Approximation of points in the plane by generic lattice orbits

We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma<\mathrm{SL}_2(\mathbb{R})$ acting linearly on $\mathbb{R}^2$. Our method gives bounds that are uniform for almost all orbits.


Introduction
Let Γ ⊆ SL 2 (R) be a lattice and for each T > 0 let Γ T = {γ ∈ Γ : γ ≤ T } with γ = tr(γ t γ) the Hilbert-Schmidt norm. For any u ∈ R 2 \ {0} and T > 0 consider the finite orbit Γ T u where Γ acts linearly on R 2 . The limiting distribution of these orbits as T → ∞ was extensively studied in [Led99,Nog02,GW07] and shown to be equidistributed with respect to a suitable measure (depending on u). In particular, for a generic point u ∈ R 2 , the orbit Γu is dense in R 2 , and hence any point v ∈ R 2 can be approximated by orbit points (when Γ is co-compact all orbits are dense).
To measure how well a point v ∈ R 2 can be approximated by orbit points in Γu, in analogy to similar problems in Diophantine approximations, Laurent and Nogueira [LN12a] defined two exponents µ Γ (u, v) and µ Γ (u, v) as follows.
Definition 1. The critical exponent µ Γ (u, v) is defined as the supremum of all α > 0 such that the set {γ ∈ Γ : γu − v < γ −α } is unbounded. The uniform critical exponent, µ Γ (u, v), is defined as the supremum over all α > 0 such that Γ T u ∩ B 1/T α (v) = ∅ for all sufficiently large T . Here B δ (v) = {u ∈ R : u − v ≤ δ} denote small norm ball with respect to some fixed norm on R 2 .
In [LN12a], Laurent and Nogueira studied these exponents for Γ = SL 2 (Z) and gave very precise estimates depending on the Diophantine properties of the slopes of Date: February 25, 2018. Dubi Kelmer is partially supported by NSF grant DMS-1401747. 1 u and v. In particular, their analysis implies that for almost all u, v ∈ R 2 \ {0} one has that 1 3 ≤ µ SL 2 (Z) (u, v) ≤ µ SL 2 (Z) (u, v) ≤ 1 2 and that µ SL 2 (Z) (u, v) ≥ 1 3 holds for every target v and any u with a dense orbit. In particular, this implies that Moreover, in [LN12b] they showed that for any lattice Γ, the upper bound µ Γ (u, v) ≤ 1 2 holds for any u with a dense orbit and a.e. v ∈ R 2 , so that µ Γ ≤ 1 2 for any lattice. Another approach for this problem was given in [MW12], where Maucourant and Weiss gave an effective version of the equidistribution result of Γ-orbits, building on effective equidistribution of unipotent flows on Γ\ SL 2 (R). In particular, their results imply the following lower bound for the critical exponents of a generic orbit: For any lattice Γ in SL 2 (R), for almost every u ∈ R 2 (respectively, for all 144 . Here τ = τ (Γ) ∈ [0, 1/2] measures the spectral gap for Γ, and in particular, τ (Γ) = 0 for Γ = SL 2 (Z) (see section 2.4 below for more details on the spectral gap).
Recently Ghosh, Gorodnik, and Nevo [GGN14, GGN15] studied a similar problem, in a more general setting, regarding rates of approximation of Γ-orbits on homogenous spaces X = G/H with Γ a lattice in a semisimple group G and H a closed subgroup. Their approach again builds on effective equidistribution results for the H action on Γ\G, but using the mean ergodic theorem instead of a pointwise ergodic theorem. A striking feature of their result is that it provides in many cases optimal rates of approximations. In this note we borrow some of their ideas, as well as ideas of [Kel16] for proving an effective mean ergodic theorem for actions of unipotent groups, and [GK15] relating mean ergodic theorems to shrinking target problems, to give bounds for the critical exponents of a generic orbit. Our main result is as follows: Theorem 1. Let Γ ⊆ SL 2 (R) be a lattice with spectral gap τ = τ (Γ). Then (1) For any v ∈ R 2 \ {0} for almost all u ∈ R 2 (2) For almost all u ∈ R 2 for any v ∈ R 2 \ {0} we have µ Γ (u, v) ≥ 1−2τ 5 . Remark 3. When the representation of G on L 2 0 (Γ\G) is tempered (in particular for Γ = SL 2 (Z)) we have that τ (Γ) = 0 and the first part implies that 1 3 ≤ µ Γ ≤ µ Γ ≤ 1 2 recovering (2). This is slightly better than the bound µ Γ ≥ 1 6 claimed in [GGN15] to be obtained by similar methods. For Γ a congruence lattices, using the best known bounds on the spectral gap, we get that 25 96 ≤ µ Γ ≤ 1 2 . It is not unlikely that in fact µ Γ = µ Γ = 1 2 (independent of the spectral gap), however, proving this seems beyond our abilities at the moment.
Remark 4. We point out a subtle difference between the first part of our result, which holds for any target point but only for generic orbits, vs. the results of [LN12a], that hold for any dense orbit, but the exponent depends on the slopes of the target point and the orbit. In particular, for Γ = SL 2 (Z), if the target point v ∈ R 2 has an irrational slope which is a Liouville number, the results of [LN12a, Theorem 2 (iii)] imply that µ SL 2 (Z) (u, v) ≥ 1 4 for almost all u, while we get µ SL 2 (Z) (u, v) ≥ 1 3 . On the other hand, if the slope of v is rational then [LN12a, Theorem 2 (ii)] imply that µ SL 2 (Z) (u, v) ≥ 1 2 for almost all u, which is best possible. Remark 5. In the second part of our result, the bound for the critical exponent is weaker because we require that the orbit of a single point u will approximate every target point simultaneously. Here the analysis of [LN12a] imply that almost , which is slightly better. However, our result holds for any lattice, and moreover, the method of proof generalizes to deal with the general problem of lattice action on homogenous spaces, thus answering the question of uniformity on a co-null set of orbits raised in [GGN14].
Remark 6. One can also consider the same problem for the action of lattices Γ ⊆ SL 2 (C) acting on C 2 . There have been a few results in this case: 2 , more generally, the work of for Pollicott [Pol11] give a lower bound for µ Γ for any co-compact Γ in SL 2 (C). The methods of this paper could also be generalized to handle this case as well to show that c Γ ≤ µ SL 2 (Γ) ≤ µ SL 2 (Γ) ≤ 1 2 for some explicit value of c Γ depending on the spectral gap for Γ.

Preliminaries and notation
2.1. Notation. We write A ≪ B or A = O(B) to indicate that A ≤ cB for some constant c. If we wish to emphasize that constant depends on some parameters we use subscripts, for example A ≪ ǫ B. We also write A ≍ B to indicate that A ≪ B ≪ A.

2.2.
Coordinates. Let G = SL 2 (R) and consider the Cartan decomposition G = NAK with N unipotent, A diagonal, and K compact. We will use the following coordinates In the coordinates g = n x a y k θ the Haar measure of G is dg = dxdydθ y 2 . Letn x = ( 1 0 x 1 ) and letN = {n x : x ∈ R}. For any g ∈ G apart from a set of measure zero we can also write g = n x an x ′ and the Haar measure in these coordinates is given by dg = dxdydx ′ y 2 . 2.3. Norms. Fix a basis B = {X 1 , X 2 , X 3 } for the Lie algebra g of G. Given a smooth test function ψ ∈ C ∞ (Γ\G), define the "L p , order-d" Sobolev norm S p,d (ψ) as Here D ranges over monomials in B of order at most d and D acts on ψ by left differentiation (e.g., Xψ(g) = d dt (ψ(ge tX ))| t=0 ). This definition depends on the basis, however, changing the basis B only distorts S p,d by a bounded factor.
2.4. Spectral gap. The group G acts on the upper half plane H = {x + iy : y > 0} by linear fractional transformation preserving the hyperbolic metric. The (self adjoint extension of the) hyperbolic Laplacian △ = −y 2 ( ∂ 2 ∂x 2 + ∂ 2 ∂x 2 ) acts on L 2 (Γ\H), and its spectrum consists of a discrete part 0 < λ 1 ≤ λ 2 < . . . and a continues part contained in [ 1 4 , ∞) and spanned by Eisenstein Series (when Γ is non uniform). We say that Γ has a spectral gap τ = τ (Γ) ∈ [0, 1/2], if λ 1 ≥ 1 4 − τ 2 . When Γ is a congruence group, Selberg's eigenvalue conjecture states that τ (Γ) = 0. This is known for Γ = SL 2 (Z) (as well as some other congruence groups of small level). The best known bound for a general congruence lattice is τ (Γ) ≤ 6 64 [KS03]. On the other hand, if Γ is not a congruence lattice it is possible to have τ (Γ) arbitrarily close to 1/2. 2.5. Decay of matrix coefficients. Given a lattice Γ ⊆ G let µ denote the G invariant probability measure on Γ\G. The group G acts on the right on the space L 2 (Γ\G, µ) via π(g)ψ(x) = ψ(xg), and for any two functions ψ, ϕ the corresponding matrix coefficient is For ψ, ϕ ∈ L 2 0 (Γ\G) (the space orthogonal to the constant function) the corresponding matrix coefficients go to zero as g → ∞, and the rate of decay is related to the spectral gap of Γ as follows (see [Ven10,Section 9.11]): For any smooth where τ = τ (Γ) measures the spectral gap for Γ.

Proof of main results
For the proof, we first use the duality of the Γ action on G/N ∼ = R 2 \ {0} and thē N action on Γ\G to reduce the problem to a shrinking target problem for a unipotent flow. Then we prove an effective mean ergodic theorem and use it to give a partial solution for the shrinking target problem. Combining these results will give the proof of Theorem 1 3.1. Reduction to a shrinking target problem. To define our shrinking targets, fix v = ( v 1 v 2 ) ∈ R 2 and assume that v 1 v 2 = 0. For small δ ∈ (0, 1/2) consider the set The following lemma shows that these shrinking targets B δ (v) are stable under small perturbation in v.
The shrinking target problem is then to determine how fast can targets B δ k shrink so that the finite orbits O k (x) = {xn l : |l| ≤ k}, keeps hitting them. The following lemma connects this shrinking target problem to the critical exponents (cf. [GGN15, Proposition 3.2]).
Proof. Assume that γ i ∈ Γ has γ i → ∞ and satisfies γ i u − v ≤ γ i −η . Let δ i = γ i −η , then for each i ∈ N there is k i ∈ Z such that γ i gn k i ∈ A δ i and hence xn k i ∈ B δ i . Moreover, since γ i gn k i ∈ A δ i and A δ i is contained in a compact set depending only on v, comparing norms we see that γ i ≍ g,v n k i ≍ k i . So there is a constant c > 0 (depending on g and v) such that xn k i ∈ B c/k η i . Now, for any α < η we have that c k η i ≤ 1 k α i for k i sufficiently large and so from some point xn k i ∈ B 1/k α i and indeed the set {k : xn k ∈ B 1/k α } is unbounded.
For the second statement, assume that for all T ≥ T 0 there is |k| ≤ T with xn k ∈ B T −η . Then there is γ k ∈ Γ with γ k gn k ∈ A T −η , hence, γ k u − v ≤ T −η . Also, as before, since γ k gn k ∈ A T −η comparing norms we get that γ k ≍ n k ≍ k so there is c > 0 (depending on g, v) such that γ k ≤ cT . SettingT = cT , andT 0 = cT 0 , assuming thatT 0 is sufficiently large so that (T 0 /c) −η ≤T −α 0 , we get that for all 3.2. Solution of the shrinking target problem. In this section we prove the following result, giving a partial solution to the shrinking target problem.
Theorem 4. Fix v ∈ R 2 with v 1 v 2 = 0 and let B δ = B δ (v) be as above. Then (1) If η > 1/2 then for almost all x ∈ Γ\G the set {k ∈ Z : Remark 9. This is a partial result because, even in the optimal setting when τ = 0, the lower bound η > 1/2 in (1) is much larger than the upper bound η < 1−2τ 3 in (2). It is reasonable that the correct upper bound is also η < 1/2 but we are not able to show this here. We note that for similar shrinking target problems, when the shrinking targets are spherical (i.e, right-K invariant), by a similar argument one can get a lower bound that is the same as the upper bound. In fact this is shown for unipotent flows on more general homogenous spaces [Kel16]. We also note that the exponent in (3) is even smaller because we require a much stronger form of approximation, that is, that a single orbit O k (x) approximate simultaneously all target points in Ω.
Our main tool for the proof will be an effective mean ergodic theorem for the unipotent flow u t =n t on Γ\G (we use the notation u t to indicate that the same results holds for any unipotent flow). For any T > 0 let β T denote the averaging operator on C ∞ c (Γ\G) given by Since the unipotent flow is ergodic, the mean ergodic theorem implies that β T (ϕ) − Γ\G ϕdµ) → 0 as T → ∞ for any ϕ ∈ L 2 (Γ\G). Using the decay of matrix coefficients we show the following effective result.
Using the effective mean ergodic theorem as a variance estimate, we can estimate the measure of points whose orbit miss a small set B δ . Explicitly, we show Proof. Let ρ ∈ C ∞ c (R) be positive supported in (−1/2, 1/2) with mean one. Define a function f δ ∈ C ∞ c (G) by and let F δ ∈ C ∞ c (Γ\G) be the corresponding Γ-invariant function, Clearly f δ is supported on A δ and F δ is supported on B δ . Moreover, since A δ ⊆ A 1/2 is contained in some fixed compact set, there is some C > 0 (depending only on v) such that A δ is contained in a union of C fundamental domains for Γ\G. Consequently, we also have that and similarly S 2,1 (F δ ) ≍ v 1. With these estimates, Proposition 5 implies that On the other hand, since β T (F δ )(x) = 0 for all x ∈ C T,δ we can bound from bellow from which the result follows.
We now go back to the shrinking target problem and give the Proof of Theorem 4. First, noting that µ(B δ ) ≍ δ 2 if we take δ k = k −η with η > 1/2 the series converges and hence by the easy half of the Borrel-Cantelli lemma, for almost all x ∈ Γ\G we have that {k : xn k ∈ B δ k } is bounded.
Next, for the lower bound for a fixed target point v, assume that δ k = k −η with 0 < η < 1−2τ 3 and let C v ⊆ Γ\G denote the set of all points such that for any and note that, since the orbits O k (x) are increasing sets and the targets B δ k (v) are decreasing, we have that We thus have that By Proposition 6 we can estimate µ(C 2 l ) ≪ ǫ 1 2 l(1−2τ −3η+ǫ) and hence µ( Now, from our assumption 1 − 2τ − 3η > 0 and taking ǫ = 1−2τ −3η 2 we can estimate µ( Finally, for the uniform bound, let δ k = k −α with 0 < η < α < 1−2τ 5 and for each . Moreover, we choose our points so that for k = 2 l the set {v k,i } m k i=1 contains all points v k ′ ,i with k ′ ≤ k. Now, let C Ω ⊆ Γ\G denote the set of all points such that for any T ∈ N there is k ≥ T with O k (x) ∩ B δ k (v k,i ) = ∅ for some 1 ≤ i ≤ m k , that is, By Proposition 6, for each i = 1, . . . , m 2 l+1 we can estimate, µ(C 2 l ,i ) ≪ ǫ 1 2 l(1−2τ −3α+ǫ) , and since for each l there are ≍ 2 2αl such sets we can bound Taking ǫ = 1−2τ −5α 2 > 0 we get that µ(C Ω ) ≪ T −ǫ for all T > 0 and hence µ(C Ω ) = 0. Now let x ∈ Γ\G, x ∈ C Ω and let v ∈ Ω. For each k let v k,i be δ k close to v. Then for all sufficiently large k, 3.3. Conclusion. Now combining the shrinking target results in Theorem 4 with Lemma 3 we get estimates on the critical exponents giving the Proof of Theorem 1. Let 0 = v ∈ R 2 and (perhaps after replacing v by γv for some γ ∈ Γ) we may assume that v 1 v 2 = 0. For any δ ∈ (0, 1/2) let A δ = A δ (v) and B δ = B δ (v) be as above.
First to show that for almost all u ∈ R 2 we have µ(u, v) ≤ 1/2 fix some η > 1/2 and let U ⊆ R 2 denote the set of all u ∈ R 2 such that there is a sequence γ k with For each u ∈ U let g u = u −1 2 u 1 0 u 2 and let U ⊆ G be defined by U = {g unx : u ∈ U, |x| ≤ 1/2}. Let η > α > 1/2, then by the first part of Lemma 3, for any g ∈ U, the set {k : xn k ∈ B k −α } is unbounded, and hence {Γg : g ∈ U} ⊆ {x ∈ Γ\G : {k : xn k ∈ B k −α } is unbounded}.
By the first part of Theorem 4 the set on the right has measure zero. We thus get that the set {Γg : g ∈ U} ⊂ Γ\G is a null set. But then the set U ⊂ G and hence also U ⊆ R 2 must also have measure zero. This shows that for almost all u ∈ R 2 the set {γ : γu − v ≤ γ −η } is bounded so µ(u, v) ≤ η for almost all u ∈ R 2 . Since this holds for any η > 1/2 we get the upper bound µ(u, v) ≤ 1/2 for almost all u ∈ R 2 .
Next to show that for any v ∈ R 2 for almost all u ∈ R 2 we have µ(u, v) ≥ 1−2τ 3 fix some α < 1−2τ 3 . Let α < η < 1−2τ 3 , let δ k = k −η and let C ⊂ Γ\G be the set of all points x ∈ Γ\G such that for every T ∈ N there is k ≥ T with O k (x)∩B δ k = ∅. Then for any g ∈ G\C 0 and any u = g ( 0 1 ) by the second part of Lemma 3, Γ T u∩B 1/T α (v) = ∅ for all sufficiently large T . By the second part of Theorem 4 we have that µ(C) = 0 and hence the set C 0 = {g ∈ G : Γg ∈ C} is a null set and the set {g ( 0 1 ) : g ∈ G \ C 0 } is set of full measure. This shows that for almost all u ∈ R 2 , we have that Γ T u ∩ B 1/T α (v) = ∅ for all sufficiently large T , and hence µ(u, v) ≥ α. Since this holds for any α < 1−2τ 3 we get that µ(u, v) ≥ 1−2τ 3 for almost all u ∈ R 2 . Finally, for the uniform bound, let Ω ∞ = {v ∈ R 2 : |v 1 | ≥ 1, |v 2 | ≥ 1}. Recall that any orbit Γv is either dense or a lattice, and hence must intersect Ω ∞ , and since µ(u, v) = µ(u, γv) it is enough to consider target points v ∈ Ω ∞ . Next, since we can write Ω ∞ = Ω i as a union of countably many compact sets (all bounded away from the axis), it is enough to show for each i ∈ N, for almost all u ∈ R 2 we have that µ(u, v) ≥ 1−2τ 5 for all v ∈ Ω i . This follows from the third part of Theorem 4 by the same argument as above.