Spectral gaps without the pressure condition

Jean Bourgain, Semyon Dyatlov

Published 2016-12-29Version 1

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 $\delta\leq {1\over 2}$. This is the first result of this kind for quantum Hamiltonians. Our proof follows the strategy developed in Dyatlov-Zahl. The main new ingredient is the fractal uncertainty principle for $\delta$-regular sets with $\delta<1$, which may be of independent interest.