Spectral gaps, additive energy, and a fractal uncertainty principle

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.Comment: 85 pages, 10 figures. To appear in GAF

What is good mathematics?

Some personal thoughts and opinions on what ``good quality mathematics'' is, and whether one should try to define this term rigorously. As a case study, the story of Szemer\'edi's theorem is presented.Comment: 12 pages, no figures. To appear, Bull. Amer. Math. So