Density and location of resonances for convex co-compact hyperbolic surfaces

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X=\varGamma\backslash \mathbb {H}^{2}$\end{document} be a convex co-compact hyperbolic surface and let δ be the Hausdorff dimension of the limit set. Let ΔX be the hyperbolic Laplacian. We show that the density of resonances of the Laplacian ΔX in rectangles \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigl\{ \sigma\leq \mathrm {Re}(s)\leq\delta,\ \big\vert \mathrm {Im}(s)\big\vert\leq T \bigr\} $$\end{document} is less than O(T1+τ(σ)) in the limit T→∞, where τ(σ)<δ as long as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma>{\frac {\delta }{2}}$\end{document}. This improves the previous fractal Weyl upper bound of Zworski (Invent. Math. 136(2):353–409, 1999) and goes in the direction of a conjecture stated in Jakobson and Naud (Geom. Funct. Anal. 22(2):352–368, 2012).