Resonances for open quantum maps and a fractal uncertainty principle

We study eigenvalues of quantum open baker's maps with trapped sets given by linear arithmetic Cantor sets of dimensions $\delta\in (0,1)$. We show that the size of the spectral gap is strictly greater than the standard bound $\max(0,{1\over 2}-\delta)$ for all values of $\delta$, which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus.