Quantitative Equidistribution on Hyperbolic Surfaces and Arithmetic Applications

The Wasserstein distance quantifies the distance between two probability measures on a metric space. We prove an analogue of the Berry-Esseen inequality for the Wasserstein distance on a finite area hyperbolic surface. This inequality controls the Wasserstein distance via an average of Weyl sums, which are integrals of Maass cusp forms and Eisenstein series with respect to these probability measures. As applications, we prove upper bounds for the Wasserstein distance for some equidistribution problems on the modular surface $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$, namely Duke’s theorems on the equidistribution of Heegner points and of closed geodesics and Watson’s theorem on the mass equidistribution of Hecke-Maass cusp forms conditionally under the assumption of the generalised Lindelof hypothesis.

💡 Research Summary

The paper develops a quantitative framework for measuring how fast probability measures on finite‑area hyperbolic surfaces converge in the 1‑Wasserstein metric. Starting from the classical Berry–Esseen inequality, which bounds the Wasserstein distance on the real line in terms of Fourier transforms, the author extends this idea to the non‑Euclidean setting of a quotient Γ\ℍ, where Γ is a lattice in SL₂(ℝ). The distance function ρ on the upper half‑plane and its associated function u(z,w)=sinh²ρ(z,w) are invariant under the group action, allowing the Kantorovich–Rubinstein dual formulation of W₁ to be used with Lipschitz test functions.

Two main theorems are proved. Theorem 1.5 treats the cocompact case: for any two probability measures ν₁, ν₂ on a compact hyperbolic surface, the Wasserstein distance is bounded by a term of order T⁻¹ plus a weighted ℓ²‑average of Weyl sums ∫ f dν₁−∫ f dν₂ over all Maass cusp forms f, where the weight is e^{−t_f²/T²} and t_f is the spectral parameter. Theorem 1.10 handles the non‑cocompact case (surfaces with cusps). In addition to the cusp‑form contribution, an integral over the continuous spectrum involving Eisenstein series E_a(z,½+it) appears. The theorem requires the measures to be “Y‑α‑cuspidally tight”, a mild decay condition guaranteeing integrability of Eisenstein series.

The analytic heart of the argument is Lemma 2.1, which computes the inverse Selberg–Harish-Chandra transform of the Gaussian‑type weight h(t)=e^{−t²/T²}·e^{1/(4T²)}. The resulting kernel k(u) is non‑negative, integrates to 1/(4π), and satisfies ∫ k(u) arsinh√u du≪T⁻¹. These properties translate the spectral weight into a spatial decay that can be combined with the dual formulation of W₁.

Armed with these inequalities, the author tackles three arithmetic equidistribution problems on the modular surface Γ=SL₂(ℤ)\ℍ. First, Duke’s theorems on Heegner points (negative discriminant) and closed geodesics (positive discriminant) are made quantitative. Unconditionally, the Wasserstein distance between the empirical measure ν_D and the Haar measure ν satisfies W₁(ν_D,ν)≪_ε |D|^{−1/12+ε}. Assuming the Generalized Lindelöf Hypothesis (GLH), the exponent improves to −1/4, matching the best known subconvex bounds for related L‑functions. Second, the paper addresses mass equidistribution of Hecke‑Maass cusp forms (quantum unique ergodicity). Under GLH, the Wasserstein distance between the mass measure ν_g=|g|² dμ/∫|g|² dμ and ν is bounded by ≪_ε t_g^{−1/2+ε}, where t_g is the spectral parameter of g. This provides an explicit rate where previous ergodic proofs gave only qualitative convergence.

Overall, the work bridges optimal transport, spectral theory, and analytic number theory. By introducing a Berry–Esseen‑type bound on hyperbolic surfaces, it furnishes a versatile tool for obtaining effective equidistribution results in settings where the spectrum contains both discrete (cusp forms) and continuous (Eisenstein series) components. The methodology is likely to influence future research on quantitative QUE, subconvexity problems, and equidistribution on higher‑dimensional locally symmetric spaces.