XI ZHAO
Fix a fundamental domain D of H n for Γ containing ˜ p . and define i V ( ˜ p ) def = sup { r ∈ R : B ( ˜ p , r ) ⊂ D } . Also define R def = min (cid:0) i V ( ˜ p ) / , , h (cid:1) , (3.4)and, for i ∈ Z + ,Γ i def = (cid:26) γ ∈ Γ : d (˜ p , γ ˜ p ) ∈ (cid:20) hi − h , hi + h (cid:19)(cid:27) . Theorem 3.8.
Let D ⊂ H n be a fundamental domain for Γ such that D contains the ball of center ˜ p and of radius R . Then for all T ∈ N , Z T V S T [ F ]( v ) dµ ( v ) ≤ T X s =1 [ s + Rh ] X i =1 X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) . Proof.
For fixed T ∈ N and v ∈ T V , we know that S T [ F ]( v ) = T X s =1 f t ( φ ht v ) ! T X t =1 f s ( φ hs v ) ! = 2 T X s =1 X t ≤ s f t ( φ ht v ) f s ( φ hs v ) . Now we can integrate S T [ F ]( v ) over T V and make a change of variable w = φ hs v . Since φ hs preserves the measure, we have the following: Z T V S T [ F ]( v ) dµ ( v ) ≤ T X s =1 X t ≤ s Z T V f s ( w ) f t ( φ h ( t − s ) w ) dµ ( w ) . By the fact that ˜ f t is the lift of f t , we obtain that Z T V S T [ F ]( v ) dµ ( v ) ≤ T X s =1 X t ≤ s Z T H n ˜ f s ( w ) ˜ f t ( φ h ( t − s ) w ) d ˜ µ ( w ) . Since ˜ f t = P γ ∈ Γ g t ◦ γ , we can write Z T V S T [ F ]( v ) dµ ( v ) ≤ T X s =1 X t ≤ s Z T H n X γ ∈ Γ g s ( γw ) X γ ∈ Γ g t ( γφ h ( t − s ) w ) d ˜ µ ( w ) . Recall that D is the fundamental domain of H n for Γ. This insures thatfor all w in T D , in the sum P γ ∈ Γ g s ( γw ), all terms but the one correspondingto γ = id are zero. So we have Z T V S T [ F ]( v ) dµ ( v ) ≤ T X s =1 X γ ∈ Γ X t ≤ s Z T D g s ( w ) g t ( γφ h ( t − s ) w ) d ˜ µ ( w ) . ATTICE POINTS COUNTING AND SHRINKING TARGETS 9
Making another change of variables v = − w , where − w means the pointin T D with the same projection as w and the tangent vector pointing inthe opposite direction, we deduce that Z T V S T [ F ]( v ) dµ ( v ) ≤ T X s =1 Z T D X γ ∈ Γ s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) . For fixed v ∈ T D , we know that g s ( v ) g t ( γφ h ( s − t ) v ) is zero when π ( v ) / ∈ ˜ B s or π ( φ h ( s − t ) v ) / ∈ γ − ˜ B t , which implies that g s ( v ) g t ( γφ h ( s − t ) v ) vanisheswhen | h ( s − t ) − d ( ˜ p , γ − ˜ p ) | > r t + 2 r s . Since we know that 2 r t + 2 r s < R , we can conclude that g s ( v ) g t ( γφ h ( s − t ) v )vanishes when t is outside of the interval (cid:20) s − d ( ˜ p , γ − ˜ p ) h − Rh , s − d ( ˜ p , γ − ˜ p ) h + 4 Rh (cid:21) . Therefore, for any v ∈ T V and any s ∈ N , (cid:8) ≤ t ≤ s : g s ( v ) g t ( γφ h ( s − t ) v ) = 1 (cid:9) ≤ Rh . (3.5)Furthermore, the integral is zero if (cid:12)(cid:12) d ( ˜ p , γ − ˜ p ) − hi (cid:12)(cid:12) > R > r t + r s forall i . Hence this integral vanishes when | hi − h ( s − t ) | > R , i.e. when hi − h ( s − t ) > R or hi − h ( s − t ) < − R. In particular, we see that the quantity g s ( v ) g t ( γφ h ( s − t ) v ) is zero if hi > hs + 6 R > h ( s − t ) + 6 R, i.e. i > s + 6 Rh .
By the above fact and the fact that the union of all Γ i is Γ, we have Z T V S T [ F ]( v ) dµ ( v ) ≤ T X s =1 X i ≥ X γ ∈ Γ i s X t =1 Z T D g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) . = 2 T X s =1 [ s + Rh ] X i =0 X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) . (cid:3) Now let us define c R = 6 R + 2 h and split the estimate of Theorem 3.8 into two parts: Z T V S T [ F ]( v ) dµ ( v ) ≤ T X s =1 [ c R ] X i =1 X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v )+ 2 T X s =1 [ s + Rh ] X i =[ c R ] X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) . (3.6)3.4. A bound on the first part of (3.6) . It is not hard to estimate thefirst part.
Theorem 3.9.
There is constant c , only depending on R and h , such thatfor all T ∈ N T X s =1 [ c R ] X i =0 X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( φ h ( s − t ) v ) d ˜ µ ( v ) ≤ c T X t =1 r nt . 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 s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) ≤ Rh . (3.7)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: s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) ≤ Rh g s ( v ) . This allows us to write T X s =1 [ c R ] X i =0 X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) ≤ N Rh T X s =1 Z T D g s ( v ) d ˜ µ ( v ) . Since R T D g s ( v ) d ˜ µ ( v ) is equivalent to r ns , up to a multiplicative constant,there exists some positive constant c , depending only on R and h , such that T X s =1 [ c R ] X i =0 X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) ≤ c T X t =1 r nt . (cid:3) ATTICE POINTS COUNTING AND SHRINKING TARGETS 11
A bound on the second part of (3.6) .Theorem 3.10.
There exist constants c and c , only depending on R , suchthat T X s =1 [ s + Rh ] X i =[ c R ] X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) ≤ c T X s =1 r ns [ s − R − X t =[ s + c h ln r s − Rh − r n − t + c T X s =1 r ns s X t =1 r nt , where c is as in Theorem 3.3.Proof. Let us fix s and produce an upper bound on R T D P st =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ).This requires the following observations:(1) (3.7) tells us that P st =1 g s ( v ) g t ( γφ h ( s − t ) v ) ≤ Rh for any s and v .(2) We know that | d ( ˜ p , γ − ˜ p ) − hi | < R , i.e. hi − R < d ( ˜ p , γ − ˜ p ) < hi + 2 R. Therefore, i ≥ c R ≥ R +2 h implies that d ( ˜ p , γ − ˜ p ) > hi − R > R + 2 − R > . Hence, we know that the distance between the centers of B ( ˜ p , r s )and B ( γ − ˜ p , r t ) is greater than 2. Thus by Theorem 3.7, the mea-sure m of the set of discrete geodesics intersecting both B ( ˜ p , r s )and B ( γ − ˜ p , r t ) is bounded by 2 c r n − s r n − t e − ( n − hi − r s .(3) Moreover, D contains the ball of center ˜ p with radius 3 R . So weknow that for fixed v , { z ∈ Z h : g t ( φ hz v ) > } ≤ Rh .(4) In addition, notice that g s ( v ) g t ( γφ h ( s − t ) v ) is not zero only if | hi − h ( s − t ) | < R. This implies that s − i − Rh < t < s − i + 6 Rh .
Now since ( r t ) is decreasing, for all i ≥ c R = R +2 h , we have that Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) ≪ r ns r n − s − i − Rh − e − ( n − hi . Therefore, for all i ≥ c R , we obtain that X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) ≪ N i r ns r n − s − i − Rh − e − ( n − hi , where N i is the number of elements of Γ i such that the integrated functionis not zero. Now we can consider the sum over all s and i ≥ c R : T X s =1 [ s + Rh ] X i =[ c R ] X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) ≪ T X s =1 [ s + Rh ] X i =[ c R ] r ns r n − s − i − Rh − e − ( n − hi N i Our goal now is to estimate N i . Recall Theorem 3.3, which allows us toestimate i ( r ) when hi ≥ max( − c ln r, r + t ) for some constants c , t > r = r s − i − Rh − . Indeed, since ( r t ) is decreasing and we haveassumed that r t ≤ R <
1, it follows that − ln r s ≥ − ln r s − i − Rh − and r s − i − Rh − < R .Now let us define V s def = max (cid:18) − c ln r s h , t h (cid:19) + c R . (3.8)Then i ≥ V s implies that hi ≥ max (cid:0) − c ln r s − i − Rh − , r s − i − Rh − + t (cid:1) . Meanwhile, we also know that g s ( v ) g t ( γφ h ( s − t ) v ) is not zero only if d ( ˜ p , γ − ˜ p ) ∈ [ ih − r s − i − Rh − , ih + r s − i − Rh − ] , where i is such that γ − ∈ Γ i . Therefore N i ≤ { γ : d (˜ p , γ ˜ p ) ∈ [ hi − r s − i − Rh − , hi + r s − i − Rh − ] }≤ c e ( n − hi +1) r s − i − Rh − when i ≥ V s .By applying the fact that ( r t ) is decreasing, we have the following: T X s =1 [ s + Rh ] X i =[ V s ] r ns r n − s − i − Rh − e − ( n − hi − N i ≪ T X s =1 r ns s X t =1 r nt . When i < V s , we will use the counting lattice point estimate (Theo-rem 3.1) to conclude that N i ≪ e ( n − hi . Recalling the definition of V s ,see (3.8), we know that the assumption i < − c h ln r s implies that i < V s .Meanwhile, since ( r t ) is decreasing, we have that T X s =1 [ V s ] X i = c R r ns r n − s − i − Rh − e − ( n − hi − N i ≪ T X s =1 r ns s X t =[ s + c h ln r s − Rh − r n − t . ATTICE POINTS COUNTING AND SHRINKING TARGETS 13
Putting it all together, we conclude that T X s =1 [ s + Rh ] X i =[ V s ] X γ ∈ Γ i Z T D s X t =1 g s ( v ) g t ( γφ h ( s − t ) v ) d ˜ µ ( v ) ≤ c T X s =1 r ns s X t =[ s + c h ln r s − Rh − r n − t + c T X s =1 r ns s X t =1 r nt . (cid:3) Completion of the proof of Theorem 2.2.
Proof.
Recall that so far we have Z T V S T [ F ]( v ) dµ ( v ) ≤ c T X s =1 r ns + c T X s =1 r ns s X t =[ s + c h ln r s − Rh − r n − t + c T X s =1 r ns s X t =1 r nt . Now let us take C = c /h , C = 2 + 6 R/h , and let us assume (1.6), i.e.that there exist
C, s such that s X t =[ s + c h ln r s − Rh − r n − t ≤ C s X t =1 r nt when s ≥ s . Then we can write Z T V S T [ F ]( v ) dµ ( v ) ≤ c T X s =1 r ns + c s − X s =1 r ns s X t =[ s + c h ln r s − Rh − r n − t + C · c T X s = s r ns s X t =1 r nt + c T X s =1 r ns s X t =1 r nt ≤ c T X s =1 r ns + c T X s =1 r ns s X t =1 r nt + c Since R T V I T [ F ] dµ ( v ) is equivalent, up to a multiplicative constant, to T P s =1 r ns s P t =1 r nt , and with the assumption that R T V I T [ F ] dµ ( v ) → ∞ , T → ∞ ,one can easily conclude that S T [ F ] I T [ F ] is bounded in L -norm. (cid:3) Proof of Theorem 1.2
Recall that we are given a non-increasing sequence r t which tends to 0 as t → ∞ and such that ∞ P r nt = ∞ and in addition satisfying (1.4), that is, forsome C , s > − ln r s r s ≤ C s when s ≥ s . (4.1) We need to show that this sequence satisfies condition (1.6). This will bean easy consequence of the following lemma:
Lemma 4.1.
Under the above assumptions, for any C , C > there exist C , T > such that s X t =[ s + C ln r s − C ] r n − t ≤ C s X t =1 r nt when s ≥ T. Proof.
By (4.1), − C ln r s ≤ C C sr s when s ≥ s . (4.2)Take s such that − ln r s ≥ C C when s > s . This and (4.2) imply that C ≤ C C sr s and 2 C C sr s ≥ − C ln r s + C . (4.3)Since r s is non-increasing, (4.3) implies that2 C C sr [ s + C ln r s − C ] ≥ − C ln r s + C when s > max( s , s ). Due to the fact that 0 < r s < r [ s + C ln r s − C ] <
1, wehave that2 C C sr [ s + C ln r s − C ] ≥
12 ( − C ln r s + C )(1 + r [ s + C ln r s − C ] − r s );thus2 C C sr [ s + C ln r s − C ] ≥ ( − C ln r s + C )(1 + r [ s + C ln r s − C ] − r s ) ≥ ( − C ln r s + C )(1 − r s ) + ( − C ln r s + C ) r [ s + C ln r s − C ] . Therefore, when s > max( s , s ), we obtain that(2 C C s + C ln r s − C ) r [ s + C ln r s − C ] ≥ ( − C ln r s + C )(1 − r s ) . (4.4)Now take s > r s < C C when s > s , and let T def = max( s , s , s ).Then (4.3) implies that, when s > T , s ≥ C C ( − C ln r s + C ) . Thus, by adding 2 C C s + (2 C C + 1)( C ln r s − C ) to both sides, weconclude that, when s > T (2 C C + 1)( s + C ln r s − C ) ≥ C C s + C ln r s − C . Now let us define C = 2 C C + 1.Then we have that, when s > TC ( s + C ln r s − C ) ≥ C C s + C ln r s − C . which, in view of (4.4), implies C ( s + C ln r s − C ) r [ s + C ln r s − C ] ≥ ( − C ln r s + C )(1 − r s ) . ATTICE POINTS COUNTING AND SHRINKING TARGETS 15
Since r [ s + C ln r s − C ] >
0, the above inequality implies that, when s ≥ T , C ( s + C ln r s − C ) r n [ s + C ln r s − C ] ≥ ( − C ln r s + C )(1 − r s ) r n − s + C ln r s − C ] . (4.5)On the other hand, since r s is non-increasing, one will notice that C ( s + C ln r s − C ) r n [ s + C ln r s − C )] ≤ C s + C ln r s − C )] X t =1 r nt , and s X t =[ s + C ln r s − C ] ( r n − t − r nt ) = s X t =[ s + C ln r s − C ] (1 − r t ) r n − t ≤ ( − C ln r s + C )(1 − r s ) r n − s + C ln r s − C ] . Therefore, by (4.5), we have that, when s ≥ T , s X t =[ s + C ln r s − C ] ( r n − t − r nt ) ≤ C s + C ln r s − C ] X t =1 r nt , and hence s X t =[ s + C ln r s − C ] r n − t ≤ C s + C ln r s − C ] X t =1 r nt + s X t =[ s + C ln r s − C ] r nt ≤ C s X t =1 r nt . (cid:3) This shows that (1.4) implies (1.6), and finishes the proof of Theorem1.2. (cid:3)
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