  Quasiperiodic potentials can be used to interpolate between localization and
delocalization in one dimension. However, little is known about the stability
of quasicrystalline phases to long-range interactions. In this work, we study
repulsive ultracold dipolar fermions in a quasiperiodic optical lattice to
characterize the behavior of interacting quasicrystals. We simulate the full
time evolution of the typical experimental protocols used to probe
quasicrystalline order and localization properties. We extract experimentally
measurable dynamical observables and correlation functions to characterize the
three phases observed in the noninteracting setting: localized, intermediate,
and extended. We then study the stability of such phases to repulsive dipolar
interactions. We find that dipolar interactions can completely alter the shape
of the phase diagram by stabilizing the intermediate phase, mostly at the
expense of the extended phase. Moreover, in the strongly interacting regime, a
resonance-like behavior characterized by density oscillations appears.
Remarkably, strong dipolar repulsions can also localize particles even in the
absence of quasiperiodicity if the primary lattice is sufficiently deep. Our
work shows that dipolar interactions in a quasiperiodic potential can give rise
to a complex, tuneable coexistence of localized and extended quantum states.
