Bounds on the fractal uncertainty exponent and a spectral gap

We prove two results on Fractal Uncertainty Principle (FUP) for discrete Cantor sets with large alphabets. First, we give an example of an alphabet with dimension δ ∈ (12,1) where the FUP exponent is exponentially small as the size of the alphabet grows. Secondly, for δ ∈ (0,12] we show that a similar alphabet has a large FUP exponent, arbitrarily close to the optimal upper bound of 12-δ2, if we dilate the Fourier transform by a factor satisfying a generic Diophantine condition. We give an application of the latter result to spectral gaps for open quantum baker's maps.