Fractal Uncertainty for Transfer Operators

Abstract

We show directly that the fractal uncertainty principle of Bourgain–Dyatlov [3] implies that there exists σ > 0 for which the Selberg zeta function (1.2) for a convex co-compact hyperbolic surface has only finitely many zeros with Re s≥1/2−σ⁠. That eliminates advanced microlocal techniques of Dyatlov–Zahl [6], though we stress that these techniques are still needed for resolvent bounds and for possible generalizations to the case of nonconstant curvature.