Local average in the Hyperbolic sphere problem

Technical Summary: Local Average in the Hyperbolic Sphere Problem

Problem Statement
The paper addresses the hyperbolic lattice point counting problem for the Picard group $\Gamma = \text{PSL}_2(\mathbb{Z}[i])$ acting on the three-dimensional hyperbolic space $\mathbb{H}^3$. The central object of study is the counting function $N(X, z)$, which enumerates the number of lattice translates $\gamma z$ within a hyperbolic ball of radius $\cosh^{-1} X$ centered at $z$.

The classical pointwise asymptotic is given by $N(X, z) = c_\Gamma X^2 + O(X^{3/2})$. While the main term $c_\Gamma X^2$ is well-understood, the exponent $3/2$ in the error term has not been improved for any point $z$ or for groups other than the Picard group. The authors investigate whether averaging the counting function over a local region (via a smooth test function) can yield a smaller error exponent, conditional on specific spectral hypotheses.

Methodology
The authors employ the spectral theory of automorphic forms to analyze the local average of the counting function. The methodology proceeds through the following steps:

1. Spectral Expansion: The counting function is expressed via the Selberg transform of the indicator function of a ball. This leads to a spectral expansion involving the eigenfunctions $u_j$ of the hyperbolic Laplacian and the Eisenstein series.
2. Local Averaging: To eliminate the dependence on the specific eigenfunctions $|u_j(z)|^2$, the authors integrate the expansion against a smooth, compactly supported test function $f$ on the quotient manifold $\Gamma \backslash \mathbb{H}^3$.
3. Hypothesis Application: The proof relies on two conditional hypotheses regarding the spectral properties of $\Gamma$:
   • Hypothesis QV (Quantum Variance): A bound on the variance of the spectral measures $d\mu_j(z) = |u_j(z)|^2 d\mu(z)$ against the uniform measure. Specifically, it assumes a bound of order $T^{q+\epsilon}$ for the sum of squared deviations of spectral integrals up to height $T$.
   • Hypothesis STX (Spectral Exponential Sum): A bound on the exponential sum $S(T, X) = \sum_{0 < r_j \le T} X^{ir_j}$, where $r_j$ are the spectral parameters. This sum is central to the Prime Geodesic Theorem.
4. Approximation and Smoothing: To handle the discontinuity of the indicator function, the authors construct smooth approximations $k_\pm$ from above and below using hyperbolic convolution. This allows for the application of the spectral expansion with better decay properties in the Selberg transform.
5. Error Balancing: The final error term is derived by balancing the contributions from the quantum variance bound (which depends on the smoothing parameter $\eta$) and the spectral exponential sum bound (which depends on the truncation parameter $T$).

Key Contributions and Results
The primary result is Theorem 1.1, which establishes an asymptotic formula for the local average of the counting function:
$$\int_{\Gamma \backslash \mathbb{H}^3} N(X, z)f(z) d\mu(z) = \frac{\pi}{2} \bar{f} X^2 + O_{f, \epsilon, \theta, q} \left( X^{\frac{6-4\theta}{5-4\theta} + \epsilon} + X^{\frac{2q}{q+1} + \epsilon} \right)$$
where $\bar{f}$ is the average value of $f$, $q \in [1, 3]$ is the exponent in Hypothesis QV, and $\theta \in [0, 1/4]$ relates to the subconvexity exponent in Hypothesis STX.

• Improvement over Pointwise Bounds: The paper demonstrates that under these hypotheses, the error exponent in the local average can be strictly smaller than the pointwise bound of $3/2$.
• Specific Exponents:
  • If Hypothesis QV holds with $q=1$ (the conjectured optimal value) and Hypothesis STX holds with the strongest known estimate ($\theta=0$), the error term is $O(X^{6/5+\epsilon})$.
  • Using the convexity bound for STX ($\theta=1/4$) and $q \le 5/3$, the error is $O(X^{5/4+\epsilon})$.
• Comparison with Dimension 2: The authors note that in the two-dimensional case (modular group), the quantum variance bound is a theorem (Luo and Sarnak), and the spectral exponential sum has been studied extensively. In three dimensions, the quantum variance bound remains a conjecture, though the spectral sum has been analyzed in the context of the Prime Geodesic Theorem.

Significance and Claims
The paper claims to provide a rigorous framework for improving the error term in the hyperbolic sphere problem via local averaging, contingent on established but unproven spectral conjectures.

• Conditional Nature: The authors explicitly state that their improvement is conditional. They do not claim to prove the quantum variance bound for the Picard group but rather show that if it holds (along with the spectral sum bound), the error term improves significantly.
• Optimality: The paper suggests that if Hypothesis STX holds with exponents $(2, 0)$ and Hypothesis QV holds with $q=1$, the resulting error term would be optimal, analogous to the known results in the two-dimensional case.
• Lower Bounds: The authors acknowledge existing lower bounds (Phillips and Rudnick) which suggest that the remainder cannot be smaller than $\Omega(X^{(n-1)/2} (\log \log X)^{(n-1)/2n - \epsilon})$. For $n=3$, this is roughly $X^1$. The authors' results approach this lower bound under the strongest assumptions, suggesting the local average captures the "true" order of magnitude of the error better than the pointwise case.

The work serves as a bridge between the spectral theory of automorphic forms in three dimensions and the geometric problem of lattice point counting, highlighting the specific role of quantum variance and spectral exponential sums in controlling the remainder term.