Title: Fourier dimension and spectral gaps for hyperbolic surfaces,
Abstract: We obtain an essential spectral gap for a convex co-compact hyperbolic
surface $M=\Gamma\backslash\mathbb H^2$ which depends only on the dimension
$\delta$ of the limit set. More precisely, we show that when $\delta>0$ there
exists $\varepsilon_0=\varepsilon_0(\delta)>0$ such that the Selberg zeta
function has only finitely many zeroes $s$ with $\Re s>\delta-\varepsilon_0$.
The proof uses the fractal uncertainty principle approach developed by
Dyatlov-Zahl [arXiv:1504.06589]. 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 $\Gamma$ are nonlinear,
together with estimates on exponential sums due to Bourgain which follow from
the discretized sum-product theorem in $\mathbb R$.