The Bowen-Ruelle conjecture predicts that geodesic flows on negatively curved manifolds are exponentially mixing with respect to all their equilibrium states. Dolgopyat pioneered a method rooted in the thermodynamic formalism that settled the conjecture for surfaces. Soon after, Liverani developed an intrinsic functional analytic analog of Dolgopyat's method allowing to settle the case of Liouville measures in higher dimensions, while simultaneously producing more information on the rates of mixing. Despite these important breakthroughs, the conjecture remains open in general, even in the case of measures of maximal entropy. In this talk, we will discuss a method for extending the functional analytic approach to deal with non-smooth invariant measures in a concrete algebraic setting. The key ingredient is a reduction of the problem to one regarding Fourier transforms of dynamically defined measures which we address using new machinery in additive combinatorics. The talk will not assume prior knowledge of these topics.