Geometric and Functional Analysis

, Volume 27, Issue 4, pp 744–771 | Cite as

Fourier dimension and spectral gaps for hyperbolic surfaces

  • Jean Bourgain
  • Semyon DyatlovEmail author


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 \({{\rm Re} s > \delta-\varepsilon_0}\). 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 \({\Gamma}\) 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 \({\mathbb{R}}\).


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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