Geometric and Functional Analysis

, Volume 26, Issue 4, pp 1011–1094 | Cite as

Spectral gaps, additive energy, and a fractal uncertainty principle

  • Semyon DyatlovEmail author
  • Joshua Zahl


We obtain an essential spectral gap for n-dimensional convex co-compact hyperbolic manifolds with the dimension \({\delta}\) of the limit set close to \({{n-1\over 2}}\). The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors–David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.


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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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