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 ${\epsilon }_0={\epsilon }_0(\delta )>0$ such that the Selberg zeta function has only finitely many zeroes $s$ with $\Re s>\delta -{\epsilon }_0$. 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 $\mathbb{R}$.

arXiv comment28 pages, 4 figures. Added Figure 1 on page 2 and made other minor changes. To appear in GAFA

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