Local average in the Hyperbolic sphere problem

TL;DR

This work analyzes a local average of the hyperbolic lattice-point counting function for the Picard group acting on ${\mathbb H}^3$, seeking to reduce the remainder in the circle problem under conditional hypotheses. The authors combine a spectral expansion with a quantum-variance bound to suppress cusp-forms, and leverage a bound on a spectral exponential sum to control oscillatory terms. Under Hypotheses QV (exponent $q\in[1,3]$) and STX (exponents $(7/4+\theta,1/4)$, $\theta\in[0,1/4]$), the averaged count has main term $\frac{\pi}{2}\bar{f} X^2$ with remainder $O_{f,\epsilon,\theta,q}\left(X^{\frac{6-4\theta}{5-4\theta}+\epsilon} + X^{\frac{2q}{q+1}+\epsilon}\right)$, improving the classical $O(X^{3/2})$. The methods connect hyperbolic lattice-point problems with quantum-variance and exponential-sum techniques, offering a refined understanding of local averages in three-dimensional hyperbolic geometry. The results illustrate how conditional analytic inputs can sharpen error terms in higher-dimensional lattice problems, with potential implications for related spectral and geometric counting problems.

Abstract

We consider a local average in the hyperbolic lattice point counting problem for the Picard group $Γ$ acting on the three-dimensional hyperbolic space. Compared to the pointwise case, we improve the bounds on the remainder in the counting, conditionally on a quantum variance estimate for Maass cusp forms attached to $Γ$. We also use bounds on a spectral exponential sum over the Laplace eigenvalues for $Γ$, which has been studied in the context of the prime geodesic theorem and for which unconditional bounds are known.

Theorems & Definitions (18)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 3.1
• proof
• Lemma 3.2