Fourier dimension and spectral gaps for hyperbolic surfaces
Name
1704.02909.pdf
Description
Accepted version
Size
586.6 KB
Format
Adobe PDF
Checksum (MD5)
37d953479c9752b00bbab06cfadcd11f
Author(s)
Bourgain, Jean
Dyatlov, Semyon
Date Issued
2017
Journal
Geometric and Functional Analysis
Publisher
Springer Nature
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/S00039-017-0412-0
