Quantitative Equidistribution on Hyperbolic Surfaces and Arithmetic Applications

Technical Summary: Quantitative Equidistribution on Hyperbolic Surfaces and Arithmetic Applications

Problem Statement
The paper addresses the problem of quantifying the rate of equidistribution for sequences of Borel probability measures on finite area hyperbolic surfaces, specifically the modular surface $\Gamma \backslash \mathbb{H}$ where $\Gamma = \mathrm{SL}_2(\mathbb{Z})$. While classical equidistribution theorems (such as Duke's theorems on Heegner points and closed geodesics, and the Quantum Unique Ergodicity conjecture for Hecke–Maaß cusp forms) establish the convergence of measures to the Haar measure, they often lack explicit rates of convergence. The author seeks to quantify this convergence using the 1-Wasserstein distance ($W_1$), a metric from optimal transport theory that measures the "cost" of transporting one measure to another. The central challenge is to derive an effective upper bound for $W_1$ in the setting of hyperbolic surfaces, analogous to the Berry–Esseen inequality in probability theory on $\mathbb{R}$ or the torus, which relates the Wasserstein distance to the decay of Fourier coefficients (or spectral data).

Methodology
The core methodology involves establishing a "Berry–Esseen type" inequality for the 1-Wasserstein distance on hyperbolic surfaces. The approach proceeds through the following steps:

1. Dual Formulation: The author utilizes the Kantorovich–Rubinstein duality theorem, which expresses the 1-Wasserstein distance as the supremum of the difference in integrals of 1-Lipschitz functions against the two measures.
2. Spectral Decomposition: To estimate these integrals, the author approximates 1-Lipschitz functions by smooth functions ($C^\infty$) and expands them in the $L^2(\Gamma \backslash \mathbb{H})$ basis. This basis consists of Maaß cusp forms (discrete spectrum) and, in the non-compact case, Eisenstein series (continuous spectrum).
3. Kernel Construction: A crucial technical tool is the construction of an automorphic kernel $K(z, w)$ derived from the inverse Selberg–Harish-Chandra transform of a specific Gaussian-like weight function $h(t) = e^{-(t^2+1/4)/2T^2}$. This kernel allows the author to smooth the measures and relate the Wasserstein distance to a weighted average of spectral coefficients (Weyl sums).
4. Handling the Continuous Spectrum: In the non-compact case (e.g., the modular surface), the presence of Eisenstein series requires the measures to satisfy a "cuspidally tight" condition (decaying sufficiently fast near the cusp) to ensure integrability.
5. Arithmetic Input: For specific arithmetic applications, the abstract spectral bounds are combined with deep results from analytic number theory. Specifically, the author employs Waldspurger's formulas and the Watson–Ichino triple product formulas to express the relevant Weyl sums (integrals of cusp forms and Eisenstein series against the arithmetic measures) in terms of central values of $L$-functions.

Key Contributions and Results

• General Berry–Esseen Inequalities:

  • Theorem 1.6 (Cocompact Case): Establishes an upper bound for $W_1(\nu_1, \nu_2)$ on a cocompact hyperbolic surface. The bound is controlled by a smoothed sum over Maaß cusp forms of the squared differences of their Weyl sums, weighted by $e^{-t_f^2/T^2}$.
  • Theorem 1.12 (Non-cocompact Case): Extends the result to finite area non-compact surfaces (like the modular surface). The bound includes an additional integral term over the continuous spectrum involving Eisenstein series. This theorem requires the measures to be $Y^{-1/2-\delta}$-cuspidally tight.

• Arithmetic Applications on the Modular Surface:

  • Duke's Theorem (Heegner Points and Closed Geodesics): Applying the general inequality to Duke's equidistribution problems, the author proves that for fundamental discriminants $D$, the 1-Wasserstein distance between the measure of Heegner points/closed geodesics and the Haar measure satisfies:
    $$W_1(\nu_D, \nu) \ll_\varepsilon |D|^{-1/12 + \varepsilon}$$
    Unconditionally. Assuming the Generalized Lindelöf Hypothesis (GLH), this improves to:
    $$W_1(\nu_D, \nu) \ll_\varepsilon |D|^{-1/4 + \varepsilon}$$
  • Mass Equidistribution of Hecke–Maaß Cusp Forms: Addressing the Quantum Unique Ergodicity (QUE) conjecture, the author provides a quantitative rate for the equidistribution of the mass $|g|^2 d\mu$ of a Hecke–Maaß cusp form $g$ with spectral parameter $t_g$. Assuming the GLH, the bound is:
    $$W_1(\nu_g, \nu) \ll_\varepsilon t_g^{-1/2 + \varepsilon}$$

Significance and Claims
The paper claims to provide the first effective, quantitative bounds for the 1-Wasserstein distance in these specific arithmetic equidistribution problems on hyperbolic surfaces. By bridging optimal transport theory with the spectral theory of automorphic forms, the work translates abstract equidistribution theorems into concrete rates of convergence.

The author notes that while other methods exist to quantify equidistribution (such as shrinking target problems or ball discrepancy), the Wasserstein distance offers a distinct metric perspective. The results are conditional on the Generalized Lindelöf Hypothesis for the strongest bounds, which is standard in this field given the reliance on subconvexity bounds for $L$-functions. The paper explicitly states that the techniques involving Eisenstein series in Theorem 1.12 do not directly apply to mass equidistribution problems for holomorphic cusp forms in the weight aspect, as those rely on incomplete Eisenstein series and sieve-theoretic methods rather than the full spectral expansion used here.

The work builds upon recent extensions of Berry–Esseen inequalities to the torus and compact Lie groups (Bobkov, Ledoux, Borda) and adapts the framework to the non-Euclidean geometry of hyperbolic surfaces, utilizing the specific spectral properties of the Laplacian on $\Gamma \backslash \mathbb{H}$.