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Spectral gaps, additive energy, and a fractal uncertainty principle

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Abstract

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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  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA, 02139, USA

    Semyon Dyatlov & Joshua Zahl

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  1. Semyon Dyatlov
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Correspondence to Semyon Dyatlov.

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Dyatlov, S., Zahl, J. Spectral gaps, additive energy, and a fractal uncertainty principle. Geom. Funct. Anal. 26, 1011–1094 (2016). https://doi.org/10.1007/s00039-016-0378-3

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