Improved fractal Weyl bounds for hyperbolic manifolds

We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $\delta$ of its limit set. More precisely, we show that as $R\to\infty$, the number of resonances in the box $[R,R+1]+i[-\beta,0]$ is $O(R^{m(\beta,\delta)+})$, where the exponent $m(\beta,\delta)=\min(2\delta+2\beta+1-n,\delta)$ changes its behavior at $\beta=(n-1-\delta)/2$. In the case $\delta<(n-1)/2$, we also give an improved resolvent upper bound in the standard resonance free strip $\{\mathrm{Im}\ \lambda\ > \delta-(n-1)/2\}$. Both results use the fractal uncertainty principle point of view recently introduced in [arXiv:1504.06589]. The appendix presents numerical evidence for the Weyl upper bound.