Spectral gaps, additive energy, and a fractal uncertainty principle

TL;DR

This paper establishes a spectral gap for convex co-compact hyperbolic manifolds near a critical dimension, linking it to the additive energy of the limit set and introducing a fractal uncertainty principle with microlocal techniques.

Contribution

It introduces a new microlocal approach and a fractal uncertainty principle to connect spectral gaps with geometric properties of limit sets.

Findings

Spectral gap size depends on additive energy estimates.

Additive energy relates to Ahlfors-David regularity constants.

New microlocal methods enable analysis of fractal structures.

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 $(n-1)/2$. 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.