  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.
