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

Let $X$ be a convex cocompact hyperbolic surface, and let $δ$ denote the Hausdorff dimension of its limit set. Let $N_X(σ,T)$ denote the number of resonances of $X$ inside the box $[σ, δ] + i[0,T]$. We prove that for all $σ> δ/2$, we have \[ N_X(σ,T) \ll_εT^{1 + δ- 2(2σ- δ) + ε}. \] This strengthens