  By using a combination of algebraic, geometric, and dynamical techniques,
together with input from higher dimensional Diophantine approximation, we give
a complete characterization of all linearly repetitive cut and project sets
with cubical windows. We also prove that these are precisely the collection of
such sets which satisfy subadditive ergodic theorems. The results are explicit
enough to allow us to apply them to known classical models, and to construct
linearly repetitive cut and project sets in all pairs of dimensions and
codimensions in which they exist.
