Fourier dimension and spectral gaps for hyperbolic surfaces

Abstract

© 2017, Springer International Publishing AG. We obtain an essential spectral gap for a convex co-compact hyperbolic surface M= Γ \ H2 which depends only on the dimension δ of the limit set. More precisely, we show that when δ> 0 there exists ε0= ε0(δ) > 0 such that the Selberg zeta function has only finitely many zeroes s with Re s> δ- ε0. The proof uses the fractal uncertainty principle approach developed in Dyatlov and Zahl (Geom Funct Anal 26:1011–1094, 2016). The key new component is a Fourier decay bound for the Patterson–Sullivan measure, which may be of independent interest. This bound uses the fact that transformations in the group Γ are nonlinear, together with estimates on exponential sums due to Bourgain (J Anal Math 112:193–236, 2010) which follow from the discretized sum-product theorem in R.