Exponential mixing via additive combinatorics

We prove that the geodesic flow on a geometrically finite locally symmetric space of negative curvature is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The approach is based on constructing a suitable anisotropic Banach space on which the infinitesimal generator of the flow admits an essential spectral gap. A key step in the proof involves estimating certain oscillatory integrals against the Patterson-Sullivan measure. For this purpose, we prove a general result of independent interest asserting that Fourier transforms of measures on R d \mathbb {R}^d , which do not concentrate near proper affine subspaces, enjoy polynomial decay outside of a sparse set of frequencies. As an intermediate step, we show that the L q L^q -dimension ( 1 > q ≤ ∞ 1>q\leq \infty ) of iterated self-convolutions of such measures tend towards that of the ambient space. Our analysis also yields polynomial bounds on the Patterson-Sullivan mass of neighborhoods of certain proper subvarieties of the boundary at infinity which are saturated along the vertical foliation.