Joint equidistribution on the product of the circle and the unit tangent bundle of the modular surface

TL;DR

This paper proves joint equidistribution of primitive rational points and horocycle orbits on a product space involving the modular surface, using spectral methods to clarify error bounds related to the Ramanujan conjecture.

Contribution

It introduces a spectral method to establish joint equidistribution results with explicit error bounds on the modular surface product space.

Findings

Error bounds are sharp given current progress on Ramanujan conjecture

Joint equidistribution holds for primitive rational points and horocycle orbits

Spectral methods effectively analyze distribution in modular surface products

Abstract

We use spectral method to prove a joint equidistribution of primitive rational points and the same along expanding horocycle orbits in the products of the circle and the unit cotangent bundle of the modular surface. This result explicates the error bound in a recent work of Einsiedler, Luethi, and Shah \cite[Theorem $1.1$]{ELS}. The error is sharp upon the best known progress towards the Ramanujan conjecture at the finite places for the modular surface.