Fractal uncertainty for transfer operators

We show directly that the fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] implies that there exists $\sigma > 0$ for which the Selberg zeta function for a convex co-compact hyperbolic surface has only finitely many zeros with $\Re s \geq \frac12 - \sigma$. That eliminates advanced microlocal techniques of Dyatlov-Zahl [arXiv:1504.06589] though we stress that these techniques are still needed for resolvent bounds and for possible generalizations to the case of non-constant curvature.