20,864 research outputs found
Spectral gaps, additive energy, and a fractal uncertainty principle
We obtain an essential spectral gap for -dimensional convex co-compact
hyperbolic manifolds with the dimension of the limit set close to
. 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
Multifractal analysis of the irregular set for almost-additive sequences via large deviations
In this paper we introduce a notion of free energy and large deviations rate
function for asymptotically additive sequences of potentials via an
approximation method by families of continuous potentials. We provide estimates
for the topological pressure of the set of points whose non-additive sequences
are far from the limit described through Kingman's sub-additive ergodic theorem
and give some applications in the context of Lyapunov exponents for
diffeomorphisms and cocycles, and Shannon-McMillan-Breiman theorem for Gibbs
measures.Comment: 23 pages, to appear in Nonlinearity; small changes made according to
comments from the referee
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