Improved fractal Weyl bounds matching improved spectral gaps for hyperbolic surfaces and open quantum maps

Abstract: We prove a new fractal Weyl upper bound for the high-energy distribution of resonances of convex co-compact hyperbolic surfaces which matches the improved spectral gap given by Fourier decay. This improves upon the fractal Weyl bound of Dyatlov which matches the Patterson-Sullivan spectral gap. We also give a new resolvent estimate improving the ones given by Dyatlov-Zahl and Dyatlov. Analogous results are obtained for quantum open baker's maps, improving an estimate of Dyatlov-Jin, where we also give an improved fractal Weyl bound matching a spectral gap given by additive energy estimates. We refine known methods for proving fractal Weyl bounds which reduce the problem to an estimate of a certain determinant function; however, we use a different determinant function which allows us to make sharper estimates by applying the methods of proof of the fractal uncertainty principle in each setting.