Spectral gaps without the pressure condition

Abstract

For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the dimension $\delta$ of the limit set, in particular we do not require the pressure condition $\delta \le \frac{1}{2}$. This is the first result of this kind for quantum Hamiltonians. Our proof follows the strategy developed by Dyatlov-Zahl [arXiv:1504.06589]. The main new ingredient is the fractal uncertainty principle for $\delta$-regular sets with $\delta <1$, which may be of independent interest.

39 pages, 5 figures. Added explanations of the proof (especially for Theorem 4) and revised according to referee's comments. To appear in Ann. Math

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