  We consider here a nonlocal phase transition energy in a periodic medium and
we construct solutions whose interfaces lie at a bounded distance from any
given hyperplane.
  These solutions are either periodic or quasiperiodic, depending on the
rational dependency of the normal direction to the reference hyperplane.
  Remarkably, the oscillations of the interfaces with respect to the reference
hyperplane are bounded by a universal constant times the periodicity scale of
the medium.
  This geometric property allows us to establish, in the limit, the existence
of planelike nonlocal minimal surfaces in a periodic structure.
  The proofs rely on new optimal density and energy estimates. In particular,
roughly speaking, the energy of phase transition minimizers is controlled, both
from above and below, by the energy of one-dimensional transition layers.
