Local average in the Hyperbolic sphere problem

We consider a local average in the hyperbolic lattice point counting problem for the Picard group $\Gamma$ 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 $\Gamma$. We also use bounds on a spectral exponential sum over the Laplace eigenvalues for $\Gamma$, which has been studied in the context of the prime geodesic theorem and for which unconditional bounds are known.