Spectral gaps without the pressure condition


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 by Dyatlov and Zahl. The main new ingredient is the fractal uncertainty principle for $\delta$-regular sets with $\delta\lt 1$, which may be of independent interest.


Jean Bourgain

School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540

Semyon Dyatlov

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 01239