Improved fractal Weyl bounds for convex cocompact hyperbolic surfaces and large resonance-free regions

TL;DR

This paper improves bounds on the distribution of resonances for convex cocompact hyperbolic surfaces, leading to larger resonance-free regions and advancing the understanding of spectral properties in hyperbolic geometry.

Contribution

It strengthens existing fractal Weyl bounds for resonances and introduces new estimates using transfer operators and oscillatory integrals.

Findings

Established sharper bounds on resonance counting function.

Proved existence of large resonance-free regions within a specific strip.

Combined methods from previous approaches with new oscillatory integral estimates.

Abstract

Let $X$ be a convex cocompact hyperbolic surface, and let $\delta$ denote the Hausdorff dimension of its limit set. Let $N_X(\sigma,T)$ denote the number of resonances of $X$ inside the box $[\sigma, \delta] + i[0,T]$. We prove that for all $\sigma > \delta/2$, we have \[ N_X(\sigma,T) \ll_\epsilon T^{1 + \delta - 2(2\sigma - \delta) + \epsilon}. \] This strengthens the previously established "improved" fractal Weyl bounds due to Naud \cite{Naud14} and Dyatlov \cite{Dya19}. Moreover, this result implies that for every $\epsilon > 0$, there exist resonance-free rectangular boxes of arbitrary height within the strip \[ \left\{\, s \in \mathbb{C} : \tfrac{3}{4}\delta + \epsilon < \mathrm{Re}(s) < \delta\, \right\}. \] Our proof combines Naud's approach \cite{Naud14} with the refined transfer operator machinery developed by Dyatlov-Zworski \cite{DyZw18}, as well as a new estimate for oscillatory integrals that arise naturally in our analysis.