Spectral gaps, additive energy, and a fractal uncertainty principle

Abstract

Abstract. We obtain an essential spectral gap for n-dimensional convex co-compact hyperbolic manifolds with the dimension δ 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.In this paper we study essential spectral gaps for convex co-compact hyperbolic quotients M = Γ\H n . To formulate our result in the simplest setting, consider n = 2 and take the Selberg zeta function [Bo07, (10.1)] ), where δ ∈ [0, 1] is the dimension of the limit set of the group (see (5.2)). Therefore Z M has finitely many singularities in {Im λ > − max(0,, Naud [Na] obtained the stronger statement that Z M has finitely many singularities in {Im λ > −β} for some β strictly greater than 1 2 − δ. Naud's result, generalized to higher dimensional quotients by Stoyanov [St11], is based on the method of Dolgopyat [Do] and does not specify the size of the improvement. Our first result in particular gives explicit estimates on the value of β when δ = 1 2 : Theorem 1. Let M = Γ\H 2 be a convex co-compact hyperbolic surface. Then for each ε > 0, the function Z M has finitely many singularities in {Im λ > −β + ε}, where β = 3 8 1 2 − δ + β E 16 , β E := δ exp − K(1 − δ) −28 (1 + log 14 C) .

Mathematical study of scattering resonances

“…All of these results assume that P E ( 1 2 ) ≤ 0. A new method for obtaining improved resonance free regions, applicable also when P E ( 1 2 ) > 0, was introduced by Dyatlov-Zahl [75]. It is based on a fractal uncertainty principle (FUP) which in [75] was combined with an investigation of additive structure of limit sets (see 3.31) to obtain an improved gap for quotients \H 2 , with δ( ) ≈ In particular that produced the first resonance free strips when P E (…”

“…The general principle of [75] and [74] for quantization of κ M,A goes as follows. In the notation of (3.38) and (3.39) we say that …”

“…This is the first result about gaps for quantum Hamiltonians for any value of the pressure P E ( 1 2 ) defined in (3.29). The proof is based on the fractal uncertainty principle of [75] and fine harmonic analysis estimates related to the Beurling-Malliavin multiplier theorem, see [175] and references given there. (in the case of = 0 it is the mass of the black hole)-see Fig.…”

An introduction to fractal uncertainty principle

“…Theorem 3.5 [DZ16,DZ17]. Let M = Γ\H 2 be a convex co-compact hyperbolic surface and Λ Γ ⊂ R be the limit set of the group Γ. Denote by Λ Γ (h) the h-neighborhood of Λ Γ .…”

Falconer distance problem, additive energy and Cartesian products

“…• In the case when dim H (A) = dim H (B), we use a recent result due to Dyatlov and Zahl [1] to show that when A is Ahlfors-David regular, the additive energy of A at scale t −1 ,…”