  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 $\delta \in (\frac12,1)$ where the FUP exponent is exponentially
small as the size of the alphabet grows. Secondly, for $\delta \in (0,\frac12]$
we show that a similar alphabet has a large FUP exponent, arbitrarily close to
the optimal upper bound of $\frac12-\frac\delta2$, 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.
