Sparse Equidistribution of Unipotent Orbits in Finite-Volume Quotients of $\text{PSL}(2,\mathbb R)$, with appendices

In this note, we consider the orbits $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ in $\Gamma\backslash\text{PSL}(2,\mathbb R)$, where $\Gamma$ is a non-uniform lattice in $\text{PSL}(2,\mathbb R)$ and $u(t)$ is the standard unipotent group in $\text{PSL}(2,\mathbb R)$. Under a Diophantine condition on the intial point $p$, we can prove that $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ is equidistributed in $\Gamma\backslash\text{PSL}(2,\mathbb R)$ for small $\gamma>0$, which generalizes a result of Venkatesh (Ann.of Math. 2010). We will compute Hausdorff dimensions of subsets of non-Diophantine points in Appendix A, using results of lattice counting problem. In Appendix B we will use a technique of Venkatesh (Ann.of Math. 2010) and an exponential mixing property to prove a weak version of a result of Str\"ombergsson (J Mod Dynam, 2013), which is about the effective equidistribution of horospherical orbits.


Introduction
The theory of equidistribution of unipotent flows on homogeneous spaces has been studied extensively over the past few decades. Furstenberg [F73] first proved that the unipotent flow on Γ\ PSL(2, R), where Γ is a uniform lattice, is uniquely ergodic. In [D78] Dani classified ergodic invariant measures for unipotent flows on finite volume homogeneous spaces of PSL(2, R), and using this result Dani and Smillie [DS84] proved that any non-periodic unipotent orbit is equidistributed on Γ\ PSL(2, R) for any lattice Γ. The proof of Oppenheim Conjecture due to Margulis [M89] by proving a special case of Raghunathan's conjecture drew a lot of attention to this subject. Soon afterwords, Ratner published her seminal work [R90a, R90b, R91a] proving measure classification theorem for unipotent actions on homogeneous spaces as conjectured by Raghunathan and Dani [D81]. Using these results, Ratner [R91b] proved that any unipotent orbit in a finite volume homogeneous space is equidistributed in its orbit closure; see also Shah [Sh91] for the case of Rank-1 semisimple groups.
Ratner's work has led to many new extensions and number theoretic applications of ergodic theory of unipotent flows. One of these results, which is related to this paper, was the work by Shah [Sh94]. In that paper, Shah asked whether {pu(n 2 )|n ∈ N} is equidistributed in a sub-homogeneous space of PSL(2, Z)\ PSL(2, R), where u : R → PSL(2, R) is the standard unipotent 1-parameter subgroup u(t) = 1 t 0 1 .
In this direction, recently Venkatesh published a result about sparse equidistribution ( [V10], Theorem 3.1). There he introduced a soft technique of calculations by using discrepancy trick, and proved that if Γ is a cocompact lattice in PSL(2, R) and γ > 0 is small, then for any point p ∈ Γ\ PSL(2, R) we have In other words, in the case of Γ\ PSL(2, R) being compact, the equidistribtion holds for the sparse subset {n 1+γ |n ∈ N}.
In this paper, we will consider the sparse subset {n 1+γ |n ∈ N} and orbits of {u(n 1+γ )|n ∈ N} in Γ\ PSL(2, R), where Γ is a non-uniform lattice. We want to prove a sparse equidistribution theorem similar to Shah's conjecture [Sh94] and the work of Venkatesh [V10]. To deal with the complexity caused by initial points of unipotent orbits, we will introduce a Diophatine condition (see section 3). Now we state the main theorem in this paper.
Remark 1.1. From the proof of the main theorem, we will see that the constant γ 0 = min{β/(2κ j + 8)|j = 1, 2, . . . , k} for some constant β > 0 which comes from the spectrum information of the Laplacian on Γ\H. Now let Γ be a subgroup of finite index of PSL(2, Z). Then we have the following corollary of the main theorem, which will be explained in section 3.
To prove the main theorem, we shall use the technique of Venkatesh in [V10] and Strömbergsson's result in [S13] about effective version of Ratner's theorem on Γ\ PSL(2, R). However, since Γ\G is not compact, all the effective results we obtain in this note will depend on initial points, and hence the estimates get out of control when we combine these results. To overcome this difficulty, we will introduce a Diophantine condition. With the help of this Diophantine condition along with the notion of (C, α; ρ)-good functions, we will be able to control the rates of these effective results. In section 2, we list the concepts and theorems that we need in this paper. In section 3, we define the Diophantine condition we need in our proofs and deduce Corollary 1.1 from the main theorem. In section 4, we will study dynamics of a special class of orbits in Γ\G. The dynamical properties of these orbits will help us control the rates of the effective results in this paper. Since we are dealing with the noncompact case of Γ\G, and also for the sake of completeness, we include the technique of [V10] in section 5. We will finish the proof of the main theorem in section 6.
It may be interesting to explore the relation between the techniques used in this work and those developed in the work of Sarnak and Ubis [SU14], where they have described the limiting distribution of horocycles at primes.

Prerequisites
Throughout this note, if there exists an absolute constant C > 0 such that f ≤ Cg, then we write f ≪ g. If f ≪ g and g ≪ f , then we use the notation f ∼ g. We denote G = PSL(2, R) and Γ a non-uniform lattice in G. Let For any element a ∈ A, we denote α(a) = s.
One of the ingredients in our calculations is the effective version of the mixing property of unipotent flows in Γ\G. The following effective version is proved by Kleinbock and Margulis [KM99]. For f ∈ C k (Γ\G) we let f p,k be the Sobolev L p -norm involving all the Lie derivatives of order ≤ k of f . Note that f ∞,0 is the supreme norm of f . Theorem 2.1 (Kleinbock and Margulis [KM99]). There exists κ > 0 such that for any f, g ∈ C ∞ (Γ\G), we have Another ingredient in the calculations is the effective version of Ratner's theorem proved by Strömbergsson [S13]. To state the result, we will introduce some notations in [S13]. We know that G acts on the upper half plane H by the action a b c d · z = az + b cz + d and we have the standard projection of Γ\G to the fundamental domain of Γ in H π : Γ\G → Γ\H by sending Γg to Γg(i). We define the geodesic flows on Γ\G by where d H (·, ·) is the hyperbolic distance on Γ\H.
Theorem 2.2 (Strömbergsson [S13]). For all p ∈ Γ\G, T ≥ 10, and all f ∈ C 4 (Γ\G) such that where s > 0 is a number depending on the spectrum of the Laplacian on Γ\H and r = r(p, T ) = T · e −dist(g log T (p)) . The implied constants depend only on Γ and p 0 .

The Diophatine Condition
In this section, we will introduce a Diophantine condition for points in Γ\G. For G = PSL(2, R), we consider the sets N Ω A α K where and K = SO(2). As is known, for the non-uniform lattice Γ, there exist σ j ∈ G and a connected bounded subset Ω j ⊂ R (1 ≤ j ≤ k) with the following property We will fix σ j (1 ≤ j ≤ k). Note that in the upper half plane H, each σ j corresponds to a cusp η j , i.e. σ j (i∞) = η j , and η 1 , η 2 , . . . , η k are the inequivalent cusps of Γ\H. Let Γ j = Γ ∩ σ j N σ −1 j . Let π j be the covering map π j : Γ j \G → Γ\G.
Now consider the usual action of G on R 2 and let e 1 = 1 0 In this way, we obtain k maps m j (j = 1, 2, . . . , k) whose images are all in R 2 \ {0}. For our purposes, we will identify a vector in v ∈ R 2 with its opposite vector −v. Using these notations, we can give the following definition of Diophantine condition of a point p ∈ Γ\G.
It is straightforward to verify that if g ∈ AN then the Diophantine types of p and pg are the same; although the choices of µ j , ν j > 0 in the above definition may differ. The hausdorff dimension of the complement of the set of points of the Diophantine type (κ 1 , κ 2 , . . . , κ k ) is given in Theorem 7.1. Now we can deduce Corollary 1.1 from the main theorem.
Proof of Corollary 1.1. If Γ is a subgroup of finite index of PSL(2, Z), then we can pick Note that for each m j , we have If there exist constants ζ > 0, µ, ν > 0 such that for any (m, then p is Diophantine of type (ζ, . . . , ζ) by the definition above. In particular, if a/c ∈ R is a Diophantine number of type ζ, i.e. there exists C > 0 such that for m/n ∈ Q, then condition (1) holds because when |an − cm| is sufficiently small, Hence, Corollary 1.1 follows from the main theorem.
In order to prove the main theorem, we have to analyze the map m j : Γ j \G → R 2 for each j. The following lemma is well known. The reader may refer to [DS84]. We will denote B d the ball of radius d around the origin in R 2 .
Lemma 3.1 ([DS84] Lemma 2.2). For each j with the maps π j : Γ j \G → Γ\G and m j : Γ j \G → R 2 \ {0}, there exists a constant d j > 0 such that for any p ∈ Γ\G there exists at most one point of m j (π −1 j (p)) which lies in B d j . Remark 3.1. We will fix these d j for j = 1, 2, . . . , k throughout this note.
Lemma 3.2. For any p ∈ Γ\SL(2, R), we have Proof. Recall that η j (1 ≤ j ≤ k) are the inequivalent cusps of Γ\H. For each 1 ≤ j ≤ k, we fix a small neighborhood C j of η j in Γ\G such that C 1 , C 2 , . . . , C k are pairwise disjoint. We observe that it suffices to prove the lemma for p ∈ C j (j = 1, 2, . . . , k) since the complement of C j is compact. Let p ∈ C j for some j ∈ {1, 2, . . . , k}. Let α j > 0 be such that π j maps σ j N Ω j A α j K isomorphically to C j . Then we can pick a representative for p in σ j N Ω A α j K, say σ j n p a p k p . By definition we know that On the other hand, in the upper half plane, the point corresponding to σ j n p a p k p is equal to Since σ j is fixed and n p is in the compact set N Ω j of N , we obtain dist(p) ∼ ln α(a p ) 2 and hence e dist(p) ∼ e ln α(ap) 2 = α(a p ) 2 = 1 d(p) 2 .

(C, α; ρ)-good functions in presence of the Diophantine condition
This section will be important in the proof of the main theorem. First, we need a modified version of the concept of (C, α)-good functions (see [KM98] for the definition of (C, α)-good functions).
Definition 4.1. A function f (x) is said to be (C, α; ρ)-good if for sufficiently small ǫ > 0 and any Now we shall begin to study a special class of functions and prove that they are (C, α; ρ)-good for some C, α and ρ > 0. Note that we restrict these functions to the domain [1, ∞).
Then there exists C > 0 such that Here the constant C depends only on ρ, κ, µ, ν and γ.
Case 1: ab = 0. In this case, by the definition of Diophantine condition, we know that a = 0 and hence Note that f (1) ≥ ρ, and this function is positive and increasing. Then it is automatically (C 1 , α 1 )good for any C 1 , α 1 > 0.
For the rest of this section, we turn to the dynamics on Γ\G. For later use, we give the following definition.
Proof. We consider the model in section 2 with the maps m j and π j . Then the image of p x where (a, b) runs over all points in m j (π −1 j (p)). By definition, what we need to prove is equivalent to the following which is equivalent to the following We denote by P the subset m j (π −1 j (p)). For (a, b) ∈ P , let I j (a,b) (j = 1, 2, . . . ) be all the maximal connected subintervals in [1, T ] such that for any x ∈ I j (a,b) the point of p has norm ≤ ρx − 1 4 + 1 κ j +4 . Since x ≥ 1, Lemma 3.1 implies that all the intervals {I j (a,b) |(a, b) ∈ P, j = 1, 2, . . . } are pairwise disjoint. Therefore, we have Because of this, to prove the lemma, it suffices to prove the following or to prove that the function has (C, 1/2; ρ)-good property. This follows immediately from Lemma 4.1.
To conclude this section, we give the following proposition, which is crucial in our proof of the main theorem. It is the discrete version of Lemma 4.2.

Calculations
In this section, we shall apply the technique of Venkatesh to obtain some effective results about averaging over arithmetic progressions. It is very similar to [V10], where Venkatesh proved the sparse equidistribution theorem for Γ being cocompact. Since in our setting Γ is non-uniform, and for the sake of self-containedness, we include the details of the calculations in this section. We will follow the notations in [V10]. Throughout this section, we fix an arbitrary point q ∈ Γ\G. For a character ψ : R → S 1 , we define (t))dt for f on Γ\G.
Lemma 5.1. Suppose that Γ\G f dµ = 0. Then there exists a constant β > 0 such that Here r = r(q, T ) = T · e − dist(g log T (q)) (4) and the implicit constant is independent of ψ. Proof.
(C.f. [V10]  We want to find γ 0 > 0 depending on κ 1 , . . . , κ k such that for any 0 < γ < γ 0 , the main theorem holds. Note that by Taylor expansion, for any M ∈ N and k ∈ N, Therefore, if M is sufficiently large and γ < 1/2, then the sequence is approximately equal to the arithmetic progression By Proposition 4.1, we know that for any ǫ > 0 and any N > 0, We proceed as follows. We pick the first element M 1 ∈ N which lies in B. Then we take Next we pick the first element M 2 ∈ N which appears after P 1 and lies in B, and we take Then we pick the first element M 3 ∈ N which appears after P 2 and lies in B, and so on. In this manner, we get pieces P 1 , P 2 , . . . in N and by our choices of M 1 , M 2 , . . . , we know that B ⊂ P 1 ∪ P 2 ∪ . . . and hence for any N > 0 Now we consider each of the pieces P i . From the discussion above, we know that {n 1+γ |n ∈ P i } is approximated by the arithmetic progressioñ 1 |P i | n∈P i f (pu(n 1+γ )) → 0 (8) as i → ∞. By formula (5), the proportion in [1, N ] which is not covered by P i 's is small relative to N . Also observe that for the P i 's which intersect [1, N ], their lengths are small relative to N . Therefore, by (8) Let ǫ → 0 and we complete the proof of the main theorem with γ 0 = min{β/(2κ j + 8)|j = 1, 2, . . . , k}.
Since this is not the main focus of the paper, we will not give a proof here. Note that the Diophantine type remains constant on any weak unstable leaf of {g t } t>0 . Therefore by Theorem 7.1 the set of non Diophantine points on any strong stable leaf has zero hausdorff dimension.