Quantitative Equidistribution of Horocycle Flows on Homogeneous Spaces of Infinite Volume

We establish an effective equidistribution theorem for horocycle flows on geometrically finite quotients of rank-one semisimple Lie groups with infinite volume. Unlike the finite volume lattice setting, where the Haar measure governs the asymptotics, the limiting distribution in the infinite volume case is described by the Burger-Roblin measure, which is infinite but Radon. By leveraging the spectral gap of the Laplacian on the limit set and the exponential decay of matrix coefficients for the associated unitary representation on L2(G, BMS), we derive polynomial error terms for the orbital averages of smooth compactly supported observables. The proof synthesizes thermodynamic formalism, specifically the properties of Patterson-Sullivan conformal densities, with the renormalization dynamics of the geodesic flow. We further discuss the implications for the Hausdorff dimension of exceptional sets and the shrinking target problem in hyperbolic geometry.