Fourier dimension and spectral gaps for hyperbolic surfaces

• Published in: Geometric and functional analysis Vol. 27; no. 4; pp. 744 - 771
• Main Authors: Bourgain, Jean, Dyatlov, Semyon
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.07.2017; Springer Nature B.V
• Subjects: Analysis; Decay; Fourier analysis; Mathematics; Mathematics and Statistics; Theorem proving
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-017-0412-0

We obtain an essential spectral gap for a convex co-compact hyperbolic surface M = Γ \ H 2 which depends only on the dimension δ of the limit set. More precisely, we show that when δ > 0 there exists ε 0 = ε 0 ( δ ) > 0 such that the Selberg zeta function has only finitely many zeroes s with Re s > δ - ε 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 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 .