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

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 \[ N_X(σ,T) \ll_εT^{1 + δ- 2(2σ- δ) + ε}. \] 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 \[ \left\{\, s \in \mathbb{C} : \tfrac{3}{4}δ+ ε< \mathrm{Re}(s) < δ\, \right\}. \] 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.

arXiv commentSeparation lemma has been revised; minor corrections and improvements made throughout