Rationals on long closed horocycle flows

In this paper, we investigate the distribution theorem of rational points on the closed horocycles. Sarnak proved that the long closed horocycles are equidistributed on the modular surface. We embed rational points on such horocyles on the modular surface and its tangent bundle, and study their distribution problem.


Introduction
In homogeneous dynamics one can take a finite Lie group G and a lattice Γ (a lattice Γ in G is a discrete subgroup such that there is a fundamental domain F Γ of the left Γ -action on G with finite left Haar measure), and consider the natural actions of subgroup H of G on the quotient Γ \G. The quotient Γ \G is a homogeneous space of the form {Γ g : g ∈ G} and H acts by right multiplication.
Let H be the upper half-plane and Γ = SL(2, Z) be the modular group and M := Γ \H be the modular surface. The closed horocycles of length 1/y become equidistributed on the unit tangent bundle T 1 M of the hyperbolic surface M of finite area, as y → 0 [8]. That is, for any f : We have in particular T 1 M = Γ \T 1 H. The asymptotic behavior of averages along closed horocycles in finite volume has been investigated by the several authors, see Zagier [10], Sarnak [8], Eskin and McMullen [4], Hejhal [7] and Strömbergsson [9]. Their approaches vary from exploit mixing to harmonic analysis and to Ratner's classification of measures invariant under the unipotent flows.
Let Γ \H be non-compact, that is, it has at least one cusp. For any y > 0, the curve {(p/q + iy, 0) | p/q ∈ Q} is an example of an orbit of the horocyclic flow on T 1 H. See section 2 for details.
We denote Z × q by the multiplicative group of integers modulo q. Our investigation is concerned with the equidistribution of the point set as q → ∞ and y → 0. We embed rational points on closed horocycles on the modular surface Γ \H and later its unit tangent bundle. To do this we exploit Sarnak's equidistribution theorem on the modular surface [8].
Our analytic way of handling equidistribution questions is to use bounds for Fourier coefficients. We will seek whether the rational points on horocycle flows are equidistributed in a suitable measure theoretical sense. We addressed this issue by exploiting Weil's bounds on Kloosterman sums.
Assume f : R × H → R is given. We also assume f is periodic in the first variable, i.e. f (ξ + 1, z) = f (ξ , z), and modular with respect to the second variable. That is, f (ξ , γz) = f (ξ , z) for all γ ∈ Γ . Note that this yields This means that f is periodic, and we can expand f in a Fourier series where e(t) := e 2πit . The Fourier coefficients are given bŷ We can identify f naturally as a function on T × M . Let and define the space of functions f with finite K σ ( f ) as We denote Euler's totient function by ϕ(q), that is The proof of theorem is contained in Section 3. The proof follows from the classical bounds on Kloosterman sums. We later extend the result of Theorem 1 to the unit tangent bundle of the modular surface Γ \ SL(2, R), namely Theorem 2 below.
As a higher dimensional variant of Equation (10) of the distribution of rational points on horospheres has been investigated in [3]. However their ergodic way of proof differs from our number theoretical proof. It is interesting to note that their ergodic-theoretic proof only works in the higher dimensional setting. Limit theorems for rational points on the metaplectic horocycle and their applications to the incomplete Gauss sums are investigated in [2].
We will now extend Theorem 1, replacing H by G := SL(2, R). We use notation M 1 stands for Γ \G. We now consider the distribution of the point set as q → ∞ and σ fixed and y → 0 in the unit tangent bundle As mentioned earlier, f is periodic, which means that Remark.In Theorem 1 and 2, we have studied the limit distribution of rationals on horocycle flows with σ fixed.

Preliminaries
We use the common notation for z = x + iy ∈ C, with the real part Re z = x and the imaginary part Im z = y. Let us take matrix g = a b c d with determinant 1, i.e. ad − bc = 1. This forms a group under matrix multiplication. We call this group the unimodular group and denote it by G := SL(2, R).

Möbious transformation is a map
where z ∈ C. Notice that −g ≡ g.
The complex plane splits into three SL(2, R)-invariant subspaces under the action (14), the upper half-plane, the real linê R and the lower upper half-plane. We focus on the upper half-plane all throughout. The complex upper half-plane H is defined as A generalized circle is a circle or a straight line inĈ. A horocycle is a horizontal line or a circle joiningR tangentially. It is important to note that M(H) takes horocycles to horocycles.
The main objects of the hyperbolic plane are the straight lines and semicircles orthogonal to the real axis, both defined in terms of Euclidean objects in C. The upper half-plane H is equipped with the metric We can find the shortest curve between given any two points of H with respect to the metric above, in fact the distance between these points is measured along with the unique geodesic. Let ρ : [a, b] → H be piecewise C 1 path such that We denote hyperbolic distance d H (z, w) by the shortest length of length(ρ) for each pair of points z, w ∈ H. Moreover (H, d H ) is a metric space.
Let A ⊂ H. Then the hyperbolic area of the set A is given by the integral The hyperbolic length and area are invariant under the action of Möbious transformation M(H).
Define the discrete subgroup of G, the modular group The generators of this group are the elements 1 1 0 1 , 0 −1 1 0 . The Dirichlet fundamental domain, which is a subset of a given topological space which contains exactly one point from each of the orbits of the group action on that space, of We define the unit tangent bundle T 1 H of the hyperbolic plane H by (z, θ ), where z is a complex coordinate in H and θ ∈ R\2πZ is the angular variable measuring the orientation of an arrow attached to the complex coordinate z. We can simply identify T 1 H with H × (R/2πZ). The definition of an action of the projective special linear group PSL(2, R) on T 1 H is then given by g : where gz corresponds the usual action of fractional linear transformation and the transformation of the angle θ comes from the transformation property of a tangent vector dz under g ∈ G, β g z = arg(cz + d). Left multiplication also becomes an action of G on H × [0, 2π) that allows us to write where a b c d ∈ Γ . The positive horocycle flow and the negative horocycle flow on the unit tangent bundle and PSL(2, R) → PSL(2, R), respectively. For more details about homogeneous dynamics we refer Bekka and Mayer's book [1].

Proof of Theorem 1
We will need following lemma in the proof of the theorem. Let us first define Kloosterman sum for integers m, n and q ≥ 1, wherep denotes the multiplicative inverse of p mod q.
Lemma 1. Given ε > 0 and for all m, n, q ∈ N, there is a constant C ε > 0 such that Proof. The bound given in Lemma 1 is a direct consequence of the Weil bound for Kloosterman sums [5], |K(m, n; q)| ≤ τ(q) gcd(m, n, q) where τ(q) is the number of positive divisors of q. It is well known that upper bound for τ(q) [6, Chapter XVI] is c ε q ε , for any ε > 0. By using standard bound for ϕ(q) ε q 1−ε for any ε > 0 and Weil's bound on Kloosterman sum (25), we have 1 for any ε > 0.
We are now ready to supply the proof of Theorem 1.
Proof. Integers p and q are given, we claim that there is γ = a b c d ∈ Γ , such that where pp ≡ 1 mod q. To prove this, notice that by fractional linear transformations. By choosing c = q and d = −p, we reduce the left hand side of (30) to Since ad = 1 + bc, ap ≡ −1 mod q and a = −p. Because f is invariant under Γ , we simply have We now use the Fourier expansion, then the left hand side of above equation equals This is hence equal to ∑ m,n∈Zf m,n 1 σ K(m, −n; q) where K(m, −n; q) is Kloosterman sum. The formula (32) then reads, Hence, The second inequality follows from the Weil's classic bound on Kloosterman sums, see Lemma 1.

Proof of Theorem 2
The Iwasawa decomposition gives a convenient parametrization of G. It is defined as follows: For given g = a b c d ∈ G there is a unique x, y, φ such that a b c d = 1 x 0 1 with x ∈ R, y ∈ R >0 and φ ∈ [0, 2π).
The curve {(x + iy, 0) | x ∈ R} is an example of an orbit of the horocyclic flow on T 1 H. Γ is non-compact as it has a cusp at infinity, and that curve will then be closed in the unit tangent bundle T 1 M 1 , with length 1 y .
Proof. For given p, q, we have a b c d By substituting c = q and d = −p, and φ = 0 we clearly have the angle We therefore get a b c d Since f is invariant under Γ , then it reads