  We study the fractal uncertainty principle in the joint time-frequency
representation, and we prove a version for the Short-Time Fourier transform
with Gaussian window on the modulation spaces. This can equivalently be
formulated in terms of projection operators on the Bargmann-Fock spaces of
entire functions. Specifically for signals in $L^2(\mathbb{R}^d)$, we obtain
norm estimates of Daubechies' time-frequency localization operator localizing
on porous sets. The proof is based on the maximal Nyquist density of such sets,
and for multidimensional Cantor iterates we derive explicit upper bound
asymptotes. Finally, we translate the fractal uncertainty principle to discrete
Gaussian Gabor multipliers.
