Fourier dimension and spectral gaps for hyperbolic surfaces

By Jean Bourgain, Semyon Dyatlov. Geometric and Functional Analysis, 2017

83 citations recorded by the source corpus.

We obtain an essential spectral gap for a convex co-compact hyperbolic surface M=Γ\H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M=\Gamma\backslash\mathbb H^2}$$\end{document} which depends only on the dimension δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta}$$\end{document} of the limit set. More precisely, we show that when δ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta > 0}$$\end{document} there exists ε0=ε0(δ)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon_0=\varepsilon_0(\delta) > 0}$$\end{document} such that the Selberg zeta function has only finitely many zeroes s with Res>δ-ε0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Re} s > \delta-\varepsilon_0}$$\end{document}. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Gamma}$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bb0fdf4cff8cae4328223d3ba483b1680b90ffd4

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In Applied Mathematics

  1. K. Pravda-Starov
  2. Karine Beauchard
  3. Philippe Jaming
  4. Eduardo Cerpa
  5. Enrique Zuazua
  6. Olivier Glass
  7. Jean-Michel Coron
  8. Eugenia Malinnikova

In Mathematical Physics

  1. Philippe Jaming
  2. Karine Beauchard
  3. K. Pravda-Starov
  4. Eduardo Cerpa
  5. Olivier Glass

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