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

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 $\sigma >\delta \mathrm{/}2$, we have $$N_X(\sigma ,T){\ll }_{\epsilon }{T}^{1+\delta -2(2\sigma -\delta )+\epsilon }\mathrm{.}$$ 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 $$\{s\in \mathbb{C}:\frac{3}{4}\delta +\epsilon <\mathrm{R}\mathrm{e}(s)<\delta \}\mathrm{.}$$ 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.